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Analyses of a group of pre-tests for students of first year college mathematics

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ANALYSES OF A GROUP O F :P R I E S T S
FOR STUDENTS OF FIRST YEAR^COLLEGE MATHEMATICS
hj
Henry Vernon Price
A dissertation submitted in partial fulfillment of the
requirements for the degree of Doctor of Philosophy,
In the Department of Mathematics,
In the Graduate College of the
State University of Iowa
August$ 1940
P ro Q u e st N u m b e r: 10311004
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ii
C o p *-2»—
The writer takes this opportunity to express
his sincere appreciation to Professor E* W*
Chittenden for his guidance in the preparation
of this thesis and also for making available
both materials and IOTA help*
Acknowledgment
la also due to Professor E* F* Lindquist for
his friendly criticisms and suggestions*
in
TABLE OF CONTENTS
CHAPTER
I.
PAGE
IHTKODtJCTXON * # # # # #
The pre-tests
#
# * . * „ . . *
* * « * « • * * • •
*
........
5
Analyses and results • * » * » « • * * • » »
Some suggestion® for further research
IX*
RELATED HESEAHCH
5
• « •
14
..........................
15
The report of the Committee on Tests • • » •
15
Test scores and success in college • • • » «
17
Studies of the Iowa Placement Examinations «
IS
The effect of additional preparation In
mathematics upon retention and college
............
achievement
21
Variability In item difficulty from school
to school
III*
• » • * • » • • • • » » » • • •
24
ANALYSES OF THE 1955 PRE-TESTS...........
25
Analyses made by the Cooperative Test
Service
• • « • • * * • •
• ! » * * . *
•
25
Preparation groups * » « ♦ * • • • * • • *
#
55
Partial and absolute scores
» * * • • • * *
55
Distributions of scores made by preparation
groups * « * • • » *
....................
57
Effect of preparation on first semester
grades • • • • • « • • • • • • • • * • • •
45
iv
CHAPTER
IV*
PAGE
ANALYSES OF THE 193© PRE-TESTS
48
Analyses made by the Cooperative Test
Service
* * * * * # • « * » * • * • • • *
48
Distributions of scores made by preparation
groups
• * * * ..........
• * . * . • •
Relative difficulty of the 1936 forms
V*
51
« •
«
$6
Mean scores mad® at each institution • • •
•
37
#
@2
ANALYSES OF THE ITEMS OF THE 1935 BRE-TESTS
Analyses made by the Cooperative Test
Service
• « * • • • • •
Selection of samples
........
@2
.......... .. , « *
»
65
*
g©
Difficulty indices for preparation groups
The standard error of a percentage as com*
puted from a stratified sample
81
Comparison of the difficulty Indices found
for preparation groups # •
84
............
Effect of preparation upon knowledge of the
concepts and processes covered by the 1935
pre-tests
........ * ...........
39
Comparison of difficulties of related Items.
The predictive power of individual Items •
VI*
ANALYSES OF THE ITEMS OF THE 1936 PRE-TESTS
93
*
95
# 105
1935 items which were repeated in the 1936
pre-tests
............
Cooperative Test Service analyses
105
» * . «
* 112
V
CHAPTER
PACE
Item dls crimination within preparation
groups « * # ♦ • * « , » * * . . » * » . * •
Difficulty indices for preparation groups
114
*
11©
Comparison of difficulty indices found for
preparation groups
......................
124
Effect of preparation upon knowledge of the
concepts and processes covered by the 1936
pre-tests
VII*
«•«
..........
• • * • * « »
THE DIFFICULT! OF AH ITEM FROM TEAR TO YEAH
127
#
151
•
131
Comparison of 1935 and 1936 difficulty
Indices
« • » • * * * . . ,
Comparison of difficulty indices found for
high school freshmen « * « » * «
BIBLIOGRAPHY
.......
................. ..
APPENDIX...............................................
142
150
154
vi
LIST OF TABLES
TABLE
1#
PAGE
Humber of Students at Each of Six Institutions
to Whom the 1935 Tests Were Administered » # *
2*
Distributionof Pro-*Test Scores for Form A, 1935
3*
Distribution
4*
Distribution of Pre-Test Scores for Form C* 1935
5#
Correlations between Scores on the 1935 Pre-Tests
of
Pre-Test Scores
for Form B,
and First Semester Grades* at Each Institution
6*
25
26
193527
28
31
Humber of Students at Each Institution and in
Each Preparation Group to Whom Form A* 1935
Was Administered • # « « * • * • * • » . • » *
7*
34
Humber of Students at Each Institution and in
Each Preparation Group to Whom Form B* 1935
Was Administered » » » • • » * « » • • • « • »
8*
34
Humber of Students at Each Institution and in
Each Preparation Group to Whom Form C, 1935
Was Administered # * » « » • * * » • « • • • •
9*
Total Distribution of Partial Scores by
Preparation Groups * Form A f 1935
10#
* ...
39
Total Distribution of Partial Scores by
Preparation Groups
11#
Form B f 1935
* •
*• » *
40
Total Distribution of Partial Scores by
Preparation Groups ** Form C f 1935
12#
54
42
Stability of Differences Observed between
Preparation Groups on the 1935 Tests *
*• * *
44
vi!
TABLE
15#
PAGE
Percentages of* Students Earning Various Grades
in 1935
14#
• * • • * * * . « • * * • «
........
46
Correlations between 1936 Test Scores and First
Semes ter Grades in College Mathematics, at
Each Institution * • • « * • * • • • • • « • •
15#
Distributions of* Scores for
16*
Distributions of
the 1936 Tests • • •
P ^ * Pg
« • • • • *
and Pg as Found for Form A, 1936* and
Form B, 1936 • « * • * « « ..................
33
Comparison of Arithmetic Means Found for Form A*
1936* and for Form B, 1936
19*
51
Comparison of Means of the Preparation Groups
?1*
18*
49
Scores for Forms A and B (1936)
by Preparation Groups P^,
17*
49
................
57
Mean Scores on the 1956 Pre-Tests for the Prepa­
ration Groups P^, P ^ , and Pg at Each
Cooperating Institution
20*
* * * « * * * • • « •
58
Mean Scores for Form A* 1936* at Each Cooperating
Institution for Each Half Tear of Total Prepay
ration in Mathematics
21#
* * » • • • • • • • • •
@0
Mean Scores for Form B, 1936, at Bach Cooperating
Institution for Each Half Year of Total Prepa­
ration In Mathematics
22#
* * • » » * * « • • * »
Distributions of Validity and Difficulty Indices
Found for the 1935 T e s t s ............
23*
60
Humber of Papers in Each 1935 Sample
64
66
viii
table
page
24*
25#
Humber of Papers in Each 1936 Sample
Item Difficulties for Each Preparation Group
f l*
26#
*%k*
F o m
A * 1935
*
...............
p2 ~ Form
» » # * » # # *
P2 * Form C, 1935 • • • • « • » •
.........
33
87
Items of Form C, 1955, which Yield Significant
Differences between Preparation Groups • • » •
34*
30
Items of Form B, 1935, which Yield Significant
Differences between Preparation Groups • • • «
33#
so
Items of Form A, 1935, which Yield Signifleant
Differences between Preparation Groups • • • •
32*
70
Distribution of Difficulty Indices for Each
Preparation Group - Form C , 1935
31#
77
Distribution of Difficulty Indices for Each
Preparation Group - Form B, 1935 • « # # * « •
30*
76
Distribution of Difficulty Indices for Each
Preparation Group - Form A, 1935 ............
29#
74
Item Difficulties for Each Preparation Group
Fl* *1%*
28*
p8 *
Item Difficulties for Each Preparation Group
pl* pl§*
27#
eg
qq
Correlations (Tetraohorlc) between Correct Solu­
tions of Individual Items in Form A, 1955, and
Satisfactory (A,B,C) First Semester Grades for
Each Preparation Group # # • * • • « » . « » •
99
ix
SABLE
35*
PAGE
Correlations (Tetrachoric) between Correct Solu­
tions of Individual Items in Form B, 1935, and
Satisfactory (A,B,C) First Semester Grades for
Each Preparation Group • • • • » » * • « • # *
36*
100
Correlations (Tetrachoric) between Correct Solu­
tions of Individual Items in Form C, 1935, and
Satisfactory (A,B,C) First Semester Grades for
Each Preparation Group » « • « « « »
37*
Items of Form A, 1936, which Were Taken from the
1955 Tests « * • *
38*
+ **«
110
•
Items of Form B* 1936* which Were Taken from the
1933 Tests
39*
• • • * « * • • » * • « • • * * •
111
Distributions of Difficulty and of Validity
Indices Found for the 1936 Tests • • * » • « *
40*
102
113
Percentage of Correct Responses to Each Item in
Each Approximate Third of the Preparation Group
and Total Distributions - Form A, 1956 • . , •
41#
116
Percentage of Correct Responses to Each Item in
Each Approximate Third of the Preparation Group
and Total Distributions - Form B, 1936 • • . «
48*
Item Difficulties for Each Preparation Group P^,
Px^, and Pg - Form A, 1936 * • * • « • • • « »
43*
117
121
Item Difficulties for Each Preparation Group P^,
P»i, and Pg - Form B, 1936
..........* «
122
X
TABLE
44#
PAGE
Distribution of Difficulty Indices for Each
Preparation Group * Form A* 1936 « • • » * # •
46#
IBS
Distribution of Difficulty Indices for Each
Preparation Group * Form B* 1936 « » • • • • *
46#
123
Items of Form A* 1936# which Yield Significant
Differences between Preparation Groups « « «
47#
« 126
Items of Form B, 1936# which Yield Significant
Differences between Preparation Groups * * .
48#
♦ 126
Comparison of Difficulties of Items Given in
1935 and Repeated in Identical Form In 1936
49*
« 162
Comparison of Difficulties of Items Given in
1935 and Repeated in Almost Identical Form
in 1936
50*
1-35
Comparison of Difficulties of Items Givon in
1936 which Involved the Same Process but Were
Hot Identical to 1935 Items
51*
* « • * « * • * «
138
Difficulty and Validity Indices for Each Item
of Form A* 1935# as Given by the Cooperative
Test Service • • •
52*
155
*
156
Difficulty and Validity Indices for Each Item
Of Form B# 1935
53#
*
Difficulty and Validity Indices for Each Item
of Form C# 1935
157
xl
TABLE
54*
PAGE
Difficulty and Validity Indices for Each 1950
Item as Given by the Cooperative Test Service
55*
159
Humber ©f Correct Responses (R), Incorrect
Responses (W), and Omissions (Q), Arising from
Each Sample, for Each Item of Form A, 1955 « •
56*
161
Humber of Correct Responses (E)* Incorrect
Responses (W), and Omissions (0), Arising from
Each Sample, for Each Itemof Form B, 1935 * *
57*
162
Humber of Correct Responses (K), Incorrect
Responses (W), and Omissions (0), Arising from
Each Sample, for Each Item of Form C, 1935 * *
58*
164
Humber of Correct Responses (R), Incorrect
Responses (W), and Omissions (0 ), Arising from
Each Sample, for Each Item of Form A, 1936 * *
59*
165
Humber of Correct Responses (R), Incorrect
Responses (W), and Omissions (0), Arising from
Each Sample, for Each Item of Form B, 1936 * «
60.
167
<r^2 = 3=2j£ 90 £■ apq
Form A, 1935
169
/
61.
0~pZ Z
4 ^npq
Form B, 1936
68.
(Tpz i
170
4^npq
Fora C, 1935
172
xii
TABLE
PAGE
63.
s
2 Z
np<j
Form A* 1956 • • « • • • • • * • • • • » » •
64#
Op
S
2 1 nP1
P o m B, 1936
65*
173
........................
175
Coaparlaon of Observed Differences between Groups
P1 and P-ti In Percentage of Correct Responses
to Each Item in Form A* 1935* and the Standard
Errors of
66 *
the Differences * ............
177
Comparison of Observed Differences between Groups
P ^ and Pg in Percentage of Correct Responses
to Each Item In Form A* 1935* and the Standard
Errors of
67*
the Differences # * * • * « • • • •
173
Comparison of Observed Differences between Groups
P^ and Pg In Percentage of Correct Responses
to Each Item in Form A, 1955* and the Standard
Errors of
63*
the Differences * » * * #
« • » . «
180
Comparison of Observed Differences between Groups
pi
pi* In Percentage of Correct Responses
to Each Item In Form B, 1955* and the Standard
Errors of the Differences
69*
• » ............* «
181
Comparison of Observed Differences between Groups
P ^ and Pg In Percentage of Correct Responses
to Each Item in Form B, 1935, and the Standard
Errors of the Differences
• » • « • • • * * *
185
xiii
STABLE
70*
PAGE
Comparison of Observed Differences between Croups
P^ and Pg In Percentage of Correct Responses
to Each Item in Form B, 1935, and the Standard
Errors of the Differences
71*
.............
.. •
184
Comparison of Observed Differences between Croups
pi ana pi§ in Percentage of Correct Responses
to Each Item In Form C, 1935, and the Standard
Errors of the Differences
79*
* * • • * • • • • •
185
Comparison of Observed Differences between Groups
P ^ and Pg In Percentage of Correct Responses
to Each Item in Form C, 1935, and the Standard
Errors of the Differences
73*
* * • • ..........
187
Comparison of Observed Differences between Groups
P^ and Pg in Percentage of Correct Responses
to Each Item in Form C, 1935, and the Standard
Errors of the Differences
74*
..........* *
188
Comparison of Observed Differences between Groups
and P ^ in Percentage of Correct Responses
to Each Item in Form A, 1936, and the Standard
Errors of the Differences
75*
* * • • « » . . » *
190
Comparison of Observed Differences between Groups
Pl| •»* P2 In Percentage of Correct Responses
to Each Item in Form A, 1936, and the Standard
Errors of the Differences
« # « « ..........*
191
xiv
TABLE
76*
PAGE
Comparison of Observed Differences between Groups
P^ end P^
Percentage of Correct Responses
to Each Item In Form A* 1936* and the Standard
Errors of the Differences
77*
•
..........* * *
192
Comparison of Observed Differences between Groups
P1 and piJ In Percentage of Correct Responses
to Each Item In Form B* 1936* and the Standard
Errors of the Differences
78*
* * *
«
194
Comparison of Observed Differences between Groups
and Pg In Percentage of Correct Responses
to Each Item In Form B, 1936* and the Standard
Errors of the Differences
79*
195
Comparison of Observed Differences between Groups
P^ and Pg In Percentage of Correct Responses
to Each Item In Form B* 1936* and the Standard
Errors of the Differences
* « • • « • • • » •
196
i
CHAPTER I
IH TBODUC TI0H
In December# 1934# the Mathematical Association
of America approved a proposal of th© Committee on Educa­
tional Testing to collaborate in the construction of tests
in first* and second-year college mathematics#
A Committee
on Testa* was thereupon appointed and given power to act
for th© Association in the prosecution of the proposal#
As a part of its program# the new committee^ constructed,
In 1933, a battery of three pre-test forma for students of
first year college mathematics*
These examinations were
revised and reassembled in two new forms in 1936#
Seven
colleges and universities administered the tests in these
two years to 4*639 students# and it is the purpose of
this dissertation (1 ) to present such analyses of the re­
sults as have been made# and to elaborate upon them?
(9) to make further analyses of a nature to be specified
subsequently! and (3 ) to point out some of the opportunities
for further research#
Items Included in th® pre-tests were selected
because of th© subject matter they Involved and not becaus©
they were thought to possess high discriminating power or
^ Ralph Be&tley# E* W. Chittenden (Chairman)#
A# H* Crathom©* L* L* Dines# and H# 8 # Everett#
® Hereafter referred to as ^th© committee*”
2
would contribute appreciably to the prediction or college
success*
That the analyses which have been made show the
pre-tests to be very satisfactory in this respect In no way
alters the fact that such virtues are only accompanying
by-products and not of first importance*
The fundamental
purpose of the pre-tests is to aid the teacher of first
year college mathematics in determining the extent to which
his students are familiar with the fundamental concepts and
processes of secondary school mathematics*
The analyses made by the writer, which constitute
by far the greatest portion of this dissertation, consist
of comparisons of the responses made to th© pre-tests by
students with different degrees of preparation in mathe­
matics*
Analyses are both statistical and graphical and
are given for both tests and it ©as*
It was expected by the committee
that the tests
would be useful for sectioning or placement, diagnosis,
prediction of subsequent performance, and would provide a
basis for the measurement of growth*
A great amount of
Information relative to the first three of these objectives
is to be found In Chapters III to vilj the fourth has no
validation In an Investigation of this type since end-tests
were not given*
3 »
^Report of the Committee on Tests,” The
American Mathematical Monthly, XLVTI (May, 1940), No* 5*
3
THE PRE-TESTS
T!m 1935 test®^ were eons true ted in three nearly
equivalent form® - Form® A, B, and C * and were administered
in October of that year to 2*630 student® In six cooperating
colleges and universities*
Each form was divided into two
parts* th© first part being composed entirely of formal
questions dealing with routine skill® and with the emphasis
on manipulation*
The second part was Intended to contribute
a subjective element and contained problems#
Many of the
items* particularly those in Fart XI, consisted of several
related parts*
Fart I contained more question® than Part II
In each of the three forms and received a time allowance
of
twenty-five minutes$ Fart II was given an allowance of
only twenty minutes*
At the conclusion of th© testing program, all
papers were returned to the Cooperative Test Service in
Hew York City where a brief statistical analysis was made
of the results*
It was the Intention of the committee to eventually
present th© pro-teats in two permanent forms*
Hence for
the purpose of revision more items were available than
could be used*
Selecting from this collection those most
* ill of the tests were published by th© Coopera­
tive Tost Service located at 500 West 116th Street,
Bow Yoazk City*
^ It was found impractical to devote more than
on© class period to the administration of the tests.
4
suitable In t o m s of the analyses provided by the cooperative
Test Service* and filling In a few gaps In difficulty with
new items* the committee produced the 1936 forms*
Two
changes in organisation were embodied In the new examina­
tions - no ^multiple** items were Included* and no separa­
tion of a test Into parts was ef fee ted*
Th© revised tests were administered In the fall
of 1936 to 2*029 students in five colleges and universities*
Four of these schools had also cooperated In th© 1935
program, and hence th© populations of students examined in
th© two years were quite similar*
As before, th© test papers
were sent to the Cooperative Test Service for analysis#
Th© content of the pro-tests was limited almost
exclusively to algebra, partly because of th© limited
administration time and partly because It Is generally
recognised that success or failure In college mathematics
is largely dependent upon preparation in algebra#
concepts were Included which
are
Very few
not a part of any
coups
In first year algebra*
Th© ©ommltt©© considered it advisable to allow
two weeks (eight lessons) for review, so the examinations
were administered uniformly on th© ninth day#
That this
review period la valuable Is brought out In a totally
unrelated paper written by Douglass (11)*
In surveying
ten studies concerned with retention of secondary school
mathematics, he concludes by making th© following
©
5
observations
It i® quit® difficult to generally® from thea®
studies* It is clear, however, that a large propor­
tion of the forgetting takes place within a very few
months and that material® forgotten within a few years
can be relearned within a few hour® of instruction*
ANALYSES AND RESULTS
The Cooperative feat Service reported th® follow­
ing statistical measures m
derivatives of the 1955
program*
1 * The reliability of each test form*
These coefficients were found to be as
follows*
for Form A, *9173? for Form B,
*91801 and for Form 0* *8906*
9* The correlation between test score® and first
semester grades*
Coefficients were reported for each school
and ranged from *42 to *74 with a median of
*80 and a mean of *59*
Similar coefficient®
for 1988 ranged from *51 to *70 with a median
of #57 and a mean of *59#
5* Item difficulties*
4* Item validities#
Th® last two sets of Indices are listed, both
for 1935 and 1986, in the appendix*
No
reliability coefficients were reported for the
1986 forms/
After th© Cooperative Test Service had completed
its analysis, the test papers were sent to Professor 32# W*
Chittenden at the State University of Iowa for further
study*
There, with th© help of several TOA students, he
set up a program of research, part of which leads up to
the present investigation#
Samples were drawn and the
Item analyses found in Tables 55, 50, and 57 of th® appendix
derived from them#
On th© basis of these samples, several
correlation studies were made*®
At about the same time, Kura T a m e r (58) made a
brief study of 300 Form A papers*
She confined her atten­
tion largely to a classification of th© errors mad® on
individual Items#
A few of her analyses, however, were
concerned with the effect of preparation upon achievement,
and these will be reviewed In the next chapter#
In the summer of 1957 the writer became Interested
w
In the pro-tests, secured the 195© test papers9 from the
Cooperative Test Service, and began th© study which consti­
tutes this dissertation#
Analyses of the tests as a whole
are given in Chapters XXX and IV, and Item analyses in
Chapters V, VI, and VII#
Test analyses embrace the ©ntir©
® Most of theso analyses are Included in th© com­
mittee report already mentioned# This report will b©
reviewed later*
^ This is not to b© construed as meaning that th©
1936 pre-tests only were studied# Both batteries were
analysed In th© same manner*
7
population involved In the study* whereas item analyses were
made on the basis of stratified staples*
The following conclusions result from the analyses
discussed in Chapters III and XV*
1# The correlation between partial and absolute
scores made on five Items of Form A* 1935
was found to b© #87 with a probable error of
*
01 #
Scores obtained by marking an Item consisting
of several parts as right or wrong are termed
partial scores are those obtained by
considering each part a© a separate item#
Seven such items are found In Form A* but only
five were scored uniformly*
A random sample
of @00 papers was used in order to determine
fee scores on these items*
This relationship is of some interest but,
because of th© small number of Items involved,
is not very reliable#
2# Groups of students with one, on© and on© half,
and two years of algebra have been designated
as ?x*
and Pg* respectively*
Th© mean
scores (partial in 1935) mad© by each of these
groups on each test form are as follows:
1225
t>
re
Bom
A
B
a
22*32
14*34
10 *43
28*28
20*51
23*08
Maximum
score
32*66
23.78
53
46
50
20*12
1933
Form
A
B
Px
22*31
20*04
hi
30*12
26*13
£g
Maximum
Score
35*31
33*4?
50
50
It has been shown statistically that all of
these differences between preparation groups
are signifleant*
Mean scores made on the two 1936 forms are
given for each preparation group in each
cooperating school*
Although not shown to be
statistically significant, th® same differences
between preparation groups were found*
These
differences are still present when total
preparation In mathematics# rather than In
algebra alone. Is considered*
The total distribution of scores made by each
preparation group Is shown on a single graph
for each 1936 form*
The position of the curves
shows that the entire distribution is moved to
a higher position on the scale by an increase
9
in preparation*
4* Assuming no appreciable differences to exist
between groups wi th the same amount of prepara­
tion who wrote the 1930 forms, it has been
shown statistically that Form A is easier than
Form B for each preparation group and for all
groups combined*
3* All first semester grades given at each insti­
tution in 1950 were classified by preparation
groups and then reduced to percentages within
each group*
in some schools, there is a
tendency for high grades to be associated with
a high degree of preparation and low grades to
b® associated with less preparation, but th®
results are not uniform*
The following oenelusl&ns result from th® analyses
discussed in Chapters ¥ and VI*
1* Difficulty Indices for each item and distribu­
tions of these Indices were computed for each
preparation group*
The distributions illustrate
as did th® graphs already mentioned, that th©
entire distribution is forced upward as prepara­
tion increases*
2* In order to compare difficulty Indices found
for the same Item but different preparation
groups, It was necessary to make extensive us©
10
of the following formula for th© standard error
of a percentage as calculated from a stratified
samples
(J^ :
^
ji niPiqi*
10 the
1=1
writer’s knowledge* this formula has not been
used before with test data*
Using this formula* differences between
difficulty indices calculated for th© same
item but different preparation groups were
tested for significance*
The following number
of items were found to show a significant
difference between th© groups listed*
In
all cases the group with more preparation
was th© superior one®
1955
Form
A
B
0
*JksLk-ps
A>
Pl*Pg
9
2
7
29
17
50
21
10
11
1936
Form
P1 Pl4
?ii,p2
Pl aP2
A
B
27
10
17
27
58
42
It is interesting to note that Form A* 1956
distinguishes principally between th© two
groups with lesser preparation* whereas
Form B* 1956 separates more frequently
11
the two groups with higher preparation*
3* The correlation (tetrachorie) between success
on an Item and satisfactory (A* B# G) first
semester grades was calculated for each 1935
item and for each degree of preparation*
The
results were not uniform# a great many items
giving widely divergent coefficients for the
three preparation groups#
Two Interpretations of this phenomenon may
be given*
an Item is too small a unit to us®
for predictive purposes# or els© different
items should be used fbr students with different
degrees of preparation#
4* The papers in the 1936 samples were arranged
in descending order according to total score#
Then each sample (also the combination of all
three samples of a given form) was divided
into three approximately equal parts#
Finally#
the percentage of correct responses to each
item in each of those thirds was calculated*
The purpose of this analysis was to determine
how well each item discriminated within a
given preparation group#
Two characteristics
of the tables stand out*
a# The Indices usually given for an item
and baaed on a heterogeneous population
are inadequate to describe the discrimi­
nation within preparation groups#
b* The difference between preparation groups
is so great that items which discriminate
for on© group seldom do so for another#
5# Groups of items pertaining to the same process
(not to be confused with the ^multiple” items
already mentioned) were constimeted in 1935
for the purpose of comparison#
As expected#
difficulty indices derived for these items
decreased as the complexity of the problems
Increased#
Chapter VII is devoted to a comparison of the
difficulty indices derived for those items which were given
in 1935 and repeated in 1936*
Forty such pairs of items
were identicals thirty-nine were almost identical* differing
only in directions or simple change of number* and fortyseven other pairs involved the same process#
The apparent
stability of the difficulty index was very marked*
relation of #924 ±
for Identical items*
A cor­
*016 was found between indices given
This coefficient is #885 dfc *024 for
the ttalmost identical1* pairs# and *761 ±
*042 for those
Items dealing with the same process*
Mention should be mad© of the fact that th©
populations in 1935 and 1936# from which these indices
were derived# were very much alike in so far as preparation
is concemecU
The percentage of students in each prepara­
tion group in each year Is as follows;
Yea**
1935
1936
Pl»
39.9
28.4
39.0
43.0
2
Other®
21 ,5
6,8
22.2
6,4
SOME SUGGESTIONS FOR FURTHER RESEARCH
Growing out of th© present study are many posslbillties fbr future investigation*
A few of these are as
follows:
X* The protest Items should he sorted according
to discrimination and predictive power for
each of the preparation groups* and new tests
experimented with ~ one for each degree of
preparation*
2* Further research should be made to check the
two methods suggested for scoring an item*
3# A very promising possibility for research is
that w£ determining the manner in which
success on a given Item affects success on
other Items*
4* A study should be made to determine the
extent to which the difficulty of an item is
is constant} that Is* to find out whether or
not it is universally true*
5* Results obtained through future use of the
pre-testa^ should be studied as a check on
the analyses of this investigation*
® The pro**teats are now offered for general use
by the Cooperative Test Service and listed in their
catalog*
15
CHAPTER II
BELATED RESEARCH
*■*
report of the Committee on Tests#1 This
report gives a general survey of those analyses made of
the pre-tests by investigators other than this writer*
Most of these analyses were in the form of ©orrelatlon
studies based on the 1953 samples*
In every case
coefficients were derived for each of three preparation
groups® and for all groups combined*
Correlations between scores made on Part I of
each test form and the corresponding scores on Part II
were reported as follows?
for Form A* th® coefficients
ranged from *43 to *83 with a median of *55 and a mean of
♦54f for Form B, the rang® was from *23 to *>06 with a
median of *45 and a mean of *46; and for Form G, the rang®
was from *48 to *7© with a median of *63 and a mean of
*61*
Th® committee interprets these coefficients as an
Indication that th© two parts measure different skills*
Another study embodied in th© report was made to
determine the predictive power of a short test#
representative Items were selected
sidered as a group*
Ten
from Form A and con­
Th© average correlation eosfficient
1 *Report of the Committee on Tests*** The
American Mathematical Monthly* XLVII (May, 1940), Mo* 5*
® Groups with 1 year, 1§ years, and 2 years of
preparation in algebra*
reported between acores on this group of Items and scores
on Fart I of the test Is *83*
Th© correlation between the
first semester grades of a group of 189 students and their
scores on these ten questions was *39*
This coefficient
is somewhat lower than those reported by the Cooperative
Test Service for scores on the entire test*
The following things were found by Turner (33)
In her study of 300 Fern A, 1935, papers#
a* The bi-serial coefficient of correlation be­
tween test scores and unsatisfactory
(B, P, or withdrawn) first semester grades is
-*51, indicating that unsatisfactory grades
decrease substantially aa the teat scores in­
crease*
Mo unsatisfactory grades were reported
for the 22 students who made scores of 39 or
more on the pro-teat*
h« There is a direct relationship between the
amount ©f preparation in algebra and the
percentage of correct responses to th© Items,
and an inverse relationship between amount of
preparation and th© percentage of Items
omitted*
The data Is given In tabular form for th®
three preparation groups already mentioned,
and no attempt was mad© to prove the differences
significant*
1.7
<s» Superior students of ninth-year algebra scored
higher on Part I of the pre-test than did any
of the preparation groups#
Th® first 33 of the 51 items In the 1936 Iowa
State Scholarship Examination in Algebra were
made to coincide with Part I of Form A*
Th©
examination was administered to 52 students who
were completing their first year of algebra
and who were competing as finalists -In a state­
wide testing program*
These students responded
correctly to 85 per cent of the Items under
consideration and Incorrectly to the remaining
15 per cent*
Bone were omitted*
2* Test scores and smeeeaa in college*
An
enormous amount of research has been carried on relative
to the prediction of success in college*
Most of these in­
vestigations* however* have made use of Intelligence exami­
nations rather than content examinations in a particular
field*
Brumbaugh (6 ) summarised a large number of these
studies in 1936* and we quote that part of his paper which
seems most significant In terms of our study*
Only a few studies have boon made of the relation­
ship between achievement in colleg© and the scores
mad,© in achievement tests such as th© College Entrance
Board Examinations* the Iowa High School Content Exam­
inations * the Iowa Placement Tests, the New York
Regents Examinations, and the Sones-Harry High-School
Achievement Tests# The data so far available indicate
that achievement tests, particularly the Iowa Placement
Tests and the Sones-Harry High-School Achievement Tests,
18
are ms efficient as intelligence tests in forecasting
suecess in college* They h&v© an additional value in
that they afford valuable information for purposes of
classifying and advising those who enter college*
In support of this conclusion Holssinger says2
"The psychological examination Indicates the student1©
general capacity to handle the various college courses|
it does not furnish standards for academic achievement*
Achievement tests* such as the Son©s-Harry Test,
measure not only achievement hut Intelligence as well*
The tmlverslty would do well to develop tests of the
achievement type as a basis for standards of entrance
and subsequent guidance In college departments#**
Studies of the Iowa Placement Bx&mlnationa*^
This group of examinations has been involved in a number
©f Investigations*^ but In very few cases have the exami­
nations themselves been studied*
The examinations were
constructed by Stoddard and others In 1925 and are In two
series «* aptitude tests and training tests*
On© test in
each series Is devoted to mathematics* and the MathematicsTraining Examination la designed for the same puipose as
are th© pro-*tests*
Th© content* however* Is considerably
different*
Stoddard (30) studied th© examinations exten­
sively, •principally for the purpose of determining and
Improving their power as predictive and placement
® A description of these examinations and their
construction is given by Stoddard (30)*
^ In a monograph published in 1928, Hammond and
Stoddard (14) review studies of the uses of these exami­
nations In a number of engineering colleges* An annotated
bibliography of most of these studies Is Included*
19
instruments*
Hi® analyses included calculations of the
reliabilities of the various tests, inter-part correla­
tions, correlations of scores made on each test and on the
parts of each test with first semester grades, and corre­
lations of th© corresponding training and aptitude tests#
Revision of the tests was then effected in accordance with
th© information secured#
Two criteria were used exten­
sively In passing judgment on items *
if (a) student
Items were eliminated
responses were all correct or all incorrect;
(b) they failed to discriminate (in the per cent of pupils
passing) those students who subsequently succeeded5 In the
course from those who subsequently failed® in the course*
In a later report describing the placement
examination®, Stoddard (Si) makes a remark which has bean
quoted many times and which, In the writer1s opinion, is
equally descriptive of th© pre-tests#
He says:
The Iowa flaeement Examinations ar© less a
prognosis test than an educative procedure* Their
aim Is not primarily to predict academic success *
but to render its "attainment more likely; that is,
to give aid in the setting up of educational condi­
tions such that sound principles of selection* classsectioning and curriculum organisation may be more
effectively applied to tfoe securing of maximum per­
formance on the part of each student#
In 1929, Miller (23) made a detailed study of the
data which had accumulated through us© of the Iowa
s Grades of A or B#
^ Grades of D, Conditioned, or Fd#
20
Placement Eliminations In 1925, 1926, and 1927.
Th© pro­
cedure© he used were essentially the same as those already
described for Stoddard1© study, and on the basis of the
Information secured he effected a further revision of the
examinations*
As a check, th© new forms were tried out and
the data studied In th© same manner as before*
Conclusions are, for the most part, confined to
technical descriptions of th© various examinations, and
ar© relatively meaningless without a more detailed descriptien of the study than is given here*
On© result, however,
Is quit© general and of considerable importance*
We quotes
Th© result® of the Investigation In ©very subject
indicate that th© best type of training examination
for placement purposes is one mad© up of Items and
parts which emphasis© the most common and fundamental
principles of th® subject rather than th© less common
and less essential*
On# further study of th© Iowa Placement Examina7
tions should be mentioned*
Hansen (15) studied samples
drawn from th© papers written in twenty-eight colleges and
universities in 1925 and 1926*
He calculated a multitude
©f intercorrelations, principally to determine how well
aptitude© ©r training in on® field predicted aptitude or
training in another field.
One of his conclusions was
that mathematics training was the best of th© group for
7 Many other studies of these examinations are
reviewed in Miller*s study* Since they have little bearing
on the present investigation, they are not repeated her©*
general predictive purposes*
In general, it may be said that th© evidence Is
in agreement that the Iowa Placement Examinations are very
useful for the purposes of prediction and placement*
4* 2 *& M M SI of additional preparation in mathe­
matics, upon retention and college achievement*
The final
grades of 291 college algebra students at th© Morgan Park
Junior College were classified by Bergen (3) according to
the number of semesters of mathematics taken In high school*
The purpose of this classification was to study th© effect
of longer training In mathematics upon achievement in
college algebra*
All students involved in the study had at
least four semesters of preparation and some had as many
as eight*
Bergen concluded that (grade) differences between
the groups of students who had four and five semesters of
preparation were negligible, but that there were large
differences when the groups who had six or more semesters
war© compared with those who had only four or five*
Schoonm&ker (28) reports a study conducted with
128 freshman students at an unnamed college for women In
1925-24 and 1924-25*
It had been the policy of th© school
to group students with only on© year of preparation in
algebra in a 11Sub Math11 class*
During the investigation
period this policy was abandoned and the sectioning was
carried out on the basis.of- scares .raid®:;tsn* the Hotz Algebra
Seales*
She reports sectioning on th© basis of ability to
be definitely th© better method in so far as conditioning
achievement is concerned*
This would imply that prepara­
tion and achievement arc not too closely related*
Douglass and Michael son (11) have reported an
investigation of 387 students of the class of 1930 In th©
College of Arts and Science and the School of Business
Administration at th® IMiversity of Oregon*
They found
a coefficient of #2 S ± *06 for the correlation between th©
number of semester credits in high school mathematics
©nd average college marks in mathematics, and list among
their eonclusions, *Prediction of success of students in
college mathematics cannot be made with any high degree
of accuracy from knowledge of the amount of high school
training in mathematics, # * #*
This study has been discounted and quit© soundly
criticised by Hart (16) who points out that average grades
were taken at the end of a long period, two years, and that
in the meantime the students involved had become quit©
homogeneous with respect to knowledge of mathematics*
Four theses written very recently at the Uni­
versity of Iowa reveal pertinent facts concerning the
effect of quantity of preparation In mathematics on reten­
tion of algebraic knowledge*
All four present analyses of
algebra tests given without warning to high school seniors*
09
Branham (5} gave the 195? Iowa Every-Fupil Test8 In ninth
year algebra to SS9 aenlors In the high schools or Peoria*
Illinois, while Allen {!}* Burch (7), and Olson (24)
administered a somewhat similar test® to 2*798 seniors in
59 Iowa high schools*
The conclusions of the four studies
are almost identical#
We quote from Burch*
The amount of retention of algebraic knowledge
varies directly with the amount of mathematics that
the students have had* Advanced algebra seems to
provide more retention than does any other mathematics
course*
Th© other three studies are in agreement with
this statement except that they all deny that a course In
plan© geometry adds to the retention*
They hasten to add
that even th© better groups do not retain enough knowledge
to Insure success In M e t i n g problems of an algebraic
nature* but offer some hope to th© college teacher by
pointing out that th© college-bound group Is definitely
superior to those not planning to continue In school#
The studies of this section are In general agree­
ment with each other and also with the writer In regard to
the following point©!
1* There Is little relationship between th©
number of semesters of preparation in
8 A description Of this testing program was given
by Lindquist (20) In 1955*
® The test is divided Into three parts and each
man reported ©n one part#
mathematics and first semester grades* unless
the preparation is six semesters or more#
2# There is a definite relationship between the
amount of preparation in algebra and retention
of algebraic skills*
V&ffiability i*^ Item difficulty from school to
school*
Two these®, written at the State Chlversity of
Iowa* are concerned with item difficulties#
Both utilised
the results of the 1953 Iowa Every-Pupil Testing Program*
and both studied the variability In Item difficulty from
school to school*
Atchison (2) used general science scores
and Robb (27) algebra scores#
Their procedure was to
calculate th© difficulty Index of each item In each of
several schools, and then to make comparisons from school
to school*
Both found large differences between schools
In the sis© of the Index of a given item* and also in the
rank of this Index in the distribution of all such Indices*
CHAPTER III
ANALYSES OF THE 1035 PRE-TESTS
©• Analyses made by the Cooperative Teat Service#
Th© three forms of the 1935 tests were administered to
2*630 students in th© ©ix cooperating Institutions men-*
tioned in th© introduction#
The following tables show
the number taking each form and the distribution of scores
which resulted#
The scores ©re well distributed over th®
rang©* with considerable difference appearing between
schools on Forms B and 0 #
Tabl© 1
NUMBER OF STOTENTS AT EACH OF SIX INSTITUTIONS
TO WHOM THE 1935 TESTS WERE AMINISTEKEB
Gollme
Form A
EstssLB.
Form.c
X
IX
XIX
XV
V
VI
330
114
462
149
444
93
85
299
103
71
376
104
-
566
565
Totals
1499
Table 8
DISTRIBUTION OF FEE-'TEST SCORES FOR FORM A, 1935
School
X
.
.
42
41
40
39
30
37
36
35
34
33
32
31
30
29
28
27
26
85
24
23
15
14
9
23
23
16
17
22
81
School
__ XI
School
111
1
1
1
4
2
2
10
7
1
1
1
1
1
1
I
2
2
1
1
1
10
6
a
4
5
7
1
X
7
2
2
5
IS
2
10
6
6
13
6
8
12
81
4
12
2
19
14
16
24
10
12
5
17
16
15
14
13
9
16
6
12
11
10
2
4
4
3
7
5
0
6
6
5
9
a
2
0
5
3
25
19
15
26
31
24
25
29
m
25
23
IS
17
14
7
8
2
12
1
3
4
9
3
3
6
9
9
5
6
6
3
4
3
2
1
Totals
.IV
1
1
3
20
School
330
114
13
1
1
8
3
2
4
7
3
7
4
6
4
4
13
10
5
9
9
4
10
@
4
9
3
3
5
3
1
3
1
1
1
1
482
149
School
7
1
I
Total
1
2
2
4
4
?
1
1
1
1
1
7
2
2
7
5
8
9
9
13
18
10
12
12
16
12
13
19
89
87
17
21
31
29
24
24
15
18
13
11
8
2
3
1
1
5
a
23
18
22
19
23
37
38
43
54
55
47
60
59
62
61
67
66
85
76
54
71
80
64
03
56
43
41
50
16
18
5
6
3
1
ijjAA
1499
Table 3
DISTRIBUTION OF FBB*TEST SCORES FOR FORM B, 1935
BOOS’©
42
41
40
39
38
37
36
33
34
33
32
31
30
29
28
27
26
23
24
23
22
21
20
19
18
17
16
13
14
13
12
11
10
School
1
.
1
1
s
School
1 X1
1
1
1
8
1
2
3
3
4
3
7
3
7
7
7
7
2
6
2
6
6
2
1
8
7
4
7
1
2
4
3
1
11
12
11
7
13
17
30
17
21
2
10
22
8
9
7
6
7
4
1
1
1
2
1
8
1
0
3
4
7
7
3
6
6
0
7
7
3
3
3
2
1
2
1
2
7
3
4
4
4
4
2
0
1
1
1
2
Total©
95
52
31
40
28
34
29
28
£1
29
11
9
10
a
1
2
I
20
17
23
17
30
23
2
4
8
5
4
3
4
4
7
7
B
12
19
10
1
9
82
17
1
1
2
10
5
Total
2
5
1
3
3
3
2
3
4
3
VI
2
3
9
6
School
3
1
2
3
School
X?
1
1
1
3
299
103
71
306
Table 4
PKS*TBST SCORES
DISTRIBUTION OP
BOfoOOl
Score
43
48
41
40
39
38
37
m
38
34
33
32
51
30
29
28
27
28
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
"W*
J
k
8
School
X?
8
8
2
3
©
4
3
5
7
7
3
1
1
3
1
8
2
5
4
5
7
7
5
7
1
11
10
3
9
1
2
10
3
20
2
6
1
2
26
83
22
33
4
20
2
1
2
2
1
1
23
27
3
6
612
a
7
5
8
6
10
12
5
9
©
22
11
2
5
10
2
3
3
85
4
2
13
14
©
Total
1
2
1
1
POKM C, 1938
1
a
8
7
6
Totals
School
XIX
FOR
376
4
5
11
11
11
19
17
17
15
18
21
28
40
37
30
40
32
31
38
19
17
25
1©
a
1
12
3
3
10
5
3
8
104
565
OQ
After such, examination of results as each Institu­
tion desired to make, the tests were sent to the Cooperative
Test Service in Hew York City where they were rescored and
subjected to a statistical analysis*
At the end of the
first semester (or quarter) of work* a list of the grades
earned by students participating in the testing program
was also forwarded to the Cooperative Test Service to he
included in the analysis*
This analysis consisted of the
computation of
a* th® power of each form to predict first
semester grades,
b* the reliabilities of the various forms,
c* the percentage of correct responses to each
item, commonly known as the ndifficulty” of
the item, and
d* the degree to which an item discriminated
between superior and Inferior students as
measured by their total score on the test*
This will be referred to as the 11validity11
of an item*
The results of the last two parts, inasmuch as
they are item analyses rather than test analyses, will be
presented in Chapter V*
Th® reliability coefficient found
for Form A was ,9173, for Form B, *9166, and for Form C,
*8926*
??n
The following table shows the relation found at
each institution between scores on th® 1935 tests and first
semester grades*
schools*
This was reported for only four of the
Three observations should b© made in connection
with this table*
First, the correlations found compare quite
favorably with those reported for similar tests*^
Stoddard (50) reports finding a coefficient of *28 for the
correlation between scores on the Iowa High School Content
Examination in Mathematics and first semester grades in
college mathematics*
He also reports coefficients ranging
from *53 to *70, with a median of *60, for the correlation
between scores on the Iowa Placement Examinations, Mathe­
matics Training, and first semester grades in college mathe­
matics s Miller (23) and Rammers (25) report coefficients
of *48 and *66 respectively for the same examinations*
Crawford and Burnham (9), who studied the relationship be­
tween scores on the Comprehensive Mathematics test of the
College Entrance Examination Board and first year grades
in mathematics mad© by the Yale freshman classes of 1933
and 1954, report a median correlation coefficient of *36*
Gilkey (15), in a somewhat similar investigation, found a
coefficient of *34 for the correlation between scores made
at New York State College for Teachers on the New York
1
Segel (29) lists several tables of such corre­
lation coefficient® on pp* 59-70 of his monograph,
*
#
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* •*
23*9
13*5
<0 ID 0* 10
M
+*v
go Oe eO Og
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as o
H
©
H
O
O
Q
oa
o
to
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m
«P
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1
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I
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4*
S3
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£
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o o
E
g‘a
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l
r*^ &i
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+«fc
£Q 05
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<$ O
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VA H
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hi
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to
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ft 0*
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ggg
p
oo
&. £V| Ce*
Fora
Fora
O ID <3 ft O
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01
6
““ F,plT**t
students*
students:
s
Advanced
He&ientary
Hi
to <0
gs§ig
VOtOtO'V
144
258
10 to to to
H oa 0> ^
*6239
*5811
31
32
Hegents l&amination in Mathematics and all college grades
In mathematics*
Ha remarks that the coefficient diminishes
as the number of years of college work increases* and
hence the correlation reported is probably less than it
would have been had first semester grades only been used
in th® study*
The second observation to b© made In connection
with Table 5 is due to th© fact that there is a definite
positive relationship* as will be demonstrated very
shortly, between th© amount of preparation in algebra and
scores on these tests*
Some of the institutions involved
in th© analysis adhere to th© policy of assigning those
students with only one year of preparation in algebra to
a special class*
Th© tendency to grade on the class
average hence effects a general increase in the ratings
assigned to members of this group in comparison with those
in th© other groups*
This* in turn, tends to force down
the correlations presented in Table 5»
The third observation is that th© rather high
correlations found for School 1 are associated with rela2
tlvely high mean scores*
Whether till3 has some signifi­
cance, or is merely a coincidence, we are unable to say*
In view of the preponderance of highly trained students
enrolled in this school, it is quit© likely that the
® A similar phenomenon was noted by Hammond and
Stoddard (14) In studies of the Iowa Placement Examinations*
33
detrimental effect mentioned in the preceding paragraph
does not have a chance to operate*
Preparation croups*
The analyses presented
and discussed thus far have been due to th© Cooperative
Test Service*
The writer has confined his attention to the
test papers written by those students who presented one,
one and one half, or two years of preparation in algebra*
The reason for this Is that these three groups are the
ones usually found in college freshman classes*
The tests
were given before the recent trend toward elimination of
college entrance requirements in mathematics had begun to
operate, and hence the typical student had on© and one half
years of preparation in algebra*
It was the one and one
half year group which the committee kept in mind whilo
constructing the tests*
Th© three preparation groups, which we shall label
^1 * ^li*
Pg, comprise almost the entire population and
are distributed among the various institutions as shown In
Tables 6 , 7, and 8 *
A comparison of these tables with
Table 1 will show how nearly equal in sis;© the two popula­
tions are*
34
Table 6
NUMBER OF STUDENTS AT EACH INSTITUTION
AND IN EACH PREPARATION GROUP
TO WHOM FORM A, 1935 , WAS ADMINI STERKD
Ins titration
I
II
III
IV
V
Totals
£l
M
Es.
22
72
45
250
75
55
497
194
19
42
18
73
346
46
145
45
291
549
Total
288
110
437
138
419
1392
Table 7
NUMBER OF STUDENTS AT EACH INSTITUTION
AND IN EACH PREPARATION GROUP
TO WHOM FORM B, 1935, WAS ADMINISTERED
Institution
I
III
XV
VI
Totals
£i
7
146
24
19
196
M
22
118
56
8
204
Total
E&
60
16
11
89
280
91
41
128
528
68
Table 8
NUMBER OF STUDENTS AT EACH INSTITUTION
AND IN EACH PREPARATION GROUP
TO WHOM FORM C, 1935, WAS ADMINISTERED
Ins tl tut ion
I
III
IV
Totals
El
Eli
5
79
35
119
14
252
60
326
Po
Total
48
35
2
85
67
366
97
530
35
8 * Partial and absolute scores.#
Before concern­
ing ourselves with th© distributions of scores made by
these preparation groups, let us observe that there are
some Items In each test which consist of several parts#
For example, item 3 of Form A Is
a.
3 (-6 ) -4(-S) »
b*
2 * 0 *
c•
0 * 3 *
a
d*
0• s
•
4
Directions accompanying the tests stated that such a ques­
tion was to be scored as right only if correct answers
were exhibited for every part; otherwise It was to b©
marked wrong*
This procedure gives rise to what we shall
term an absolute score*
However, these Items were also
scored as If th© above example were fotir separate exercises*
Scores thus obtained w© shall label partial*
The question
w© wish to raise now Is, "How do the two methods compare
in so far as ranking the student® is concerned?”
An
attempt has been mad© to answer this question by a study
of items which were subjected to both scoring procedures.
A brief examination of th© tests found on pages
6 8 , 69, 70, 71, 72 and 73 will disclose that each of the three
forms of the 1935 tests contained about th© same number of
"multiple" Items* However, since Form A was administered to more
38
students than were Forma B and 0 combined
confined to this form*
attention was
In order to select a sample with
which to work, all copies of Form A were drawn from the
three preparation groups and placed In a single pile in
a haphazard fashion*
Then, using Tippettfs table of random
numbers (32), a random sample of 200 papers was drawn*
The next logical step was to pair partial and
absolute scores*
Since only a few items were of the
"multiple” type, it was obvious that if scores for the
entire test were used, the correlation would necessarily
be spuriously high*
Hence only those Items which were
scored by both methods were considered*
Seven such Items
appear on the examination but only five could b® used;
item 35 is subjective in nature and was not scored uni**
formly, and in a great many cases part Co) was the only
part of item 30 which was scored at all*
Even with th® number of usable item® thus narrowed
to five, a complicating factor was In evidence*
Two of
the items consisted of four parts each, one of them of
three parts, and the remaining two of two parts each*
Making use of only the sum of correct responses to parts
would thus give some items more weight than others*
In
order to counteract this variability, each part of th©
four**part items was multiplied by three, each part of th®
thre@~part items by four, and each part of the two**part
3 See Table 1
Items by six#
Thus each item was credited with a composite
weight of* twelve, and the total possible partial score was
sixty#
After the partial and absolute scores were counted
and paired, a correlation table was set up and the Pearson
product-moistent coefficient of correlation computed*
This
coefficient was found to be *87^ with a probable error of
#01#
The writer interprets this to mean that if on© is
concerned with th® total distribution of scores, then th®
two methods yield approximately th® same results*
However,
it can hardly be said that th© two procedures will rank an
Individual in the same order in the distribution*
$*
groups*
%$m. of , m m m
M
mmMM&Sm
When the 1935 tests were revised and th® 1930 teats
assembled, all ^multiple1* questions were broken down and
the components used as separate items#
Consequently, since
we shall later have need to male© comparisons of responses
to those Items which were given both in 1935 and 1936, we
shall concern ourselves with partial rather than absolute
scores#
The following three tables give th© total distri­
bution of partial scores for each of the preparation groups
4 Due to the smell number of items involved, this
coefficient is not to be relied upon too heavily#
38
together with the mean (M), standard deviation (<T) and
standard error of the mean5 (d^) for each distribution.
5 _
_ I E"d®
M "*y
n(n - 1 ) * wher© d represents th© devia­
tion of a given element from Its mean* K* F# Lindquist,
Statistical Analysis in Educational Research, P* 51*
39
Table 9
TOTAL DISTRIBUTION! OP PARTIAL SCORES B5T PREPARATION GROUPS
FORM A, 1938
§&m&
51
50
40
48
47
46
46
44
43
48
41
40
m
m
37
56
33
34
33
38
31
50
89
98
27
m
m
84
83
29
21
20
19
18
17
16
13
14
13
12
11
Ex
pi&
2
%
i
9
2
1
3
I
7
3
f e ­
lt
9
9
18
10
18
18
8
22
27
84
24
24
40
29
40
35
37
25
23
14
23
12
®
6
1
1
2
5
8
7
a
5
11
15
10
88
11
14
27
IS
82
24
26
21
21
89
23
24
24
17
19
1®
10
5
9
0
0
4
5
!s
i
2
1
3
2
8
®
10
7
6
7
12
IS
14
IS
17
84
20
18
19
15
10
IS
20
18
a
12
®
s
10
5
5
7
2
2
1
5
3
1
4
8
2
^.o
Table 9 {continued)
FORK A, 1935
Score
*1
10
7
1
9
3
7
7
S
3
1
6
1
5
I
4
1
Totals
349
%
pg
497
23*33
7*43
*3®
r
«S
346
33*23
7,93
,36
32*66
7.97
.43
Table 10
TOTAL DISTRXBOTIOH OF PARTIAL SCORES BX PREPARATION GROUPS
FORM B, 1935
Score
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
&3j.
1
4
1
8
2
0
3
3
9
£s
1
I
1
1
S
3
I
S
S
1
S
I
3
1
3
3
6
6
7
6
7
11
5
9
8
3
2
6
5
10
2
4
6
8
5
10
2
41
7abla 10
FORM 8* 1955
£i
III
18
295
18
109
S6
18
16
18
21
10©
12
17
©9
11
78
189
182
82
74
2
28
1
8
1
19©
204
20*51
14*34
$#87
•42 7*48
*82
*
Room
21
20
19
IS
17
10
IS
14
IS
12
11
109
8
65
824
1
totals
oMr
«a
Is.
10s4
0
14
254
1
11
128
25*78
7*44
*6©
12
Table 11
TOTAL DISTRIBTJTXOB OF PARTIAL SCORES BT PREPARATION GROUPS
m m
Seort
4®
48
47
46
48
44
46
42
41
40
39
83
87
86
36
34
31
38
81
30
80
c, loss
£i
i
%
i
2
1
X
t
0
1
1
87
8
m
m
a
24
23
82
21
30
1©
8
7
18
7
4
9
8
xa
a
a
Totals
M
<r
«n
£2
i
m
17
16
15
14
18
12
11
10
©
£ii
«jr
7
4
@
7
3
1
a
3
1
119
19*43
6*20
#57
2
6
2
9
S
12
7
16
6
15
17
17
22
24
19
21
22
16
17
15
12
12
8
6
1
3
1
4
4
2
1
4
6
4
2
5
4
5
6
7
4
2
3
1
2
3
2
1
2
6
4
2
1
2
2
1
326
25*08
6*70
*57
1
85
29*12
7*15
.78
It is obvious from an examination of the preceding
tables that those students who entered college with two years
of high school algebra to their credit mad© higher average
scores on these testa than did those with only one and on©
half or one year of algebra#
Similarly# th© one and one
half year group responded correctly, on the average, to more
Items than did the one year group#
The question quit©
naturally arises then, ”Are these differences real or merely
due to chance?”
That Is, if th© tests were given again to a
similar group of students, such as perhaps would he enrolled
in these or similar institutions in other year®, would th©
differences again he present, or might they In some eases
vanish or even appear in th© reverse form?
In order to
answer this question, w© shall make use of the following
table# . The first four columns of this table are based on
th© results of the preceding tables and th© fifth column
was obtained by dividing each difference in means as shown
in th© third coliimn by the corresponding standard error of
th© difference® in th© fourth column*
6
*Taiff.} -
F M . + *ri* •
F# Kenney, ”Som©
Topics In
cal Monthly, XLVX (February, 1959), Bo# 2 f
pp# 61, 62#
44
fable 12
OTABIDXTST OF DIFFEKBBCES OBSERVED BKWfiBH PREPARATION GB0TJP5
01 T m 1935 TESTS
Emm
A
A
A
B
B
B
C
e
c
Preparation
Orotma
Difference
..
Jin. Means
and h i
and
**
F*i
p
rg
and
h
5*96-
*48
12*4
4*3®
*56
7*8
10*34
*64
19*1
*67
9*2
3*27
#84
3.9
9*44
*78
12*1
5*05
#68
8*3
4*04
#86
4*7
9*69
*97
10*0
h
and h i
h i and P2
Pi and *2
H
and h i
h i and P2
pi end p2
*i
Standard Error
of Difference
Critical
Katlo
Th© hypothesis we wish to test is that these various
pairs of preparation groups are actually random samples drawn
from th© same population and hence any differences found can
be readily attributed to chance*
Under this hypothesis*
all of th© differences observed are highly significant at
the 1 per cent level#
Hence we may say with assurance that
our preparation group® were derived from populations which
have distinctly different degrees of mathematical skill In
so far as this skill Is measured by the 1935 tests*
to* ICCftftfc of preparation on first m a m i a M E grades*
*0o the differences fust shown carry oyer to grades?
That is,
does th© probability of success, as measured by these grades,
increase with preparation?
In order to answer this question,
all of the 193$ grades which were reported have been assembled
in Table 13*
Gtrades given at each school, and at all schools,
are classified by preparation groups*
H represents the number
of students with a given amount of preparation in algebra,
and the numbers paired with grades are percentages*
Thus, In
school I there were $9 students with one year of preparation
in algebraf of this group, 12*5 per cent earned grades of A,
15*6 per cent grades of B, etc*
The ^miscellaneous** entry
consists of withdrawals, transfers, absentees, and th© like*
Table IS
PERCENTAGES OP STUDENTS EARNING VARIOUS GRADES IN 1935
SCHOOL I
Grad©
A
B
C
£
failure
misc.
H
1
12*5
15*6
12*5
25*0
25*0
9*4
52
SCHOOL II
Preparation
Total
l|
2m*
«
2*8
17*8
23*4
26.2
28*0
1.9
107
7*5
16.3
27.2
24*1
22*5
2*5
16*6
25*3
24*6
24*0
$20
459
6.8
2.8
Preparation
1
21.7
10*9
17.4
10.9
21.7
17.4
46
SCHOOL III
A
B
C
D
failure
mi sc*
N
Grad©
A
B
C
D
failure
misc.
1*
1
9.6
18 *2
27*5
19*8
19*5
5.8
563
1
si
13*9
18*9
25*6
20
15.1
6*6
610
2
29.2
24*0
25.0
11.5
6*3
4*2
96
22.9
14.7
11.1
16.7
16.7
18
16*5
13.8
109
8.9
45
Total
13*8
19.1
26.1
19*2
15*7
1
ii
2
Total
6*9
8.6
12 *9
21*8
22.7
38*9
20.5
3*8
5.4
185
22.6
8.5
22* 4
37.5
18.0
6.1
34.7
16*8
12 *9
6 *9
1009
101
38.7
6*5
3.2
16.1
31
Preparation
2
42*7
2*8
46*8
17*0
14*9
8*5
12.7
16.5
45*0
16*5
5*1
3*8
986
47
79
22.0
10.1
Preparation
Preparation
0
12,8
14*3
22*2
22.2
11*1
11.1
ALL SCHOOLS
6*6
12*6
21*0
84.4
15.0
31.1
8*9
SCHOOL V
li
Total
SCHOOL IV
Preparation
Grad©
2
Total
7.0
13.3
43.7
19.7
12*6
3*6
412
1
9*1
13*2
32.4
19*6
17.1
5.7
828
it
2
0.6
6*9
317
Total
11*0
6.0
12*9
18.2
29.6
19.3
15*8
4.2
994
544
2366
11.6
19*0
29.1
20.1
14.2
17.8
30.3
19.7
15.6
5.5
There is some tendency for high grades to be
associated with increased preparation* but It Is not uniform*
In general* we must agree with the research discussed in
Chapter II* and conclude that there Is very little relation*
ship between the two variables*
CHAPTER IV
ANALYSES OF THE 1930 PEE*TESTS
*1# An&lveef
bv the Cooperative Test Service.
As has already been suggested* the 1930 teats were given to
a population very much like that used for the 1930 program*
Four of the five schools administering the revised forms had
also cooperated th© previous year* end the fifth school was
quite similar to one of those withdrawing from the group*
Th© new tests were slightly shorter and easier
than the first set* a fact which was known {and intended)
by the test committee* and which is illustrated by th© diat­
tributions in Table 10*
This table was constructed by the
Cooperative Test Service and Includes all of the papers
written*
Other analyses made by the Cooperative Test Service
reveal. In general* the same things as were found for th®
1930 tests *
Correlations of test scores and first semester
grades at each institution are given In Table 14 below*
-19
Table 14
C0RRELATI0H8 BBTWEEH 1930 TEST SCORES
AED FIRST SEMESTER GRADES IE COLLEGE MATHEMATICS,
AT EACH INSTITOTIOH
Sigma
Fom
I
X
A
B
120
141
XX
A
B
III
XT
Sohool
Seeres
EL
1*007
1*138
0.84
6*50
*65
*70
461
449
1*324
1*730
8*40
7*45
.51
*53
A
B
023
143
1*144
1.194
4*20
3*62
*56
.51
A
B
S3
87
1*011
1*700
8.47
*57
.69
tE. (BUS
A ©emp&rison of fables 8 ana 14 reveals that
193S and 1930 seta possess about the same predictive power.
Table 10
DISTRIBUTIOHS OF SCORES FOR THE 1930 TESTS
Frequencies
Seore
50
49
48
47
46
45
44
43
40
41
40
39
38
87
36
Fens 4
0
5
3
10
11
0
10
18
14
25
30
36
29
40
45
Fern
1
1
1
2
8
6
12
20
7
18
15
20
17
24
50
Table 1© (eon.timed)
§§m&
38
34
33
32
©1
m
89
28
2?
26
2S
24
28
22
21
20
19
18
IT
16
10
14
18
12
11
10
9
8
7
6
5
4
3
t
Totale
Frequencies
Fo ot B
SmLA
44
m
m
40
36
44
48
30
36
33
m
34
28
31
29
28
22
19
14
14
20
10
14
©
©
4
S
3
20
29
30
a
41
39
42
40
36
02
m
rf
Of
36
63
34
46
3©
37
36
28
20
18
21
20
14
9
12
10
4
4
2
1
1
1
0
2
6
1
1015
1014
51
i&*
M^ovtps»
M
w®m£§Mm
Th© corresponding distribution® for each of th©
preparation groups
and Pg* together with the
arithmetic mean# standard deviation#, and standard error of
the mean of eaeh distribution# are as follows*
Table 16
DISTRIBUTIONS OF SCORES FOR FORMS A AHI) B
m PREPARATION © R O W S Px # $ % y and Pg
ore
SO
49
48
47
46
4§
44
43
49
41
40
39
38
37
36
35
34
33
39
31
30
29
28
87
26
25
24
23
22
21
l a l
ix
M
I 14
3
1
11 75
22 13f7
10
14
16 17
18 21
50
22
553 86
23
5
119 24
20
31
88 16
21
7 18
14 14
18 18
12
12
11 16
IS 11
5a
Xi
5a
63
86
14
10
80
14
14
11
15
10
15
12S
9
4o5
a
6
6
213
1
Cl9S6>
t e J
Total
112
99
79
16
13
25
95
30
29
37
42
41
35
46
40
54
40
45
88
55
31
34
33
26
88
27
Ex
M
i
i
is
1
634
65
4
8
11
17
14
12
17
11
20
2s4
55
7
11
10
16
15
16
23
19
19
22
17
25
14
18
28
21
25
Ez
121
64
10
10
1284
136
11
13
109
IS
11
13
IS
9
6
8
53
S
31
Tet<
112
74
11
15
6
16
13
19
16
23
23
29
28
38
40
37
58
39
34
48
53
33
SO
33
46
52
Table 16 (continued)
A
SfifiEft
£i
Sat
20
is
11
19
10
10
11
S
8
2
9
S
10
7
s
2
2
2
2
1
2
IS
17
IS
IS
14
13
12
11
10
9
£t
O
7
6
5
4
3
2
Tofcala
M
ff*
11
s
6
2
2
Q
P
Total
2
1
1
1
1
1
20
20
13
14
8
19
14
14
18
24
IS
IS
10
11
10
8
0
1
2
1
10
13
13
13
7
7
a
11
8
7
1
2
4
4
8
4
12
8
1
5
9
3
3
K
©
1
1
2
28
10
2
1
2
X
2
940
322
220
2
2
9
1
406
35
36
35
26
25
17
20
20
12
3
2
1
1
460
Total
M
2
8
1
1
254
El
2k
825
953
2&8B. 3M8* 30&L4 29,075 2X0€> £€026 35*466 25*955
7*0
6*9
8*7
63
9a
7a
s*i M
*3 5
*46
*28
*35
•49
*00
*40
53
As for tli© 1935 tests, wo wish to determine
whether or not these differences in means are significant*
We make use of the following table*
Table 17
COMPARISON OF MBAIS OF THE PREPARATION GROUPS
P ^ , and Pg
AS FOOT!) FOB PORK A, 1956, AM) FORM B, 1936
P^na
A
A
A
B
B
B
Preparation
0rouB3
?1
« “«
and
Pj and
Pi and
px|? **»a
Pi and
Pg
Pg
Pi£
p2
Pg
Difference
In Mean®
7*875
5*128
13*003
6*08©
7*340
13*426
Standard Error
of Biffex*©noe
Gritleal
Ratio
•62
*57
*68
*53
*58
*61
12*7
9*0
19*1
11*5
12*7
22*0
With critical ratios as large as these, there can
be little doubt that, as was true of the 1935 tests, a elg*
nlfloant increase in mean score accompanies an Increase,
from one to two years, in algebra preparation*
The follow­
ing graphs of the distributions given In Table 16 illustrate
these differences very clearly*
It is to be remembered that the committee construct­
ed the tests essentially for the one and on© half year groups*
The graphs illustrate how well they accomplished their &!mj
th© crests of the
distributions fall near the center of
th© rang®, the crest© of th© P^ distributions are somewhat
lower on th© scale, and th© crest® of th© Pg distribution®
are higher*
For th© purposes of comparison, th® ordinate®
are given in percentage® of the total distributions*
seaoos XIV jo ^^©0
Made in U. S. A.
-SEgjggpP
No. 5780E— 20 Squares
Scores
Form
per
In c h
a to
M
i
saa:oog
xiV
J°
CJ.U8Q
5 6
13, IfAamCT difficulty of. tha 3S5§. £SEM.*
Another ch&r&e taris tic of Table 16 which should he observed
is that the mean score for Form A Is higher than the mean
score for Form B for every comparable mroms*
This is a
strong indication that Form B is th© more difficult of
the two#
{Indeed, one gets this impression by simply ex­
amining the distribution®#}
We do not know that the two groups of students
who wrote Forms A and B respectively differed essentially
in mathematical ability; neither do we know that they were
alike*
Since no effort was made to pair any particular
type of student with either test form, it is reasonable
to assume that no appreciable difference® existed#
Under
such a hypothesis we compare th© test forms for difficulty
in the ©am© manner a© w© compared th© preparation groups
for ability#
Table 18 was constructed in exactly th© same
manner a® Table 17 except that, of course, the differences
in means refer to mean® found for th© two forms*
A fifth
column has been added which gives the probability of the
occurrence of such a difference when th© groups compared
are actually random samples of th© same population*
Table 13
COMPARISON OP ARITHMETIC MEANS POUND POE
POEM A, 19Be, AMD POEM B» 1936
VIftarenoe la
Qroup
Mmmna (MA-*B)
*1
*lh
ps
Totals
Critical Probability
.Ratio .. of Occurrence)
2,m
*65
3*5
4*060
*49
8*3
1*848
*64
2*9
3*480
*41
8*3
B m
Form A m
standard Error
of Difference
*0005
0
*0038
0
th© probability that th© two-year group finds
difficult as Form B is only about four on©**
thousandths* which w© consider too small to be of consequence,
and hence * unless marked differences in ability exist between
the two groups under consideration, Form, A f» definitely
easier than Form B*
14. Mean scores made * £ aagh Afiy&SL&iMSfi.
following question now arisess
«»
are th© differences found
between th© means of th© preparation groups characteristic
only of th© entire population* or are they present at indi­
vidual institutions as well?
Due to the small number of
individuals Involved in some of the sub-groups * it is im­
possible to give a definite answer to this question, but
the following data Indicates that, for the most part, th©
differences hold for Individual schools*
58
Table 19
MEAH SCORES OH TEE 1936 FIB*TESTS FOE THE PREPARATION GROUPS
*1*
F9 AT m m COOPEBATIHG INSTITUTION
School
X
Frepaa*©tior%, Group
I
Mean
£
Mean
H
hi
3
33
36*00
34*82
5
39
20*00
30*31
P2
96
36*15
100
34,94
*X
hi
ps
194
24V
37
20*05
28*04
38*00
813
247
26
18*52
24*67
29*32
III
«R»aA*4i»
h
hi
?8
31
101
15
26*38
30*92
54*60
73
42
11
20*07
26,91
32*00
XV
P1
hi
*8
20
24
36
33*35
33*00
33*00
19
37
30
28*63
27*81
30*71
V
pl
hi
6
62
40
39*00
33*05
Ml
00*00
29*90
55*29
II
P2
30*43
41
51
The one-year groups at School X* and perhaps th<
two-year groups at School III and th® one-year groups at
School V, are so small as to give quite unstable results
It is to b© noted that th® larg© groups at School II reveal
quite substantial differences In mean scores between the
preparation groups * particularly between groups Pi and ^lj-*
Relatively ©mall numbers Involved at School XV could readily
account for the fact that no differences are found between
groups*
59
That the char ac ter is tie of differences between
preparation group® is a function of the m o u n t of prepara­
tion in mathematics» aa well as preparation in algebra
alone, may be seen by am examination of fables 20 and 21*
Sot® that the information blank on the first
page of each test3, asks for the number of semesters of
preparation in each of the mathematics courses usually
taught in high school*
In computing the composite prepa­
ration of each student, the following procedure was
observedt
algebra preparation was accepted as given by
the students on these blanks* plane geometry preparation
was considered as constituting on# year, and solid geometry
and trigonometry were counted as on# half year each*
Th®
composite preparation was then the sum of these parts#
It is this preparation which appears under th# heading
19Total Preparation in Years11 In th# tables*
1 See pages 68, 69, 70, 71, 7S, 73, 106, 107,
108, and 109,
so
Table SO
tSBAH SCORES FOR FORM A, 1980. AT EACH COOPERATIBG IHSTITUTIOH
FOR EACH HALF TEAR OF TOTAL FREPAKATIQB IE MATHEMATICS
1
§£lttsl
i
it
E
3?
M
H
ix
8
M 19*5
i
28
H
in
M
H
I?
M
1
as
I
v
M
I
§t
ISO
IS *01
1
56
84
32*58
SO
152
27*75
25*05
it
m
£
33*12
77
36*83
75
30*58
22
34
24
26*21
62
30
37
31*05
15
32*8
0
37*33
7
32*46
14
31
80
53*28
13
35*62
16
35*06
4
32*5
36
52*04
41
54*81
13
38*15
14
41
Table SI
MEAH SCORES FOR FORM B, 1986. AT EACH COOPERATING IKSTITUTIOS
FOE EACH HALF TEAR OF TOTAL FREPARATIOH IE MATHEMATICS
1
1
H
M
II
I
III
K
M
XV
M
H
M
V
*
M
WMM
&
10
27e8
18
30*78
28
32*79
87
84*93
150
67
23*85 88*81
74
27*16
18
81*07
64
18*80
26
23*62
80
m
7
26*57
7
34*86
14
28*14
10
28*37
54
27*59
16
29*8
10
36*5
5
25*8
10
29*11
41
32
14
53*64
88
86*04
1
27
176
17*50
3
14*33
1
0
1
82
1
25
at
si
&
1
ox
In order that no misunderstanding may aria© from
tli® last three tables* let it be clearly understood that the
differences cited have not been represented as sianlfleant*
The facts have been presented simply as evidence la support
of the belief that* had enough students been Involved in
each of the various subgroups* the same significant differences
would appear as were found for the populations represented
in fable 16*
62
CHAPTER r
ANALYSES OP THE ITEMS OF THE 1935 FEE-TESTS
IS* M&lirms made by the Cooperative Teat Service*
Th© Items used In th© 1935 tests were studied by the Co­
operative Test Service according to the following plan: of
Form A* a random one third was selected from each of th®
highest and lowest three hundred papers*
Since there were
nearly fifteen hundred papers* this sampling constituted
a random selection from th© highest and lowest fifth*
In
th® cases of Forms B and C* th© highest and lowest one
hundred paper® were examined*
In computing Item difficulties, the percentage
of correct answer® to an item was determined for each fifth
mentioned In the preceding paragraph and the two percentage®
averaged*
Difficulty indices derived by this method h&v©
been found by the Cooperative Test Service to correlate
very closely with difficulty Indices calculated by using
a random sample of the whole population*
Th© item valldltes were computed by th© Coopera­
tive Test Service by finding the percentage of correct
answers to each Item for the highest and lowest fifth (or
some random fraction of th© highest and lowest fifth)*
These percentages wen© plotted on coordinates In th© units
used and the validities computed In terms of the distances
of th© resulting points from the diagonal y * x*
83
Validities of 2 or more are considered by Ben B* Wood* the
director of th© Cooperative Test Service, to be satisfactory
when calculated in this manner#
Difficulty and validity indices for each item
In th© 1935 tests are listed In th© Appendix#
Heeoll that
some of the items on Fart XX of each test are subjective
In nature#
These were not scored by the Cooperative Test
Service and hence no Indices are available#
Distributions
of difficulty and validity indices for each test are given
In Table 22*
G4:
Table 22
DI3TBXBUTIGBS OF VALIDITY AID DIFFICULTY IKDICBS FOOTD
FOB THE 1955 TESTS
PjfficttltT
Form B
5b Ss s
95-100
90—94
85-89
80-84
75-79
70-74
@5-69
60-04
55-5®
BO-54
45-4®
40-44
55-59
50-54
SB-9®
SO- 24
15-1®
10-14
5-9
0-4
1
5
S
5
5
5
1
5
4
@
5
5
5
5
1
5
S
1
1
S
1
4
5
1
2
5
0
1
4
5
4
5
1
S
2
1
1
1
1
2
4
5
5
4
2
1
1
5
S
5
«
5
5
5
5
1
1
Validity
M i m
14
IS
11
10
9
8
7
0
6
4
5
Form A
Form B
Form
1
2
6
5
4
6
5
9
6
8
4
5
1
4
5
5
7
7
10
2
1
5
7
8
9
10
4
7
2n view of th© criterion used by the Cooperative
Test Service, these Items are ell valid*
Form B does not
contain enough easy items, but for the other two forms dif­
ficulty indices are spread quite uniformly over the entire
range*
Considering the relatively small number of items
in the tests, the distributions of these indices are very
satisfactory*
16*
Selection off samples*
In order to make a
detailed analysis of the items of the pre-tests, it was
necessary to make use of samples*
Th© procedure which was
used in drawing these samples is as follows*
all papers
not falling into on© of th© thro© preparation groups
F ^ , and Pg were separated out*
Those remaining were then
divided into subgroups according to test form and prepara­
tion*
Thus there were nine such subgroups In 1935 and six
in 1936*
Papers in each subgroup were arranged by institu­
tions and then alphabetically within ©ach institution group*
One sample was drawn from ©ach of th© subgroups
and the procedure used in building a sample is aa followss
if fifty papers were to be selected from on© hundred, alter­
nate papers would be taken? if fifty were to be selected
from two hundred, every fourth paper would b® taken, et
cetera*
The net result was, then, a representative, or
stratified, sample each of whose components was a random
sample from some institution*
"The sl&es of samples chosen by the above method
are as follows*
66
Table
23
NUMBER OR PAPERS IN EACH 1935 SAMPLE
Preparation
Form
A
B
C
fi
?Ur
50
50
50
100
50
50
50
50
50
Tattle 24
NUMBER OP PAPERS IN EACH 1936 SAMPLE
Sam
A
B
*i
100
100
%
100
100
1O0
100
It is to b@ observed that approximately the same
number of papers were used to each of th® two years % also
that the 1935 sample eonsI®ting of students with on® and
one half years of preparation in algebra, and who took Form
A, Is larger than the others chosen for that year®
The
reasons for this are that more students took Form A than
th® other two combined, and because the group with on® and
one half years of preparation to algebra, being the most
typical, Is naturally th® center of Interest#
17# Difficulty Indices for preparation aromas♦
One of the thing® w© are interested In la determining th©
effect of preparation in mathematics upon student responses
S7
to problems*
to view of the manner In which the subject
matter covered by the tests was selected, this must, for the
most part, be confined to preparation In algebra#
It has
been demonstrated In Chapters III and IV that Increased
preparation results in correspondingly increased mean scores
on the tests*
dividual Items?
How does it affect the difficulty of the In-*
We could expect that in some oases a con**
cept would be mastered to a sufficient extent by those with
one year, or one and one half years, of preparation that
further study would not cause an appreciable improvement#
In other eases, of course, we should expect additional prepa­
ration to carry additional achievement with it*
Th® following tables give th© difficulty of each
item for each preparation group as computed from the 1955
samples*
The data bear out th© expectations voiced In the
preceding paragraph#
In a very few Instances the Indices
decrease as th® preparation Increases*
these differences
However, non© of
is large enough to b© significant*
A column entitled ^tot&X** is Included In each
table, Its entries being merely th© averages of the corre­
sponding Indices found In th® columns listed for preparation
groups*
It Is exhibited In order to call attention to the
closeness with which It approximates th© column of Coopera­
tive Test Service difficulties which were computed for a
sample of th© entire population*
Two observations should be made in connection
C
O
O
P
E
R
A
T
IV
EM
A
T
H
E
M
A
T
IC
ST
E
S
TF
O
RC
O
L
L
E
G
ES
T
U
D
E
N
T
S
QS
Pre-Test for First Year Students
Experimental Form A - t 9 J 5
Please print:
Name
____________ -
_________
Last
Date
First
Class_________________________ Age
Middle
Date of Birth
Yrs. Mos.
College or University______________________________
Classification (check one):
Liberal A r ts____
City
Sex_________
M . or F .
Engineering___
Pre-Professional
In the following spaces Indicate the number of years that you have studied each of the following
subjects (one semester = 1/2 year;
one quarter = 1/3 year;
one summer session = 1/4 year)
Algebra
Plane
Geometry
Solid
Geometry
Trigonometry
In High School
leneral Directions: Do not turn this page until the examiner tells you to do so. This examina­
tion consists of two parts and requires 45 minutes of working time. There Is a time limit for
sach part. You are not expected to answer all the questions In either part in the time limit,
lowever, if you should finish Part I before the time limit Is up, go on to Part II. No questions
nay be asked after the examination has begun. Use the blank spaces at the left of the page for
four calculations, and write your answers on the lines at the right of the page.
Part
Minutes
I
25
II
20
Total
Score
45
Copyright, 1935, by The Cooperative Test Service.
Printed in U. S. A.
All rights reserved.
G £'
-3 -
Part I
Directions: In all
answers in simplest
any square roots to
(Time: 25 minutes)
£
x
questions of this part, carry out the Indicated operations and give the
algebraic orarithmetical form and without parentheses* Do not reduce
decimal fractions • Write the answers in the spaces to the right of the page.
x
^
1. 2 i 5 _
3 * 6
3
2. 2 + 1 _
4
3
6
3(-6) - 4(-5) =
3. a)
c
. . . .
b)
2 + 0 ~
...............
y
c)
0 * 3 =
...............
6
d)
0 ^
4
4. Add
...........
3 - 5y/2
0
. . Sum =
2 + V?
5. 2v/5 •
3 y /7
=
1
................
6. Add
x - 4y + 1
3x + 4y - 6
4x + 3y
7. Add
.06ra + .03n - 5.Op
- .30m + «46n + 2.4p
-----------
Sum =
8. 3x ■- [2 - (5x + 2)] =
...
7
9. (4r - St) + 3(t - t») =
10. Expand and collect terms:
(2a2 - 7a - 9) . (5a - 1) =
r
■jo
11. ab ^ I + I ) =
a
b /
12. Write with a single exponent:
a • a2 • a^ =
...........
1
13. Solve for x:
3x - 5 = 8x + 10
</
f!
. -jr =
14. Solve for W:
P
|w - W = 3
O
. . W = ___________________________
(J
15. Solve for y:
2y _ 2y + 1 _ 1
3
5
~ 3
16- Solve for T:
T _ MT - G
L — -—-
.t =
17. Expand and collect terms:
(n + l)(n2 + l)(n2 - 1)
Go on to the next page*
i!
- 3 Reduce to lowest terms:
q2 - 9
"■-*^
3 3
q2 - 8q + 15
Reduce to lowest terms:
c2 - 5c - 10
c2 - 5c
(c - 2)^
*
3
3
c - 2
Perform by long division:
(2x3 - 9x2 + llx - 3) 4.{2x - 3)
Simplify:
(x - y)(y - z)(z - x)
(x - y )(x - z)
,
^ (r~
Simplify:
6ab + 12b 2
3a2 + 6ab
____________________________
i/- Lf
Express as a single fraction:
2m
+
m +1
1
m -2
3
3
^
^
.......
Express as a single fraction:
3
2xy
_ 4y - x
4x2y^
\y/a. + x + Vx)
If -y
• (VaT + x - \/x) j___________________________
Va2 - x2 A/a2 - x 2 + — 7
'
Vsa2 - x2
a) Compute
b)
8I
- 5
Express by use of a
fractional exponent:v a ------------------------
a,
/
—
Change to simplest radical
form:
^
,
3 *
Solve for a (the value of b
is not required):
3a + 7b « 7
5a + 3b = 29
.
a = ---------------------------Solve for S:
C - (1 - S2 ) - S2
. S = ____________________________
Solve for x:
3x + 7 =
_ x + 6
x - 3
One root of the equation
2r2 + 3r = 2
is r - -2.
Find the other .............
If t = -*04 A.B.(vy
, and
(A + B)r2
F * 144, f = 81, r = 2,
A = B = 120, find the exact
numerical value of t . . . t -
Number right, Score
2. ^
r ' ->
^
Part II
Directions:
34.
4
(Time: 20 minutes)
Write the answers to the following questions in the spaces to the right of the page.
Write formulas for each of the following statements, using the given letters:
a) The volume V of a rectangular solid is the product of its length x,
breadth y, and height
........ ............................................................. .
b) The volume V of a sphere is foxir-thirds the product of tt and the cube of
the radius
...............................................................
c) The volume V of a right circular cone is one-third the product of tt, the
altitude h, and the square of the radius r of the base..................
d)
35.
The perimeter p of a regular polygon is the product of the number n of
its sides by the length s of one side..................................
Construct a formula for the quantity desired In each of the following cases.
Define by picture or words any letters you introduce for dimensions.
a) The volume of a rectangular box with square ends.
Volume =_
b)
The product of the sum and difference of two numbers.
Product =_
c) The cost of paving a street in the shape of the shaded
part of Fig. 1. The street Is of uniform width and
lies on two sides of a square block.............. Cost =_
yy
I
Pig, 1
36. If the radius of a. sphere is doubled, by what number is the volume multiplied?
37. If m books cost n dollars:
a) What will one book c o s t ? ...................
b)
38.
What will x books cost?
...................
If W and w are weights balanced at distances A, a from a fulcrum at F, the
law of the lever can be expressed by the equation, WA = wa.
A
a) If w = 20, W = 40, a = 10, find A ...........
w~|
Ekl
p
b) Express W In terms of A, w, a.
c)
Fig. 2
39.
-
I f
A = |
a
3*
find - .....................
w
One number Is four times another and the sum of the numbers is 240.
a) Write this statement in the form of an equation in one unknown.
b) Write this statement as two simultaneous equations In two unknowns.
c ) Find the numbers.
40.
Fig. 3
d
.................................................
The sum of three numbers is 150.
The first number is
twice the second and twenty more than the third.
Find the third number................................
_ 41. In Fig. 3, BC is parallel to ED, AB = 7, BE = 5,
£
ED = 6. Find B C ................................
42. By definition a radian Is "an angle which, if Its vertex is placed at the
center of a circle, intercepts an arc equal to the radius." How many radians
are there In an angle at the center of a circle of radius 4 Inches which
intercepts an arc of 12 inches? ................................................
43. The base of a pyramid is the face of a cube.
The altitude of the pyramid is
equal to the edge of the cube. Find the volume of the cube If the sum of the
volumes of the two solids is 8 cubic inches.....................................
44. Consider the operation: Multiply a number by 2 and subtract 4. Perform this
operation on an unknown number x. Perform the operation on the answer.
Repeat. Find a number such that the result of the third operation is 4.
. .-
Number right, Score
COOPERATIVE MATHEMATICS TEST FOR COLLEGE STUDENTS
Pre-Test for First Year Students
Experimental Form B
Please print:
-Name
_________
Date
Last
Class________________________
First
Middle
A g e _____________ Date of Birth
Yrs. Mos.
-College or University_____________________________
C i t y ____________________________ Sex_________
M. or F.
.Classification (check one):
Engineering___
Liberal Arts____
Pre-Professional
In the following spaces indicate the number of years that you have studied each of the following
subjects (one semester = 1/2 year;
one quarter — 1/3 year;
one summer session = 1/4 year)
Algebra
Plane
Ge ome try
Solid
Geometry
Trigonometry
In High School
-General Directions: Do not turn this page until the examiner
tellsyou to do so.
This examina;ion consists of two parts and requires 45 minutes of working
time. There is a time limit for
sach part. You are not expected to answer all the questions in either part in the time limit,
lowever, If you should finish Part I before the time limit is up, go on to Part II. No questions
--lay be asked after the examination has begun.
Use the blank spaces at the left of the page for
mur calculations, and write your answers on the lines at the
right of the page.
Part
Minutes
I
25
II
20
Total
Score
45
Copyright, 1935, by The Cooperative Test Service.
Printed In U. S. A.
All rights reserved.
- 2 Part I
(Time: 25 minutes)
Directions: In all questions of this part, carry out the indicated operations and give the
answers in simplest algebraic or arithmetical form and without parentheses * Do not reduce any
square roots to decimal fractions. Write the answers in the spaces to the right of the page.
1.
Divide 8 by —
7
2.
Add | and |
3. Add 2 - /®
.
and
4. Multiply 7/3
_______________________
5 + 4/3
by
. . . .
4\ / 2 ...........
5. Find the value of
3 - 2(-5) + 4(-3).. ...........
6. Remove the parentheses and combine
like terms:
5x - [Zx - y - (2x-y+3) - 6y]
7. Perform the Indicated operation:
(3L - 7K) + (L + 2K) .........
8. Write with a single exponent:
5
JZ
3
2 *2
- 2
.................
9. Solve for x;
4 - 2x = x + 1 1 ..........
10.
Solve for u*
■^u + u
11.
—
=2 u - 2
u =
Solve for y (thevalue of x Is
not required):
6x + 5y = 22
............ y
4x - y =
6
_
= _
12. Perform the indicated multiplication:
(t3 - t2 )(t2 - t - 1)
. . . .
-
13. Divide x® - x^ + 2x - 2 by x - 1.
14. Simplify:
(a - l)3 (a - 2)
(a - 2)(a + 2)(a - l)2
15. Compute the value of 9x ~ ^y Wh en
4y + 3x
x = 2, y = 3
......................
16. Add:
4a^ + 5ac^ — 3c®
8a2 - 7ac2
______ 2ac2 + 5c3
17. Add:
. Sum =-
-Olx2 + .4 x - 1
3.00x2 - .02x +
.03
Sum =_
18. Simplify:
(7 + 2)(y + 3)(-- i-— --- ^ — ) .
\y + 2
y + 3'
19. Simplify:
1
1_____
/a + x - Vx “ Va + x + y/x
Go on to the next page*
- 3 Simplify:
V(x+y)2 + 2(x-y)(x+y) + (x-y)2___________________________
Add:
x
2x
x + 4
x + 5
Reduce to lowest terms:
2S2 -o16S + 24
S^ - 36
...
Write in simplest radical form:
3^5
/aa ub
Simplify:
— x--+
Simplify:
a)
x ■
x5 ■ x 7
( X 5 )2
Solve for x:
b2 + (c - x)2 = a2 + x2 ..• x =Solve for x:
•04x — #475 = #005 — #2x
Solve for t:
... x
^ ~ 3 = 2s
t + s
t =.
Find the positive root of the
equation:
3m2 + m = 2
.............
Find the value o f :
Number right, Score
- 4-
Part II
Directions:
31.
32.
(Time: 20 minutes)
Write the answers to the following questions in the spaces to the right of the page,
Write a formula for the following statement, using the given letters:
The lateral surface S of a right circular cylinder is equal to the
product of tt, its altitude h, and the diameter D of the base. . . .
-------------------- -
Construct a formula for:
a) the entire area of the surface of a rectangular box,
b) the quotient of the sum of two numbers divided by their difference.
.
c) the cost of a sidewalk around a square block; the walk Is of uniform
width and there is no space between the walk and the curb...........
33.
If a train runs M miles in 5 hours, how many miles will It run in
K hours at the same rate of speed? ......................................
34.
If W and w are weights balanced at distances A, a from a fulcrum at P,
the law of the lever can be expressed by the equation, WA = wa.
a) If W = 40, w = 20, a =
10, find A.
fwl~
b) Express a In terms of W, A, and w
7\
F
c) If the ratio of A to a Is 4, find
the ratio of W to w .............
Fig,
35. The sum of ‘two numbers is 25. Their difference is 17. Write this
statement:
a) as an equation in one unknown ..................................
b) as two simultaneous equations
c) Find the numbers.............
36. Let a, b, c be the sides of a triangle. The perimeter p is the
If p = 100, a + b = ^c,
sum of the lengths of thd sides.
b + c = 3a, find a
37. In Fig. 2, the angle at B is a right
angle and DE is parallel to B C . If
AD = 8, DE = 6, and BC = 9, find DB.
DB =.
A
Fig. 2
38. In the tetrahedron OABC (Fig. 3), OA = AB,
OC = OB, angles OAB, AOC, and BOC are
right angles. Find CB If OA = 4.
CB =__
Fig. 3.
39. A parallelogram may be formed by drawing two equilateral triangles on a
given base with their vertices on opposite sides of the base. What Is
the perimeter, in inches, of the parallelogram If the length of the
given base is 4 inches?
.................................................
40. Let y be a number obtained from x by the following operation: multiply
x by 2 and subtract 4. Let F be the result of applying this operation
to y. Find x so that when the operation is applied to F the result
will be 1 2 .................................................................
41. The geometric mean G of two numbers x and y Is given by the formula
G = Vxy.
If the geometric mean of two numbers is 6 and one of the
numbers is 9, what is the other number?
............................
42. Find an Isosceles triangle in which the altitude equals the base and
each is numerically equal to the area................
Altitude =
Number right, Score
COOPERATIVE MATHEMATICS TEST FOR COLLEGE STUDENTS
Pre-Test for First Year Students
Experimental Form C
Please print:
Na m e _______
Date
Last
Class________________________
First
Middle
A g e _____________ Date of Birth
Yrs. Mos.
College or University_____________________________
City
Classification (check one):
Engineering___
Liberal A rts____
Sex -________
M. or F .
Pre-Professional
In the following spaces indicate the number of years that you have studied each of the following
subjects (one semester = 1/2 year;
one quarter = 1/3 year;
one summer session = 1/4 year)
Algebra
Plane
Geometry
Solid
Geometry
Trigonometry
In High School
General Directions: Do not turn this page until the examiner tells you to do so. This examina­
tion consists of two parts and requires 45 minutes of working time. There is a time limit for
each part. You are not expected to answer all the questions in either part in the time limit.
However, if you should finish Part I before the time limit Is up, go on to Part II. No questions
may be asked after the examination has begun. Use the blank spaces at the left of the page for
your calculations, and write your answers on the lines at the right of the page.
Part
Minutes
I
25
II
20
Total
Score
45
Copyright, 1935, by The Cooperative Test Service.
Printed in U. S. A.
All rights reserved.
- 2 Part I
(Time: 25 minutes)
Directions; In all questions of this part, carry out the indicated operations and give the
answers in simplest algebraic or arithmetical form and without parentheses. Do not reduce any
square roots to decimal fractions. Write the answers in the spaces to the right of the page.
1
9 . 18
14 * 21
2. 1
5
+
3
_
=
.............
............................................
4
3. 2\/3 - 3V2
........... stun
4* 6V2 * 2^5 =
...............
5. (6 - 2) - 3(2 - 5) = . . . .
7. (6u + 5v) + (4u - 6v) =
♦
II
l— I
10
i
0
02
1
1
---- 1
1
o
to
6.
•
.
»
.
_2 • r
„3 • r
_4 =
_
8. r
.............................
9. Solve for x:
5x - 24 = 6
.
,
.
1
>
!.■«■ 11■ II1
......
■'
x
10. Solve for k:
2k - I k = 1 8
5
....................
k
s=
11. Solve for n (the value of m
is not required):
5m - 2n = 7
2m + n - 1 0 ........................ n
12. Simplify:
(u + v)2 - (u - v)2
uv
13. If a = 16 and x = 9, find
Va + x
.................
-
14. If (a + b)2 = a2 + 2ab + b2 ,
find the corresponding
expression for (M + 2N)2 . .
I
3(1 - 1) * .............
n
' It
o o
+ +
w to
15. a)
.
----------------------------
ll
o|
'o
4
16.
P - q2 - 4
6p + 5 + 3q2
q2 + 3
17. lg H - 2-| K
7
8»
-
2
§«
Difference
Go on to the next page.
(3x2 - 7x + 6)(2x - 1) =
'6x2 + llxy - 10y2)-i- (2x + 5y)=.
Solve for T:
TL_j_Q = M
s
.
. T =
Solve for x:
*15*-=.11 = (x - l)2. . X
If x = y/2 and
(6 - V^)x “ 3V2y + 2 = 0,
find the value of y.
x + 9 _ x
x + 5
x
. . . y
+5 _
+8
y - *
( * - 1)
x + y
\/l215 =
The expression a + b is a
factor of a3 + b ^ . Find the
corresponding factor of
8R3 + S3 ...................
1 - t _
t + 1
t - 1
V*64a2 + (\/8a)2 + \/2a • -\/8a =
(xs - ys) (x® + y s ) = • • • Solve for x and y:
—
x
+
—
y
3 - I
=
12
= 1
y =Solve for x:
2x2 = 5x + 3. .
. x —.
Number right, Score
-
Part II
Directions:
Write the answers to the following questions In the spaces to the right of the page.
33. The perimeter of a polygon is the sum
of the lengths of the sides. Write a
formula for the perimeter of the given
polygon In each of the following cases:
a) An equilateral triangle of
side
...............
b) A square of side s
c) A rectangle of base b and
altitude a ...........
Pig.
d) A regular octagon of
side s (Fig. 1) • *
34. If x blocks of metal weigh 4 pounds, what will z blocks weigh?
35. A circular cylinder and a circular cone have equal base radii and equal
altitudes.
The volume of the cylinder is how many times greater than the
.....................................................
volume of the cone?
36. The sum of half a number and its double exceeds the number by twenty-one.
a) Write an equation equivalent to this statement ...................
b ) Find the number.
37. The sum of three numbers is 18. The sum of the second and third is twice
the first number.
The difference between the first and third numbers is
one-tbird the first.
Find the second number.............................
C
38. In Fig. 2 the angle at B is a right
angle and DE is parallel to B C .
If AD = 8, DE = 6, and BC = 9,
find A C ............................. AC
A
Fig. 2
A'
41.
-
(Time: 20 minutes)
32. Express the following statement as a formula, using the given letters.
The energy E of a particle is equal to half the product of its
mass m and the square of its velocity
..................................
40.
4
39. The triangle ABC is Inscribed In the
circle whose center is 0. AB goes
through 0. DC Is perpendicular to AB
0A = 5, BC = 6.
(Fig. 3) Find DC.
DC
Fig. 3
The average of twonumbers, x and y,
is
k. Write an expression thatgives
the value of y in
terms of x and k .................................... y
=
If y = 3 - 2x, x =
3 + 2t, and t = 3 - 2u,
express y in terms of u In
Its simplest form......................................................... y =.
42. Find a number x such that the volume of the cube of edge x Is twice the
area of the square of side x ........................................... .. .
43. If a positive number is added to ten times its reciprocal the result is
three times the number diminished by eight times its reciprocal.
Find
the number...................................................................
44. If the perimeter of a right triangle Is 70 and one of the two shorter
sides Is 20, find the length of the other.................
Number right, Score
—
74
with column four, the ntotal11 eoluam®
Sine© th© samples for
Forms B and G are all th© same sis©, this column In Tables
26 and 27 consists simply of th© indices derived by consider­
ing th© sum of th© three preparation groups as © single
sample*
This la not the case, however, in Table 25*
though th©
Al­
and Pg groups contained only 50 papers each,
th© best estimate of their true difficulty Indices is as
found*
Hence they were weighted th© same as th© P^j, group
which was based on 100 papers*
Table 25
ITEM DIFFICULTIES FOR EACH PREPARATION GROUP Pn# P-i, and P«
m m a, lose
Seam
£l
fn
£2
i
2
5a
b
e
a
4
5
6
76
84
70
92
94
94
56
46
86
56
60
66
46
50
86
06
56
18
10
48
54
sa
92
88
98
96
95
88
74
90
68
70
92
64
55
95
80
75
45
2®
80
78
82
94
92
98
100
94
84
86
98
66
84
92
84
78
98
86
70
54
54
70
78
7
a
9
10
11
12
15
14
15
16
17
18
Total
82
90
83
96
97
94
74
69
91
03
71
83
65
50
93
77
60
38
24
61
62
Cooperative
Test Be:
70
84
78
95
93
91
69
59
86
61
62
77
55
53
87
70
52
42
35
52
53
95
Table 25 (contirraed)
FORM A, 1935
M m
pl»
l£
?oJal
Coopers
Test £>©:
19
20
21
29
23
24
25
26
27a
b
28
29
m
si
52
SS
30
38
SO
16
1©
1©
©
©
1©
14
8
0
0
4
0
0
58
59
33
45
32
24
47
22
44
47
25
20
0
2©
m
6
70
7©
52
©8
44
42
7©
40
72
7©
48
30
4
30
2©
6
53
58
35
43
31
27
43
23
44
46
27
17
1
80
14
4
42
56
36
44
34
31
47
34
45
50
35
24
3
28
18
7
54a
b
e
6
35a
b
c
36
57s
b
92
76
82
SO
98
83
78
91
9©
90
90
90
95
83
83
87
98
73
81
83
*#
mm
38ab
c
39a
b
0
40
41
42
45
44
m
mm
12
54
40
54
48
20
**
Wfe
22
©5
50
78
72
29
*»
82
66
66
80
40
21
07
54
6©
07
30
«**
56
19
25
28
2
11
mm
m
m
80
40
2©
5©
B
16
24
04
49
74
58
34
mm
mm
p#
'#►
40
20
14
10
2
18
m«p
59
26
22
27
4
15
51
24
17
31
4
12
* This line indicates th© separation of Farts I
and :
Table 26
ITEM DIFFICULT IBS FOR EACH PRJ2PARATI0H GROUP P19 P.*, and P0
FORM B f 1955
^
xir
^
H
i
2
3
4
5
6
7
8
9
10
11
12
13
14
IS
16
17
18
19
20
21
22
23
24
25a
b
26
27
28
29
30
51
32a
b
e
35
34a
b
c
MM1a T T
ano
xx#
Cooperative
M
%
Tot.
Teat Service
74
82
SO
38
68
52
82
80
44
40
30
66
50
68
76
86
52
20
0
8
16
20
18
8
22
16
0
14
2
4
6
72
84
70
70
76
70
80
80
68
72
48
50
64
08
80
98
76
44
10
18
Oo
32
40
38
48
22
8
26
2
6
12
70
92
76
78
80
m
94
80
80
70
04
60
00
80
82
94
70
48
16
24
36
50
42
60
42
18
30
10
20
14
70
86
67
02
75
63
85
80
64
61
47
57
05
72
79
95
66
39
11
17
30
32
36
29
43
27
11
23
a
13
11
74
77
54
58
67
62
78
76
57
50
44
55
60
62
72
S3
64
43
26
33
37
42
41
39
49
33
26
32
13
26
17
66
88
98
84
75
an
«**
4*
mm
«N*
m
*3SrT®
mm
m
mm
*#
>#►
mm
W»
28
56
28
2
42
72
66
18
46
92
70
26
39
75
55
15
34
70
52
85
This line Indicates the separation of Parts I
*
yy,
Table 26 (continued)
FORM
Mm.
35a
b
a
fit
***
57
48
6
8
38
39
40
41
42
22
0
4
4
36
2
**■
70
14
24
12
36
4
16
4
B9 1935
£2
m
m>
68
18
22
14
60
4
IB
4
Total
Cooperative
Teat Service
«
4m
4m
32
13
18
9
36
3
13
58
22
25
14
38
4
18
9
4
Table 87
ITEM DIFFICULTIES: FOB EACH FBEPAHATIGH CROUP Pf *
FORM 0, 1935
Item
F1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15a
64
88
30
42
62
64
84
86
92
58
6
18
66
70
78
88
74
64
68
52
64
32
b
c
16
17
18
19
20
76
96
58
60
72
§6
94
94
100
78
36
26
82
94
100
90
92
80
76
64
74
48
74
94
70
84
6©
7©
90
9©
92
82
60
54
90
94
98
94
84
82
7©
86
7©
76
and P©
Total
Cooperative
Test Service
71
93
53
©2
©7
©6
89
92
©5
71
34
83
79
86
92
81
83
75
75
67
71
52
75
84
50
65
©5
58
82
90
95
71
43
39
71
7©
87
88
77
68
70
60
©8
46
78
battle 27 (continued)
POEM C, 1956
Item
21
22
25
24
26
26
27
28
29
30
31
32
33a
b
e
a
34
35
56a
b
37
38
39
40
41
42
43
44
Cooperative
Test Service
Pm-«a
P **
fx|
P
r2
20
6
10
2
4
80
4
2
12
0
8
52
20
26
4
6
40
10
16
24
12
14
68
60
26
32
6
12
56
20
42
42
26
28
44
17
23
4
7
42
11
20
20
13
17
40
24
31
3
11
47
14
29
35
17
27
84
98
SB
82
98
44
52
@6
82
81
70
84
28
39
me
52
19
37
10
35
15
16
15
3
65
81
81
75
77
32
38
49
22
55
12
39
23
27
19
6
46
68
68
56
64
12
18
«#
40
14
14
2
20
2
8
4
0
86
88
72
92
88
43
■m
52
18
40
10
38
16
12
10
2
84
24
58
18
4©
26
28
30
8
*otal
This line indicates the separation of Parts I
and XI*
*
Illustrative of th© differences between preparation,
groups found In the preceding chapter are the following dis­
tributions of difficulty indices*
It will b© observed that
there are a great number of items which were very difficult
for th© on© year groups*
Difficulties are spread somewhat
uniformly over th© entire rang© for th© one and one half year
groups, while the Items proved to b© rather easy for the twoyear groups*
More than half of th© Items in Form A disclose
difficulties of 73 or more for the two-year group*
Table m
DISTRIBUTION OF DIFFICULTY INDICKS FOR EACH PREPARATION GROUP
FORM A* 1935
Index
95-100
90-94
85-89
80—84
75-79
70-74
65-69
60—64
55-59
50—54
45-49
40-44
35-39
30-34
25-29
20-24
15—19
10-14
5-9
0— 4
p2
4
2
3
2
1
2
1
8
2
S
1
3
2
3
7
4
3
6
5
4
2
3
3
4
2
2
4
1
4
1
2
6
4
2
1
1
2
S
7
2
7
6
3
4
2
1
5
1
8
3
1
2
1
SO
Table
m
DISTRIBUTION OF DIFFICULTY INDICES FOE EACH PREPARATION GROUP
FORM B, 1935
.Siggat
95-100
90-94
85-89
80*84
75*79
70*74
05*09
60-64
55-59
50-54
45-49
40—44
35-59
50-54
85—89
80—84
15-19
10-14
0—4
*L
h k
1
3
1
1
3
2
4
1
2
1
1
5
3
3
1
0
8
£a
1
i
4
1
4
3
6
3
1
1
6
2
4
2
3
2
3
3
1
1
2
4
3
2
3
2
2
3
1
1
2
2
4
3
2
Table 30
DISTRIBUTION OF DIFFICULTY INDICES FOR EACH PREPARATION GROUP
FORM C, 1935
M
m
95-100
90—94
85-89
80-84
75-79
70-74
65—69
60—64
55-59
50-54
45-49
40—44
85-39
80-84
25-29
20-24
15—19
10-14
5-9
0— 4
£l
i
3
1
1
2
4
6
2
1
1
2
3
2
2
5
4
9
M
5
0
2
2
2
4
1
2
2
2
2
2
2
3
2
3
0
1
2
3
7
2
6
4
2
1
3
2
2
1
3
2
5
2
1
1
2
31
We shall return, now to Tables 25, 26, and 27, and
compare the difficulty indices given for each of the prepara­
tion groups*
Almost every item yields different indices for
each of the three groups and the question which arises is,
wWhich of these differences are too large to be explained
by sampling theory?n
In order to answer this question, we
must have some measure of the error involved in each of the
percentages given*
I®# ^he standard error
nuted from & stratified sample»
& percentage as com**
The standard error of a
percentage, or a proportion, as computed from a random
1
sample is given by the formula
where P is the proportion of correct responses, 0, « i-P
(or Q $ 100* P
If P is given in percentage rather than
decimal form), and H is the number of elements comprising
the sample*
If a stratified sample is used, In which the
elements selected from a given stratum are chosen at ran­
dom, the standard error is given by the formula
where
c"L? I s the variance of the percentage of correct
responses for each stratum taken about the percentage P
* 0* Udny Tula and M, 0, Kendall, An Introduction
to the Theory of Statistics, p« 351*
©f the entire sample*
The following derivation of the
formula is, except for minor changes, the same as that
given by Bowley (4)*
I»et a sample contain
ng, **.♦ ,
elements
In t strata, and let the numbers which have a certain
characteristic (number of successes on a given item) be
Pl»X* Pgn£* •*« , ®tn t
bet H *
ln thea€? strata*
+ Eg * * • » + & £
Thua
* # Pj
Pln l * PSn 2 * *** * Pfcn t *
(^) + %
How
( ^ ) + »'*» + Pfc (-jp)
^ £§— • „
and hence^
t
-
1 *
1, 2, ... , t
g
ZZ(w*-)
^m
t-1
t
s
ijt H
nipiql
O)
g
If w a* ©x35! ♦
+ **• +
where the
x *a are Independent variables and the c fa are arbitrary
— ^
*w
«
«
constants, then
tfw = ^ a <r£
*
J. F* Kenney,
11Some Topics In Mathematical Statistics,ft The American
Mathemetic al Monthly* XLVI (February, 1939), Ho* 2, pp» 81,
62*
83
Tbaa
H8 ffpz a
»iPl<l - Pi>
t
t
“ lPl * /
“ lPi
i n mi m i l
t-l
i=l
-Jfe
^
/
o
Ki*pi *•
p)
_
*>
hp
t= l
*
where
W ft *• N 0~ *
P
<T* i J
Bi (Pf * *>S
*
t= I
^Pherefore
or
^ ^
s
~~
r
{8)
Formula (8) demon®tratea that the error la less
when a stratified aample Is used than It is when a random
sample Is used (unless
0“"? *s 0)*
However* Its form Is
tr
not as convenient for our purposes as la that of (X)* It
Is to b© observed that (1) may b© written as
*** & t
*i«i
(3)
where t± Is the frequency of correct responses In stratum 1 *
34
Xf, now* P is in pereent&g© form rather than
decimal form,(3) hecomes
<rp* = -3gr / " n^ClOO Pj XIOO- 100 Pj)
=
, M
t
®
W
tSl
U
l
fi<i
.
U )
Heeall* now, that some of our samples contained
50 papers and the rest 100 papers#
For the samples of 50,
formula (4) becomes
•*‘ s
and for the samples of
4 ]UiL
t
°"p =
] T
fiqi
These are the two formulas we have used to calculate the
standard errors of the various difficulty indices#
These
measures will be found in Tables 60 to 64 in the Appendix#
19# Comparison of the difficulty indices found
for preparation groups#
In order to compare the difficulty
indices given for each of the preparation groups, it was
necessary to compute the standard error of the difference
«
between two percentages# The formula used was
<T(diff#} 2
3
yj <T2 +
, where <Tt and <F£ are the
X# F* Kenney, f,Som© Topics In Mathematical
Statistics,*1 The American Mathema11oal Monthly* XLVI
(February, 1939), So# 2, p # 62#
standard errors of the percentages Involved#
Using this
formula, the standard error of the difference was calculated
for each Item on each test and for all ways of pairing the
preparation groups*
Thus there are three such calculations
for each Item, on© involving groups
involving groups
and P ^ , a second
and Pg, and a third involving groups
P*k and P«g#
After the above calculations were made, each
observed difference In percentage was divided by the cor**
responding standard error of the difference#
All quotients
of 2*698 (the equivalent of 4 probable errors) or more
were considered as significant#
All of the differences In percentages, standard
errors of these differences, and their corresponding ratios
will be found in the Appendix In Tables 65 to 79#
Those
ratios which were large enough to be classified as
^signlfleant® are listed In the following three tables*
36
Table Si
ITEMS OF FORM A, 1935, WHICH YIELD SIGNIFICANT DIFFERENCES
BETWEEN PREPARATION GROUPS
h
m&
Item
pi&
p2
Item
5a
4
5
pi «“a ps
Item
5a
4
5
a
0
9
10
%0
11
11
14
16
16
14
16
16
17
18
10
W
10
20
22
22
26
26
27a
b
28
20
51
52
26
27a
b
28
58a
b
300
40
20
81
22
25
24
26
20
27a
b
08
00
51
50
37a
«
38b
300
# This line indicates the separation of Farts I
and 11#
Table
m
ITEMS OF FORM B, 1935* WHICH YIELD SIGBXFICAKT DIFFERENCES
BBTTOEK FREPiVHATXOH GROUPS
P1
glj|
Item
^1-j
F2
Item
gl
Item
3
4
5
4
9
10
10
11
19
23
24
25a
__________
31
34b
c
15
19
23
24
25a
b
29__________29
31
34a
34a
b
c
39
»
* This line indicates the separation of Farts I
and 11#
Table 33
ITEMS OP FORM C, 1935, WHICH YIELD SIGNIFICANT DIFFERENCES
BETWEEN PREPARATION GROUPS
Pi
Hi
Item
Hi
ps
Item
3
4
11
11
12
14
15 a
IB
20
21
28
28
30
and i
Item
3
4
10
11
12
13
14
15a
18
20
21
22
23
2©
28
29
30
31
32
33a
G
d
33d
35
38
41
43
34
35
38
3©
40
41
42
43
* This line indicates th© separation of Parts I
and XI*
20m Bff.ee t. of preparation upon toowled^e of the
concepts and processes covered by the 1935 pro-teats#
He-
eall that each teat was written in two parts and that
a
time limit was placed on each part*
The consequence of the
restricted time element was that the number of omissions
Increased very noticeably toward th© end of each part of
each test*
In an analysis of difficulty these later items
should not b© considered * th© factor which determines
success is probably speed rather than ability*
It has thus
been found necessary to eliminate from cons!deration Items
28*33 and 40*44 of Form A $ Items 20*30 and 36*42 of Form B t
and items 27*31 and 30*44 of Form 0#
The section which follows represents an effort
to assemble the concepts and processes cowered by th© pro*
tests In four overlapping groups*
In one group are found
the concepts and processes which were well*knowm by all
groups of students*
This Is a rather loos© term* but has
been defined (arbitrarily) to mean that difficulty Indices
for all three preparation groups were 70 or more*
A second group consists of the concepts and
processes which were found to be difficult for all three
preparation groups and is defined so as to include only th©
items for which no index exceeded 50*
fourth
groups
In the third and
are listed those areas which were materially
strengthened by a study of third and fourth semester algebra
respectively*
In assembling these four groups* no attempt was
made to differentiate between results secured for the
separate examinations*
All three forms were considered to­
gether and an effort was made to determine general con­
clusions*
In other words* If all Items covering a certain
concept or process In Form A were found to be easy, but
similar Items In one of th© other forms were not, then it
was not considered possible to general1b ©
ment was regarded as necessary*
- complete agree­
In no case, however strong
the evidence might appear, has a conclusion been drawn
from a single observation*
Using the procedure outlined above, th© following
conclusions were derived from Tables 25, 26, 27, 31, 32,
and 33*
A* Concepts and processes which were well-known.
1* Elementary operations of arithmetic*
By this expression Is meant addition,
subtraction, multiplication, and division
of positive Integers and fractions*
Intro­
duction of parentheses and negative Inte­
gers Increased the difficulty, but only
slightly*
S* The principle of adding exponents when a
series of terms with Identical bases are
multiplied together*
Items illustrating this principle were
very elementary, consisting of no coef­
ficient® other than 1 and of only one base*
It I® interesting to not© that th© Item
S5 ,22,2«
was more difficult than those con­
sisting of literal bases*
3* Column addition of algebraic binomials and
trinomials with integral coefficients*
There were three Items testing for this
prooeaa and in one caae the Px index wee
64*
All others were 80 or above*
This
particular exercise did not list similar
terms in the same column, and Its signifi­
cance is Illustrated by th® fact that all
three indices were lower than any other
index in this group of exercises*
4# Removal of simple parentheses followed by
addition of similar terms, when plus signs
preceded each parenthesis*
Insertion of a number before th© parenthe­
ses added to the difficulty for th©
group, and removal of confound parentheses
was considerably more difficult*
Concepts and processes which were difficult
for all groups*
1* Addition and subtraction of algebraic
fractions*
92
2 * Operations with fractions containing ir­
rational denominators#
3 * Reduction of irrational terms to simpler
form#
This Is difficult for all types of
radicands exhibited, whether they be alge­
braic or arithmetic#
4# Writing an equality of two products in th©
form of a proportion and then manipulating
the terms •
This Is illustrated by Item 38(c) of
Form A and item 34(c) of Form B*
give the lever formula WA « wa
th© ratio of A to a#
Both
and also
The student Is re­
quired to find W/w#
0 * Areas materially strengthened by a study of
third semester algebra*
1* Addition, subtraction, and multiplication
of Irrational expressions*
Th© agreement is not perfect but In eight
cases out of nine significant differences
appeared*
For the ninth Item a significant
difference was found between groups
and Pg*
2# Solution of all but th© very simplest of
th© equations Involving numerical fractions#
93
3* Solution of simultaneous equations.
Strictly speaking, we have no right to
list this process since only one observa­
tion is usable*
However, there are two
Items in the set we have agreed to disregard
because of omissions, and all three yield
significant differences*.
Hence
the concept
Is Included#
4# Under®tending of positive fractional ex­
ponents «
D# Areas materially strengthened by a study of
fourth semester algebra#
1 # Understanding of positive fractional ex­
ponents *
An examination of Tables 51, 32, and 33 will reveal
the fact that the ^
did the
groups scored significantly higher than
groups on almost
all of the items*
Since this
can be largely attributed to increased general knowledge
and mathematical power, rather than to an Improvement In
specific
areas (other than those listed under G and D above)
no detailed discussion of the processes Involved has been In­
cluded *
2 1 # Comparison of difficulties of related Items *
Before concluding this section, mention should be mad© of
certain groups of items pertaining to th© same general
94
process and which were constructed for the purpose of com­
parison*
Bom© of the®© groups, with the accompanying dif­
ficulty Indices, are listed below*
The results are as would
be expected*
Index
Form A, item
©s
Adds .
Form A, item
7s
Adds *0©m + *Q3n - 5*0p
+ «46n + 2*4p
x * 4y + 1
Bx ♦ 4y « ©
4x + 5y
Form B„ item 1©*
Add;
Form B, item 17t
Addt .01*® + «4x - 1
8,00*® - ,02x ♦ .03
Fona C, item
Solve for x:
9s
4a® + 5ae® - So5
8a2 - 7ac2
2ae2 + So®
5x - 24 * ©
Form 0 $ item 1 0 s
P1
Plf
P2
8©
90
$6
68
86
98
94
52
76
70
92 100
92
58
82
98
Solve for ks
2k- Ik Z IS
72
Index
P1
Form A ,
item13*
Item14*
Item15s
36 75 70
Solve for ys
®Z S
Fora A f item 18 1
♦ 1 a 1
5
3
Item 18*
16
45
54
10
28
34
34
73
78
30
59
70
Solve for T*
L _ MT - Q
1’
Fora A
66 80 86
Solve for W;
|w - W ; 8
Form A *
P2
Solve for xs
3x « 5 Z 8x * 10
Form A ,
Pli
Reduce to lowest terms*
q2 - 9
q2 - 3q ♦
Fora A , Item 19s
Reduce to lowest terms*
e® - 3e - 10 ± o® - 6o
(c- 2)®
e— 2
96
Ha© writer has grouped another trio of items in
order to compare methods of testing*
Index
f i l l i Ie
Form A,
item 57%
Form B, item 55:
Form C, item 54:
If m books cost a dollars:
la) what will on® book coat?
54
(b) what will x books cost?
46
65 82
50 60
If a train runs M miles
in 5 hours, how many miles
will It run in K h o w s at th®
asm® rate of speed?
If X blocks of metal weigh
4 pounds, what will Z blocks
x
weigh?
4®
*B 44
Apparently an item is made easier by breaking it
up Into parts*
2 2 * The predictIt® power of individual 1 terns»
Correlation coefficients were given in Chapter XII to show
th© relationship between the number of items solved cor*
reetly on each 1935 form and first semester grades*
NXA
students found similar correlations between first semester
grades and th© sum of correct solutions to items 2, 4, 5,
9, 15, 17, 20, 23, 29, and 52 of Form A*^
this comparison was
The purpose of
to determine on© of th© effects of
using a short test, and th© papers used were those
In the
^ These Items have no particular significance
except that th©y were considered as typical of all Items*
97
samples*
The correlations found were:
for the P1 group,
a coefficient of .305 with a probable error of *08| for the
a coefficient of *379 with a probable error of
.O0| for the Pg group, a coefficient of *554 with a probable
error of *09j and for all groups combined, a coefficient of
*390 with a probable error of *04*
These ar© somewhat
smaller than the coefficients found for the entire test*
The writer carried the above plan to Its extreme
and endeavored to find th© relationship between correct
scores on Individual items, as given by the samples, and
first semester grades*
The purpose of this study was to
determine how each Item contributed to the total correla­
tion and to discover whether or not some Item or Items
provided a key to success*
Sine© the samples were rather small for a study
of this type, first semester grades were divided Into only
two groups*
Grades of A, B, and C were labeled Rsatis­
factory” and all of th© others as "unsatlafactory*n
Faired
with these grades in a two by two table were correct and
Incorrect responses to each Item*
With a four-fold table thus constructed for each
item of each form, w© were In a position to determine the
relationship between th© two variables (that is, between
success or failure on the item and satisfactory or unsatis­
factory first semester grades) by calculating
tetr&ehoric
correlation coefficients#5
These coefficients were
determined by reading directly from a set of charts pre­
pared by Cheslre, Safflr, and Thurston©,5 and are listea
In th© following tables*
at
Th© validity of this procedure rests upon th©
assumption that th© two variables are essentially con­
tinuous and would b© found to be normally distributed If
It war© possible to classify them more exactly Into finer
groupings#
Henry B* Garrett, Statistics ip PsyeholoCT
lyjfl.
» P* 571*
^ Leon© Cheslre, Milton Baffin, and M* L#
Thurston©, Oomoutinse h^egramg for the Tetrachoric Correlation
Coefficient*
99
TahlQ
34
CORRELATIONS (TETRACHORXC) BETWE3B CORRECT SOLUTIONS OF
INDIVIDUAL ITEMS IN FORM A* 1935* AND SATISFACTORY <A,B.C)
FIRST SEMESTER GRADES FOR EACH PREPARATION GROUP
Item
%
1
2
*45
3a
h
e
d
4
5
©
7
8
9
10
3LiJL
12
13
14
15
1©
17
18
19
80
81
22
83
24
85
86
87a
hb
28
29
50
31
32
33
34a
b
and XX*
-*11
*34
*08
-*15
*29
*17
*14
-*02
*17
#41
*4©
*27
*26
a*
*21
*52
ps
♦46
*80
*55
.S3
*44
*25
♦06
•40
#26
-*11
*22
*02
*37
#37
*12
#27
*54
,31
•30
*31
— #34
•S3
.31
-.11
.31
.57
«*
,40
*47
,49
*38
*11
,02
<37
*32
*47
,53
*31
*2 ©
*5©
#53
,33
*14
*14
*33
*08
- *42
*
*»
m
w
.58
*40
*31
*26
— #18
-*02
-.27
.43
*40
*65
.10
,15
,19
#17
,51
*55
#3©
*51
*58
#5©
*33
•41
♦22
,20
— .09
#47
*57
,33
#54
♦15
.25
*37
*64
*24
*37
*50
m
•nr
*66
*40
* This line Indicates the separation of Parts I
ICO
Table 34 (continued)
P o m A, 1935
P
JL
34e
a
35a
b
c
56
57a
b
58a
b
©
59a
b
o
40
41
42
m
44
*JL
#55
-*07
.37
.60
*
■m
***
*44
*25
*3©
*36
*52
*44
*70
*43
*29
*45
*74
.IB
**
-
*93
*26
*02
*11
*80
#11
.31
*56
**#
mm
*3®
-.06
*13
*15
**
*
.09
*41
.37
*09
— .04
*35
mm
.34
-.41
.03
.87
Mft
■iw
Table 35
CORRELATIONS (TETRACHORIC} BETvVESH CORRECT SOLUTIONS OP
INDIVIDUAL ITEMS IN FORM B, 1935, AND SATISFACTORY (A,3,C)
FIRST SEMESTER OEADES FOR EACH PREPARATION GROUP
IJffla
1
2
3
4
5
6
7
8
9
10
11
12
13
h
.50
*00
*38
*22
.28
*31
*50
.10
*48
*46
-.37
.04
*04
M
*31
*46
#60
.09
-.03
— .01
*48
.21
*09
*44
.11
*38
.10
fk
-.06
.23
-.17
.13
.36
m4^*
,24
-*17
-.07
.00
.19
.24
Table 35 (continued)
Form B $ 1935
Item
11
£li
fk
14
15
10
17
IS
19
80
81
88
23
24
25a
b
20
27
28
29
50
51
32a
b
0
33
34a
b
e
56a
b
c
36
57
38
39
40
41
42
.05
-.18
#44
*39
*09
«*
*14
-.12
.21
.34
— .04
w
*41
•ia
«*
*20
*26
*14
*45
«*
*45
— *08
.50
*36
*31
*36
.02
— *04
— *09
-.13
*20
-.07
+m
•m
mm
.13
.41
.10
.14
.61
*■*
mm
.18
— *10
— *08
*33
.24
*39
.09
*24
- *18
*24
*08
-*11
.46
.32
*57
<*•
mm
**
mm
■P»
mm
*54
*55
*70
-
mr
mm
*21
-*43
**
*16
*»
*»
*13
*09
*24
*36
-
-
-.06
.21
nm
*
*22
*22
.15
.14
.03
m*
-.13
-
*13
.05
*47
.33
*24
.06
1.00
.37
1.00
# This line Indicate© the ©eparation of Fart© I
Table 36
CORRELATIONS (TETRAOHORIC) BETWEEN CORRECT SOLUTIONS OF
INDIVIDUAL ITEMS IK FORM C, 1935, AND SATISFACTORY {A*B,C)
FIRST SEMESTER GRADES FOB EACH PREPARATION GROUP
21m
i
2
3
4
S
6
7
3
9
10
11
12
13
14
23a
b
0
16
1?
IB
19
20
91
22
23
24
25
26
27
28
29
30
31
32
35a
b
c
a
-.13
-.25
.21
**#10
*32
*40
*42
.11
*35
.32
*M
.67
.4?
*21
.58
*25
.53
.40
*28
*12
*63
*54
*72
*►#21
*44
4**
m
*21
a*
**
a*
•
*31
.54
#42
*25
*73
fit
g
**.51
*14
-
.03
*46
.17
*►.29
- .11
*17
.35
.25
— .36
.51
•m
*53
**.24
.32
*15
*56
*23
*27
.49
*62
.23
-**
-.21
**#26
-*#13
*17
.14
#22
*04
*24
•45
#02
#44
*47
*10
*38
*20
*13
•m
~*1I
*41
*71
*49
#40
# 44
<w
.44
.10
*23
-.15
.60
.37
#03
.23
*60
*51
-.02
*•
#28
#04
*10
#39
.39
#25
*31
*44
.72
.41
«•»
& This line Indicates the separation of Parts I
and II*
±32
Table 36 (continued)
Form C, 1935
Item
34
35
56a
m
m
b
39
40
41
42
43
fi
*52
.10
*36
*50
*11
-*11
*53
**
*35
A./L
TCT
fli
f®
*29
-.17
*03
.57
.57
.28
*32
*59
.15
- *21
*35
*40
.30
*42
♦03
-
.45
*45
— *08
•so
*43
*13
**
Blank spaces In the tables are of two kinds*
The
first Is due to the subjective items for which no data was
secured*
The second arose in the following manners
due to
the small number of students involved, there were many times
very small numbers or zeros In some of th© compartmerta*
When this occurred, the coefficient could not, in most
cases, be read from the charts*
On two occasions, it was
read with correct but misleading results*
Th© two cases
are th© ones Involving Items 40 and 42 of Form B with the
group Pg, and the coefficients were read a® 1*0*
Th© cor­
rectness lies In th© fact that every student who solved the
problem correctly earned a satisfactory grade % the mislead­
ing factor is that th©r© w©re only two such students*
Probably th© most striking fact about these tables
Is that Items which yield unreasonably high coefficient for
104
on© preparation group seldom do for the others*
A very
good example of this is part b of Item 38 in Form A*
Th©
correlation coefficient for the one and one half year group
is *74, for the one year group It is *32, and for the two
year group it is -*04*
Thor© are many Items which have rather high coef­
ficients for one or two groups, but very few have for all
groups#
However, items 22 and 34b In Fora A and item 33b
in Form 0 are very satisfactory in this respect*
It Is to be observed that th© peculiarity noted
above does not necessarily mean that the Items are at fault*
It certainly requires more mathematical ability for a stu­
dent with only one year of preparation In algebra to master,
say quadratic equations, than It does for one with more
preparation in the subject#
In addition, success as
measured by grades is rendered somewhat Ineomparable by
registration In courses with different content*
This ap­
parent discrepancy may mean, then, that concepts whose
mastery is indicative of success in first year college math­
ematics are not the same at all for students with different
degrees of preparation*
The implication is that separate
pre-tests for each of the preparation groups are desirable*
135
GHAFTEK VI
AHALTSES OF THE ITEMS OF THE 1936 PRE-TESTS
^5# 1933 items, which war© repeated in the 1936
are-tests#
Recall that the purpose in constructing
three
experimental test forma in 1935 was to form a sufficient
number of items that, on the basis of Information secured
concerning them, enough satisfactory exercises would
evolve that two permanent tests coitld be assembled.
The 1935 Items which were repeated In the 1936
tests
are given in the following tables*
iXi® to gaps in
difficulty which were found In the distribution of experi­
mental Items, it was necessary, In constructing the 1936
forms, to Interpolate a few new items of unknown validity
and difficulty and hope that they would meet the require­
ments#
All of these Items proved to possess satisfactory
validity but very few of them yielded difficulties com?**
measurete with, the location of the Item In the test#
Item
50 of Form A was particularly Ill-placed, since, unfortu­
nately, it can b© solved by Inspection#
■f v(I
COOPERATIVE MATHEMATICS TEST FOR COLLEGE STUDENTS
Pre-Test for First Year Students
Experimental Form A - 1936
Please print;
Name________________________________________________________________________ Date
Last
First
Middle
Class_______________________ A g e ___________________ Date of Birth
Yrs.
Mos.
College or University________________________________
Classification (check one):
Liberal Arts________
City_______________________ Sex__________
M . or F .
Engineering_________
Pre-Professional
In the following spaces indicate the number of years that you have studied each of the following
subjects (one semester = l/2 year;
one quarter = l/3 year;
one summer session = l/4 year):
Solid
Ge ometry
Plane
Geometry
Algebra
Trigonometry
In High School
General Directions: Do not turn this page until the examiner tells you to do so. This examina­
tion requires 45 minutes of working time. You are not expected to answer all the questions In
the time limit. No questions may be asked after the examination has begun. Use the blank spaces
at the left of the page for your calculations, and write your answers on the lines at the right
of the page.
Score
Percentile
Right
Wrong
Omitted
Copyright, 1936, by The Cooperative Test Service.
Printed In U. S. A.
All rights reserved.
-2 Directions: In all questions, carry out the Indicated operations and give the answers in
simplest algebraic or arithmetical form and without parentheses. Do not reduce any square
roots to decimal fractions. Write the answers in the spaces to the right of the page.
1. Multiply:
0*3
2. Simplify;
4 7. 4 = . . . .
3. Simplify:
r2 • r5 • r5 =
4. Add the fractions:
— + i
3
4
5. Write a formula for the following
statement, using the given letters:
The volume V of a right circular
cone is one-third the product of
ir, the altitude b, and the square
of the radius r of the base. . ___
6. Find the value of:
..............
3 (_6) - 4(-5)
7. Remove parentheses and collect
like terms:
(3L - 7K) + (L + 2K) « . . . .
8. If (a + b)2 = a2 + 2ab + b2 ,
find the corresponding
expression for (M + 2N) . . .
9. The perimeter of a polygon Is
the sum of the lengths of the
sides. Write a formula for
the perimeter p of a rectangle
of base b and altitude a.
. .
10. If y = j , find the value of k
when y = 34 and x = 2.
k =
11. Divide 8 by 24 ..............
7
12. Compute the value of
~
4y + 3x
when x = 2, y = 3. . . . . . .
13. If a = 16 and x = 9, find the
exact numerical value of
ya + x ........................
14. Solve for x:
3x - 5 = 8x + 10
x =
15. If W and w are weights balanced
at distances A and a from a
fulcrum, the law of the lever
can be expressed by WA = wa.*If
W = 40, w ss 20, a = 10, find A,
A =_
16. Add:
3 - 5y£T
17. Express the following statement
as a formula, using the given
letters: The energy E of a
particle is equal to half the
product of its mass m and the
square of its velocity v.
Go on to the next page.
- 3 18. Write £
5 in decimal form.
• . .------------------ ja_v>
19. Multiply:
(3x2 - 7x + 6 ) (2x - 1) .
•-----------------------
20. Divide x3 - x2 + 2x - 2 by
x - 1 .......................... .......................
21.
Multiply 7y5" by
4V§~- •
22. Simplify:
5c - (2c - 3 ) J ........ ...
23. What is the interest on $600
for three months at the rate
of 5% per annum?
Interest =
$
24. The area of a triangle is
24 square units and the base
is 8 units. Find the altitude
of the triangle.
Altitude = ___
25. In an arithmetic progression
the difference between any two
adjacent terms Is constant.
Find the fourth and fifth terms
of the arithmetic progression
whose first three terms are
-3, 1, 5.
Fourth term = ___
Fifth term
= __
26. If the length of the side of
a square Is s, how long is the
d i a g o n a l ? ................. . —
27. Simplify:
ab(I +
.
—
28. One number Is four times another
and the sum of the numbers is
240. Find the numbers. . . .__
29.
Simplify:
v $
(x°)^
= .
30. Simplify:
(Va + x +
Vx) (Va + x - Vx) -
31. Solve for T:
TP t 9- =
O
m
. T =_
2
32. Compute the value of
8
. . ._
33. Solve forn (the value of
not required):
5m - 2n = 7
2m + n =
10
34. Simplify:
— S— “ 16s +
S2 - 36
35. Solve for y:
3
m is
n =_
24 .
gy. + 1 = i
5
3
y = ------------------Go on to the next page.
-
36. Multiply:
(x& - y^)(x^ + y^-)
38. If x blocks of metal weigh y pounds,
what will z blocks weigh? ..........
39. Simplify:
2x2 » 5x + 3.
X
s,
41. If y = 3 - 2x, x = 3 + 2t, and t =
3 - 2u, express y In terms of u In
its simplest form.
y =
42. The sum of three numbers Is 18. The
sum of the second and third Is twice
the first number. The difference
between the first and third numbers
is one-third the first. Find the
second number........... ..........
43. The geometric mean G of two numbers
x and y is given by the formula
G = y/xy. If the geometric mean of
two numbers is 6 and one of the
numbers is 9, what Is the other
number? .............................
44. In the figure, BC Is parallel to ED,
AB = 7, BE = 5, ED = 6. Find B C .
BC =
A
45. Consider the operation: Multiply a
number by 2 and subtract from 4.
Perform this operation on an unknown
number x. Perform the operation on
the answer. Find a number such that
the result of the final operation is
4.
46. Solve for t:
« = 2s.
t + s
t =
47. Simplify the following radical:
V^2l5 ............................ .
48. If the perimeter of a right triangle
is 70, and one of the two shorter
sides is 20, find the length of the
other................................
49. The base of a pyramid is the face of
a cube. The altitude of the pyramid
is equal to the edge of the cube.
Find the volume of the cube if the
sum of the volumes of the two solids
is 8 cubic inches.
(The volume of a
pyramid equals one-third the area of
the base multiplied by the altitude.)_
50. Two men are 42 miles apart and walk
toward each other at the rate of 3 and
4 miles an hour respectively. After
how long a time do they meet?
A33274-6
-
.----------------
37. Find the value of:--------------------
40. Solve for x:
4
Number right, Score
—-------------- —
■i\ ■V.Q
_jLo O'
COOPERATIVE MATHEMATICS TEST FOR COLLEGE STUDENTS
Pre-Test for First Year Students
Experimental Form B - 1936
Please print:
Name------------------------------------ !____________________________________Date
Last
First
Middle
Class_______________________ A g e ___________________ Date of Birth
Yrs.
Mos.
College or University________________________________
Classification (check one):
Liberal Arts________
City_______________________ Sex__________
M. or F.
Engineering
Pre-Professional
In the following spaces indicate the number of years that you have studied each of the following
subjects (one. semester = l/2 year;
one quarter = l/3 year;
one summer session = l/4 year):
Solid
Geometry
Plane
Ge ome try
Algebra
Trigonometry
In High School
General Directions: Do not turn this- page until the examiner tells you to do so. This examina­
tion requires 45 minutes of working time. You are not expected to answer all the questions in
the time limit. No questions may be asked after the examination has begun. Use the blank spaces
at the left of the page for your calculations, and write your answers on the lines at the right
of the page.
Score
Percentile
Right
Wrong
Omitted
Copyright, 1936, by The Cooperative Test Service.
Printed In U. S. A.
All rights reserved.
- 2 Directions: In all questions, carry out the indicated operations and give the answers in
simplest algebraic or arithmetical form and without parentheses. Do not reduce any square
roots to decimal fractions. Write the answers in the spaces t_o the right of the page.
1. Multiply:5 * 0 ....................
2. Find the value of:—
............ ...............
3. Find the value of:
4. Simplify:
5. Add:
_ _
81 * ^ .......
o
a • a^ • a^ .
i + £
5
4
.
. ..
6. The perimeter of a polygon is the
sum of the lengths of the sides.
Write a formula for the perimeter
p of a square of side s .......... 7. Remove parentheses and collect
like terms:
(4r - 5t) + 3(t - r)._
8. Write a formula for the following
statement, using the given letters:
The lateral surface S of a right
circular cylinder is equal to the
product of it, its altitude h, and
the diameter d of the base........
9. Divide:
— -r —
14
21
10. The altitude of a triangle is 2 ft.
less than the base b. Express the
area of the triangle in terms of b.
(The area of a triangle is one-half
the product of the base by the
altitude.)
Area = —
11. If W and w are weights balanced at
distances A and a from a fulcrum,
the law of the lever can be ex­
pressed by the equation WA - wa.
If w = 20, W = 40, A = 10, find a.
a =
12. Write the formula: The volume v of
a sphere is four-thirds the product
of it and the cube of the radius r. __
13. Solve for k:
2k -
o
= 18.
k =
14. Find the value of:
3 - 2(-5) + 4(-3)
15. Multiply
6y/2 • 2\/5.
16. An electric bill varies directly as
the number of kilowatt hours consumed,
If 45 kilowatt hours cost $3, what
will 60 kilowatt hours cost?
$_
17. If a = 4 and b = 3, find the exact
numerical value of
5
y/e^ + b® . . . __
18. The sides of the smaller triangle in
the drawing are 3, 4, and 5. In the
larger similar triangle the shortest
side is 5. What are the other sides?
(In two similar triangles any two of
the sides of one are proportional to
the two corresponding sides of the
second.)
b =___
Go on to the next page.
- 3 19 * Divide:
(6x2 + llxy - 10y2 )
20.
Simplify:
(2x + 5y) .
3x - [2 - (5x + 2 ) ]
__________________
.
21. The sum of two numbers is 25. Their
difference is 17. Find the
numbers............................. 22. Multiply and collect terms:
(2a2 - 7a - 9) • (5a - 1 ) .......... 23.
Which of the following statements
are true?
(Answer by number.)
(1) 5 + _L_ = 5 + x,
(x + 0)j
X
1
(2) 5
(3) 4 3 • 2 3 = 4;
(4) (x - l)2 = (1 - x) 2
24.
Write a formula equivalent to the
following statement: A Fahrenheit
thermometer reading, F, is 32 de­
grees greater than nine-fifths
the corresponding Centigrade
thermometer reading, C ......... .
25.
Add:
5 + 4V§
2 - V3
26. How much money must be placed at
3% simple interest for one year
to earn $12? ...................
27. Add:
.Olx2 + ,4x - 1
3.00x2 - ,02x + .03
28. Perform the indicated
multiplication:
(t3 - t2 )(t2 - t - 1).
29. Simplify:
Sum =.
.
--- 3— zJt--q2 _ 8q + 15
30. Express by use of a fractional
exponent:
31.
a3 ...............
The expression a + b is a factor
of a3 + b 3 . Find the corresponding
factor of QR3 + S3 .
32. If lemons cost A cents a dozen,
how many dozen can be bought for
D dollars? ........................
33. Solve for y (the value of x Is
not required):
6x + 5y = 22
4x - y = 6
y =
34. The average of two numbers, x and
y, Is k. Write an expression that
gives the value of y in terms of
x and k .
y =
35. Simplify:
(x +- b)2
'
\ - x2 .
b
36. Solve for T:
L =
~ G .
T
T =
Go on to the next page.
-
37. If a train runs M miles in 5 hours,
how many miles will it run in K
hours at the same rate of speed?
38. Simplify:
7a2 - x2 (v42 - x2 +
39. Simplify:
| i |
.-
-
.........
40. Find the positive root of the
equation: 3m2 + m = 2.
m =
41. If x *
and
(6 y + 2 =* 0, find
the value of y.
y =.
42. If the radius of a sphere is
doubled, by what number Is the
volume multiplied? ..............
43. Let a, b, c be the sides of a
triangle. The perimeter p is the
sum of the lengths of the sides.
If p = 100, a + b =-|c, b + c = 3a,
find a.
a =-
44. If a positive number is added to
ten times Its reciprocal the
result is three times the number
diminished by eight times Its
reciprocal. Find the number.
-
45. In the tetrahedron OABC, 0A = AB,
0C = OB, angles OAB, AOC, and BOG
are right angles. Find CB If
OA = 4.
CB =.
B
46, If £ = § and - = 1 , find
47, if* fc = *04 AB(y!P ~ y/t)
and
(A + B) r2
F = 144, f = 81, r = 2,
A = B = 120, find the exact
numerical value of t.
t =
48. Find an isosceles triangle in which
the altitude equals the base and
each is numerically equal to the
area
Altitude = _
49. Let y be a number obtained from x
by the following operation:
Multiply x by 2 and subtract 4.
Let F be the result of applying
this operation to y. Find x so
that when the operation is
applied to F the result will be
12............................. x =
50. ( Z _ J u n f . 2*-_ - i)
\x + y / \ x - y
/
B33274-6
= ,
Number right, Score
4
-
110
Table 37
ITEMS OF FORM A, 1936, WHICH WERE TAKES FROM THE 1935 TESTS
U m
i5
a64
a97
ii
12
ia
14
la
16
t e l
Form B
5c
8
2
34o
5a
7
14
33c
1
15
15
13
58a
4
34a
32
18
17
19
20
21
13
4
22
27
28
29
30
31
32
33
34
35
36
37
39
40
41
42
43
44
45
4®
47
48
49
Form
6
11
300
25a
25
20
27a
11
22
15
29
25b
84
51
41
37
41
41
44
28
25
44
43
Table 38
ITEMS OP FORM B, 1936, WHICH WERE TAKEN PROM THE 1935 TESTS
Item
1
9
4
5
Form A
1035 Item
Pom B
Form C
So
3d
12
6
7
8
9
11
12
13
14
15
17
19
9
31
1
38a
34b
34a
10
S
4
13
19
8
21
22
35c
10
23
27
28
3
17
12
29
ia
30
31
33
34
55
36
37
38
39
40
41
42
43
44
45
47
48
49
50
27b
26
11
40
24
16
33
26
23
29
22
36
36
43
38
33
42
40
24
112
24* Cooperative Tea t Service Analyses*
The dif­
ficulties and valid!ties of the 1956 items, as ©omputed by
the Cooperative Test Service, have been placed in the
Appendix*
The following table gives the distributions of
these indices*
The distributions of difficulty indices corraborate
the statement mad© in Chapter IT that the 1956 tests were
somewhat easier than the 1956 forms*
It is Important to
repeat at this time that it was the intention of the com­
mittee that this phenomenon should be true, and it Is
interesting to not© that it was accomplished In spite of
the fact that two entirely different populations were
tested*
The significance of this fact will b© revealed
in Chapter VIZ*
113
Table 39
DISTRIBtTTIOHS OP DIPPICULTY AND VALIDITY IHDICES POBKD
FOR THE 1936 TESTS
difficulty
IMS*
95-100
90-94
85-8©
80-84
75-70
70-74
65-69
60-64
55-59
50-54
45-49
40-44
35-39
30-34
25-29
20-24
15-19
10-14
5—9
0-4
£ orm A
Form B
1
9
1
1
3
S
1
5
5
3
4
3
0
1
3
2
3
2
7
1
S
7
6
5
S
1
4
5
0
1
3
2
3
2
2
1
Validity
Index
14
11
10
9
B
7
6
6
4
5
1
Form A
1
2
4
5
4
11
10
3
6
5
1
Pom B
2
6
7
9
10
B
3
4
1
114
Observe that one item in each form falls to meet
the requirement of 2 set up for satisfactory validity*
These are item 1 of Form A and item 2 of Form B*
Bach was
given in 193d and the indices reported then were 6 and 4
respectively*
That we need not be too concerned with a
discrepancy of this type is pointed out by B* F* Lindquist
{17} from whom we quotes
#**An index of discrimination* therefore should
be used with particular caution for evaluating items
In a general achievement test where the criterion em­
ployed is the total score on the test Itself* It may
be of considerable value* in such situations* in
identifying Items that contain structural or technical
imperfection®* but It is a dangerous basis upon which
to eliminate an Item if the content of that item 1®
acceptable In term© of other logical considerations *
Regardless of the criterion employed* the indices
of discrimination computed for the Items of a test on
the basis of the performance of a given group of
pupils will determine the relative effectiveness of the
Items only for that &roun of pupils or for other simi­
lar groups * "The same item In a test may show a low
index of discrimination for one group of pupils and a
high index for another* depending upon the nature of
their Instructional background* **
25* Item discrimination within preparation groups*
The above quotation suggest© the possibility that the power
of an Item to discriminate between students grouped accord­
ing to ability should be listed for each preparation group*
For this purpose* Table® 40 and 41 have been assembled*
The manner In which they were constructed Is as follows.
X and for all samples com­
Fox* each 1956 aample *
bined* th© papers were arranged In descending order accord­
ing to total score*
Bach such distribution was then
divided into approximate thirds (approximate In order that
no two papers with the same score would fall In different
groups)* and the percentage of correct responses to an item
for each of these thirds was computed*
The three groups
wax'© labeled tthlgh*” ^medium**1 and nlowH respectively* and
the corresponding percentages computed for each of these
groups are listed In the tables under the letters H ## M mt
and L t
H represents th© number of papers in each of the
groups*
For an Item to discriminate between different
levels of achievement* then* the percentages should deereas©
from column H to column M to column L - the more abrupt
th© decrease* the higher the discriminating power of the
Item*
^ Th© 1935 samples were considered too small to
subject to this analysis*
116
Table 40
PERCENTAGE OF CORRECT RESPONSES TO EACH ITEM IN EACH APPROXI­
MATE THIRD OF THE PREPARATION GROUP AND TOTAL DISTRIBUTIONS
FORM A f 1936
N Z
M m
i
2
3
4
5
0
7
S
9
10
11
10
13
14
15
16
17
18
19
SO
SI
00
S3
24
85
86
87
08
09
30
81
38
33
34
35
36
57
38
H.
P
2i
M*
L#
34
32
34
97
91
100
100
79
88
94
100
74
91
88
85
79
08
88
74
94
97
74
02
76
@5
97
60
85
21
79
79
91
56
50
63
50
59
47
50
50
38
94
63
94
91
75
78
91
81
56
63
63
78
78
73
84
50
69
97
81
70
08
36
81
34
44
0
41
59
60
9
9
28
22
22
9
28
00
16
74
15
94
SB
29
SB
71
62
20
59
59
50
41
32
47
09
24
80
55
47
15
30
50
18
06
3
15
21
26
0
0
3
9
15
9
30
6
5
H.
h k
M* L«
H.
li
M* D#
34
34
32
33
39
97 91
97 79
94 97
94 04
100 91
100 91
94 100
94 88
91 68
91 94
88 85
94 97
80 85
91 85
85 88
97 79
91 100
100 100
SB 82
91 79
88 47
68 68
91 91
97 68
88 85
56 26
85 71
100 91
97 94
91 62
94 60
91 71
79 32
82 56
68 35
94 08
59 44
56 15
91
59
94
94
69
81
72
69
31
69
66
50
47
59
59
34
53
97
59
53
44
38
81
44
66
0
22
69
53
41
22
63
25
31
3
08
25
3
97 97
94 82
100 95
100 100
97 87
97 100
100 95
94 95
85 90
100 70
91 85
85 95
85 92
100 07
01 85
97 97
94 85
97 100
@8 92
91 09
94 79
85 72
97 85
100 79
94 85
79 36
9 4 92
97 90
97 95
100 79
94 69
100 97
67 74
67 82
79 59
94 77
85 04
67 46
Total
H*
M*
L#
28
92 117
01
89
68
93
93
79
75
89
82
75
61
79
82
80
79
79
80
71
93
61
68
57
48
64
57
75
29
54
79
04
46
36
82
43
21
32
04
43
14
97
93
98
98
95
99
97
95
08
93
90
91
87
97
89
96
93
99
89
87
88
75
95
93
92
61
91
98
96
93
88
99
75
72
74
92
70
59
87
40
92
90
53
71
76
68
41
62
62
60
57
52
63
37
44
90
60
58
27
42
67
32
40
4
25
45
44
10
4
26
15
21
7
23
15
7
02
81
97
97
80
89
95
91
70
81
82
88
81
86
81
81
87
98
77
07
02
01
85
62
82
23
88
84
87
59
54
74
45
48
30
63
43
24
117
Table 40 {c©n tinu®&)
Form A $ 1936
fx
H.
39
40
41
4®
43
44
4$
46
47
43
49
50
26
47
21
9
53
9
6
©
3
0
3
29
L,
'
3
3
0
0
0
0
0
0
9
3
0
0
0
0
0
0
0
0
0
0
0
0
9 12
fii
H.
M»
L«
5©
9
9
©5 18 19
50 15 16
2© 21
©
68 56 34
47 26
©
26
0
3
21
©
0
©
24 12
18
0
0
21
3
3
©5 18 22
Total
!e
, m#
79
94
82
55
94
70
39
42
58
39
30
52
3©
67
49
10
77
46
21
13
18
13
5
3©
II. L#
21
39
18
7
43
14
7
0
11
0
0
25
63
76
©1
32
86
55
29
29
37
25
22
51
17
4
38
3
23
2
14
0
50 12
18
3
7
0
5 0
a
1
i
0
i
1
22 14
Table 41
P1BGBWTAGB OF CORRECT RESPONSES- TO EACH ITEM IN EACH APPKOXIMATE THIRD OF THE PREPARATION GKO HP AMD TOTAL DISTRIBUTIONS
FORM B* 1956
»*
i
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1©
H
H. K«
L#
28
44
2©
96
96
96
89
96
©8
8©
93
71
©1
8©
96
75
86
64
89
95
09
89
82
93
45
70
70
75
20
68
55
57
©8
23
91
82
82
93
71
71
14
4©
39
71
4
32
32
32
54
10
71
M
M,
L.
H*
fs
M4
28
40
32
31
41
9©
82
100
100
9©
89
S9
93
89
71
82
100
89
89
©8
9©
95
90
95
90
90
85
80
98
S3
45
75
83
80
83
68
90
88
88
97
6©
75
44
©9
75
47
13
59
53
59
©9
38
81
Total
L.
H.
M.
1*
28
94 113
93
9©
88
96
89
93
73
82
90
79
41
BO
81
75
82
56
93
89
85
92
72
80
34
©3
63
©1
12
54
4©
51
62
26
82
100 100 9©
94 90 82
100 98 9©
100 95 82
97 95 89
97 88 64
100 85 93
97 95 89
90 90 ©8
84 59 52
97 90 82
94 95 ©8
90 93 79
100 93 79
100 73 54
97 95 100
99
91
98
97
96
91
93
96
89
7©
90
97
93
95
82
96
118
Table 41 (continued)
Fona B. 1936
Item
H,
17
IB
19
90
21
22
93
24
25
26
27
28
29
30
81
32
m
34
55
86
37
38
39
40
41
42
45
44
45
46
47
48
49
50
86
57
86
©6
79
75
18
64
71
89
71
64
75
71
71
21
46
43
57
25
50
14
11
25
14
14
11
7
7
29
11
7
0
0
fi
M*
64
39
86
64
82
73
7
36
43
SO
52
64
52
36
55
9
25
7
16
2
16
0
5
2
2
0
0
2
7
25
2
0
0
0
29
7
59
36
57
39
0
7
25
71
25
21
32
18
7
0
7
0
0
0
11
0
0
0
0
0
0
0
4
14
0
0
0
0
03
93
89
75
96
86
36
82
06
100
82
89
m
82
79
32
79
71
86
34
75
57
32
57
18
36
11
21
14
29
7
11
4
11
IU
l*w
86
68
83
78
85
75
8
58
88
7©
68
83
60
73
48
5
33
13
23
8
28
8
47
44
75
44
83
38
©
16
50
©6
54
47
19
56
53
0
IS
3
9
0
9
0
0
0
3
0
0
8
0
19
0
a
13
5
0
5
8
3
25
0
8
0
3
6
0
0
fs,
M* ,L*
97
94
87
90
100
94
48
90
90
100
87
100
90
100
97
55
90
74
87
71
87
94
55
81
71
71
65
77
81
74
52
39
26
45
95
85
83
76
95
73
37
6©
78
98
78
73
85
95
S3
27
71
54
76
49
©8
46
22
©8
44
27
15
15
15
29
10
5
0
2
©2
64
82
71
82
75
32
50
86
64
71
54
64
82
57
14
46
21
32
18
21
11
11
32
11
11
4
18
21
21
0
4
0
4
Total
H. M* U
94
88
88
82
100
85
38
82
88
99
85
07
91
94
88
37
85
68
84
60
77
©7
40
73
48
45
32
38
31
48
23
18
10
19
87
65
85
77
81
75
19
55
80
81
67
65
67
73
58
13
37
22
35
15
52
10
9
19
9
7
4
10
9
22
3
5
0
2
44
30
©7
46
68
49
4
20
38
69
37
40
31
35
39
3
IS
3
©
0
13
O
1
0
1
0
0
2
3
18
1
2
0
0
Two characteristics of these tables are of con~
siderabl© importance#
First of all, the 11total® column,
the one given for test items when only on© is available,
is inadequate to describe the discrimination within the
preparation groups#
In some oases, of course, where the
percentage differences are about the same for all groups,
the general index is satisfactory, but where wide
divergence between groups Is found, Its reliability is
considerably diminished#
The second characteristic is that Items which
discriminate for one group do not necessarily do so for
another#
The earlier and easier Items do not discriminate
between members of the
often the
group, and
the later Items are so difficult for all members of the
group that no differentiation takes place#
Th® implication of thes© two features is Hi at
not only should indices be given for each group, but
separate tests themselves are desirable#
Further research
should b© mad© in connection with this point#
26# Difficulty indices for v reparation groups#
Corresponding to Tables 25, 26, and 27, are the following
two which give the difficulty Indices, as derived from the
samples, for each 198© preparation group#
Attention should
be called again to column four, whos® Indices ar© the
averages of th© corresponding ones given in th© first three
columns, and which Is presented solely for the purpose of
exhibiting th© degree to which it approximates th© column
of indices derived by the Cooperative Test Service*
The same remarks can be mad© of Tables 42 and
43 as were mad® of those involving th© 1935 items#
In a
few cases th© index listed for a particular group exceeds
that given for another group with mor® preparation, but
all these differences are small enough to b© entirely
accounted for by chance fluctuations#
For most of the
items there is a definite, though not necessarily signifi­
cant, Increase In difficulty accompanying an increase in
preparation*
Distributions of the difficulty indices for
each preparation group are listed in Tables 44 and 45#
Table 4£
12.1
ITEM DIFFICULTIES FOR EACH PREPARATION GROUP
pl- *1* , and Pg
FORM A* 1930
Cooperative
Total
Ei
Test Service
Sfctt
!i
hk
88
l
93
95
92
92
2
56
78
82
72
74
96
3
95
96
96
95
4
95
98
94
95
90
87
5
61
88
79
66
6
77
91
92
87
82
85
89
95
7
90
84
81
a
84
91
85
78
52
9
64
©7
84
47
10
71
85
79
82
72
70
80
85
11
78
75
71
12
81
88
80
75
m
72
89
13
71
70
14
65
79
93
79
73
yy
71
78
15
85
78
16
71
53
93
72
68
62
17
82
84
76
68
18
92
97
99
96
77
69
88
75
75
69
19
75
20
59
70
73
68
21
41
80
70
60
83
22
58
59
69
52
S3
77
89
23
83
78
80
63
38
70
24
©3
52
70
SO
72
85
85
26
8
48
29
80
31
87
45
82
61
63
54
86
89
75
28
76
54
87
29
60
76
73
©2
77
51
30
©5
55
22
50
48
21
68
00
31
07
03
32
94
32
75
47
27
65
45
46
33
47
39
32
52
57
34
38
45
35
57
81
36
61
59
79
36
64
59
46
45
66
26
45
37
24
43
29
25
38
19
28
28
4©
39
26
11
39
43
16
68
40
34
28
30
7
27
41
51
20
42
15
3
IS
24
42
49
73
22
58
43
34
25
27
46
3
44
15
23
11
10
45
1
18
10
19
2
9
46
21
15
47
29
1
14
14
8
0
6
18
48
10
7
9
49
1
12
28
17
29
50
38
31
±32
Table 45
item
Mm
1
2
3
4
5
6
7
8
©
10
11
12
13
14
IS
16
17
18
19
20
21
22
23
24
25
28
27
28
29
SO
31
32
33
34
55
36
37
38
59
40
41
42
43
44
45
46
47
48
49
50
DIFFICULTIES FOB EACH I5REPARATION GROUP
^1* ^1
P0BM B, 1936
!i
92
89
92
81
8©
43
68
88
73
87
63
60
55
69
30
85
60
35
73
69
74
64
8
56
46
80
50
52
53
41
46
10
26
IS
23
3
24
4
5
3
5
4
3
3
6
23
4
2
0
0
..fir
93
87
97
85
07
73
79
89
75
42
72
78
76
©0
SB
89
75
67
82
65
81
65
15
51
78
79
61
65
57
71
08
11
40
26
36
18
35
19
12
21
8
10
5
11
5
23
2
a
1
4
a*
99
89
98
93
94
84
92
94
84
59
90
87
a©
91
76
97
92
82
84
79
93
80
39
69
84
89
79
76
81
93
80
32
70
51
67
47
61
51
32
62
43
36
27
35
31
41
20
15
S
10
Total
95
88
96
80
90
67
80
84
77
43
75
75
73
80
55
90
70
61
80
09
83
70
21
52
69
83
63
64
64
68
61
18
45
31
42
24
40
25
16
30
19
17
12
16
14
29
9
8
3
7
^2
Cooperative
Tost Service
95
87
91
85
87
57
72
67
74
31
73
65
64
72
50
84
@8
55
73
56
77
60
24
45
62
83
54
54
55
66
57
19
49
55
44
15
29
33
20
36
27
25
15
IB
18
36
15
10
4
10
Table 44
DISTRIBUTION OR DIFFICULTY INDICES FOR EACH PREPARATION GROUP
FORM A, 1936
S M m
95*100
90*94
85*89
80*84
75*79
70*74
65*69
60*64
55*59
50*54
45*49
40*44
55*30
30*54
25-99
20*24
15*19
10*14
0*9
0*4
!i
lii
i
2
2
1
2
A
tSU
5
3
2
5
1
1
2
2
2
4
3
X
2
7
2
3
5
6
0
3
1
5
2
1
1
1
1
3
4
0
1
2
3
0
£2
5
5
8
8
4
1
4
1
1
2
3
1
I
0
1
2
2
1
0
0
Table 45
DISTRIBUTION OF DIFFICULTY INDICES FOR EACH PREPARATION GROUP
POEM B» 1936
Index
95*100
90*94
85*89
80*84
75*79
70*74
05*69
60*64
55*59
50*54
45*49
40* 44
35*59
30*54
25*29
20*24
15*19
XU 4
0*4
Ek
0
2
5
2
0
3
3
5
1
3
2
2
2
1
2
3
1
1
6
8
^8
M
1
1
5
3
0
4
4
1
3
1
0
2
2
0
1
2
3
t
3
3
9
4
8
4
1
2
2
1
2
1
2
3
3
1
1
2
?
0
These distributions reveal differences between
preparation groups in the same manner as was noted for the
1955 tests*
That is, indices increase in si£© quite
generally as th© preparation is Increased, giving the im­
pression that th© entire distribution has been forced
upward*
Th© distributions also reveal th© fact that the
195© tests were relatively easy, perhaps too easy*
FYen
for th© one*year groups, there are tw©nty*five items in
Form A and twenty* two in Foma 8 whose indices are 50 or
more#
27* Comparison of difficulty indices found for
preparation groups»
We seek, now, to determine those items
for which the differences between preparation groups are
significant, and th© procedure used is identical to that
employed for the 1955 items*
Following the same pattern
as before, the calculations have been placed in th© Appendix
and th® significant differences found are listed In th©
next two tables *
Table 46
,
ITEMS OB' FORM A, 1934 WHICH YIELD SIGNIFICANT DIFFERENCES
BETWEEN PREPARATION GROUPS
P1 and
Item
Plj 8113 f2
Item
P1 a»d pg
Item
2
2
5
B
6
9
e
9
12
16
17
21
13
14
16
21
24
.25
26
27
28
29
30
31
32
33
54
32
35
56
57
39
40
41
42
43
44
45
47
49
37
58
39
40
41
43
44
47
48
13
14
18
17
21
22
24
25
26
27
2B
29
30
31
32
33
34
53
36
57
38
59
40
41
42
43
44
45
46
47
48
49
50
JJ36
Table 47
ITEMS
OF FORM B* 193© # WHICH YIELD SIGHIFICAHT DIFFERENCES
BBTWEEH PREPARATIOK GROUPS
\
and %
Item
F1
F2
Item
Item
7
©
7
8
IX
11
6
8
10
12
12
15
IS
14
15
16
17
18
20
21
15
15
17
IB
22
23
24
25
27
30
58
40
29
30
51
32
33
54
35
3©
37
58
39
40
41
42
43
44
45
4©
47
50
23
84
85
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
30
88* B|T
t of preparation upon knowledge of the
fe&n_Q.SP.tfi. and processes covered by the 1956 p r o t e s t s *
Hslng Tables 42, 45, 46, and 47, we shall now classify the
1938 Items In the ©am© manner as was described for the 1955
items*
The groups are as followss
A* Concepts and processes which were well~known*
1* Elementary operations of arithmetic*
The agreement here la not perfect, but
It is very nearly so and hence Is included
In the list*
Of the ten items covering
these operations, only on® failed to meet
the criterion of a difficulty Index of 70
for 9well-known*w
The
group yielded
an Index of only 54 for Item 2 of Form A*
The process involved was a simplification
2* The principle of adding exponents when a
series of Items with Identical bases are
m l tipi led together*
3* Substitution of numbers for letters In
simple algebraic expressions followed by
simplification*
When extraction of roots was added to
these processes, difficulties were Increased*
A cursory examination of the responses
revealed on© common error ( \/9 ♦ 16~»3 ♦ 4)
128
made by those with one year of preparation*
4* Simple written problems commonly found in
arithmetic*
In order to illustrate the type of exerelse referred to, the three items ar©
listed below*
Item 2 3 » Form A ; What is the Interest
on $600 for three months at the rate of
5 p©r cent per annum?
Item IQ, Form B s An electric bill varies
directly as the number of kilowatt hours
consumed* If 45 kilowatt hours cost $3,
what will 60 kilowatt hours cost?
Item 2 6 , Form B s How much money must b©
placed at'^per cent simple interest
for one year to earn #12?
B* Concepts and processes which were difficult*
1* Subtraction of algebraic fractions*
C* Areas materially strengthened by a study of
third semester algebra*
1* Addition and multiplication of irrational
expressions*
Mo subtraction was involved in 1936#
2* Solution of unfamiliar problems testing
the power to follow definitions or direc­
tions*
3* Simple written problems involving linear
equations In two or three unknowns*
4* Operations involving fractional exponents*
5* Solution of quadratic equations*
129
B* Areas materially strengthened by a study of
fourth semester algebra*
1# Extraction of square roots#
2* Multiplication of irrational expressions*
3# Subtraction of algebraic fractions.
4. Evaluation of monomials with integral bases
and fractional exponents.
5. Hqp re sen tation of literal relationships
such as the follow!ngi
a* If lemons cost A cents a dosen, how
many dosen can be bought for D
dollars?
h« Th© average of two numbers, x and y,
is k. Write an expression that gives
the value of y in terms of x and k*
c* If a train runs M miles in 3 hours,
how many miles will it run in & hours
at th© same rat® of speed?
6* Solution of quadratic equations.
7# Multiplication of algebraic expressions
Involving fractions*
It is to be remembered that no process was listed
In the above groups, or the corresponding groups Involving
the 1935 Items, unless more than on© item covered th© process.
Ther© were two such items omitted from th© 1935 groups and
also from thO 1936 groups, but which exhibited the same
characteristic In both years.
That Is, these two Items
will b© classified as were the others since two observa­
tions were mad© for different years.
S' rs f,
JLo-U
Th© following item falls Into th© difficult
groups
11If x -
{2
and
(6 - /s)x - 5 ^ 2 y ♦ 2 » 0,
find th© value of y,n
3# 7
For th© other item, ”simplify
significant difference was found between the P^ and
groups#
a
131
CHAPTER VII
THE DIFFICULTY OF AK ITEM FROM YEAR TO YEAH
29# Comparison of 1935 and 1936 difficulty
Indices,
Th© purpose of this section is to investi­
gate the difficulty indices reported for those Items
which were given In 1935 and repeated In 1936#
groups of items will he ©onaidereds
Three
those which were
repeated in Identical form; thos© which were repeated
in almost Identical form, differing only In the direc­
tions Involved or through some simple change of number;
and those which do not fall into the other two groups
but which Involved th® same process or concept#
These three groups of corresponding pairs of
Items, with difficulty Indices for each, are listed
In Tables 48, 49, and 50, respectively#
The apparent
stability of th© difficulty index for a given item Is
quite noticeable#
That this agreement should, and does,
become less marked as th© items become less alike is
evident#
Table 48
COMPARISON 0F DIFFXGULTIK,s of ;ETE&S Oilr m ib loss
AMD REPEATED IB IDENTICAL FORM IB 1956
Item
£l
Pl&
*2
1*
1935
1936
0*14
A-8
70
91
94
84
94
91
88
85
76
78
8*
1935
1936
B~1
A-ll
74
70
72
80
70
as
78
78
74
75
3.
1935
1936
B-15
A—12
76
71
80
SI
82
88
79
SO
72
75
4*
1935
1936
A**13
A—14
66
@3
80
79
8©
93
77
79
70
73
5*
1935
1936
JW4
A*16
56
53
83
71
84
93
74
72
©9
68
6*
1935
1936
C<»$2
A-17
46
62
68
82
84
84
68
7©
©5
©8
n
1936
1936
B—13
A* 20
50
59
64
75
80
75
65
70
80
©8
8«
1935
1936
B**.4
A-2X
38
41
70
60
78
78
82
60
58
53
9*
1935
1936
B~25a
A-29
82
60
48
82
60
87
43
7©
49
73
10*
1935
1936
C-2Q
A-31
32
21
48
@0
7©
68
52
50
4©
48
11 •
1935
1936
0-11
A-83
6
27
36
46
60
63
34
45
45
47
•
01
H
Cooperative
Test Service
Year
1935
1936
A—15
A-36
16
21
45
3©
54
57
38
38
42
45
13*
1935
1936
0—31
A— 40
8
16
14 ,
34
28
88
17
39
27
43
14.
1935
1936
C—41
A— 41
2
7
16
27
26
51
15
28
23
30
Total
13:
Table 48 (continued)
Cooperative
Test Service
Year
Item
fl
is#
1935
1936
0-37
A—42
14
3
18
18
24
24
19
15
22
20
16*
1935
1956
B-41
A-43
4
28
1©
53
18
73
13
49
IB
42
17*
1935
1956
A-41
A*»44
14
3
25
27
2©
46
22
23
17
34
18 *
1955
1956
B-28
A*#46
2
2
2
9
10
19
5
10
13
18
19*
1935
1936
C*44
A~48
0
0
2
6
©
18
3
8
6
14
20 *
1935
1936
B~3!
B**8
@6
©8
88
89
98
94
S4
84
75
87
21*
1955
1936
G~10
B~18
58
55
72
7©
82
88
71
73
71
64
22*
1936
1956
B*5
B-14
68
©9
76
80
so
91
75
80
67
72
23*
1955
1936
B~17
B~27
52
50
76
61
70
79
8©
63
54
54
24*
1935
1936
B-12
■Q
i
x
>«*O*5Co
5@
52
56
©5
60
76
57
64
53
54
25*
1935
1936
A~27b
B~30
14
41
47
71
76
93
46
60
50
86
26*
1935
1936
C-26
B~31
30
46
40
58
56
80
42
61
47
57
27*
1955
1936
B~ll
B~33
30
26
48
40
64
70
47
45
44
49
28*
1935
1936
C~40
B-54
20
15
38
2©
48
51
35
31
39
35
29*
1936
1936
B-24
B~35
8
23
38
36
42
87
29
42
39
44
1'jgT
!e
Total
4^4
fable 48 {continued}
Cooperative
Test Service
Xf**
Item
Ei
Eli
Eg
30*
1935
1936
iW16
B-36
10
8
28
18
34
47
24
24
35
15
31*
1935
1936
B~53
B~37
28
24
42
55
46
61
39
40
34
29
32*
1935
1938
B*S9
B~40
4
8
6
81
28
62
13
30
85
36
33*
1935
1936
0-22
B- 41
8
5
20
8
26
43
17
19
24
27
34*
1935
1936
A-36
8-42
12
4
22
10
28
36
21
17
24
25
35*
1935
1936
B-86
B-43
6
3
14
5
18
27
13
12
22
16
36*
1935
1936
C-43
B-44
4
5
10
11
30
35
15
16
19
18
57*
1935
1956
B~38
IB'*"45
2
8
12
5
14
31
9
14
14
18
38*
1935
1986
A-53
B-47
0
4
6
8
6
20
4
9
7
15
39*
1935
1936
3*42
B~48
4
2
4
8
4
15
4
3
9
16
40*
1985
1936
8-40
B-49
0
0
4
1
4
8
3
3
4
4
%1rM.
Table 49
GGMPABISOB OF DIFFICULTIES OF ITEMS GIVEN IN 1955
AND REPEATED IN ALMOST IDENTICAL FORM IN 193 a
Cooperative
Teat Service
fiMwr
Item
PI
Eli
1*
1935
193©
A-3 c
A«*l
94
88
96
93
100
95
97
92
93
92
2#
1935
1936
A-2
A-4
84
95
92
94
94
98
90
95
84
90
3*
1938
1936
A~34c
A***5
82
©1
78
87
90
88
83
79
81
60
4*
1938
193©
A-3a
A«*6
70
77
88
91
92
92
83
87
78
82
5#
1938
1936
B-7
A~7
82
85
80
89
94
95
85
90
78
84
6*
1985
1930
C«,33c
A*9
56
52
72
©4
82
84
70
67
75
47
7*
1935
1930
C-13
A«»13
66
66
82
72
90
89
79
76
71
71
8§
1955
193©
B**54a
A-38 a
A-15
56
54
71
72
78
78
92
66
85
73
66
78
70
74
77
9*
1955
1956
0*18
A«*19
52
69
©4
75
8©
82
07
75
60
69
10*
1935
1936
O**0
A~22
@4
51
5©
58
76
69
65
59
58
52
11 *
1955
193©
A~ll
A~27
3©
45
55
61
78
82
56
03
83
54
18.
1985
195©
A-39 c
A~28
40
54
5©
86
80
09
59
76
51
75
13*
1935
193©
A~25
A-30
0
22
47
65
76
77
43
55
47
51
14#
1955
193©
A«*27a
A-32
1©
32
44
75
72
94
44
67
45
63
£&
Tots
Tatola 49 (eont limed)
Cooperative
hk
£s
Tot
32
32
57
44
52
32
47
42
39
12
39
24
64
42
79
20
61
35
59
A-37
16
26
22
43
42
00
27
45
33
46
1935
1936
A-*24
A-59
16
11
24
26
42
46
27
28
31
28
1935
1936
C-23
A* 47
4
1
6
14
12
29
7
15
21
1935
1936
A-43
A-49
2
2
9
12
4
7
4
1
10
1935
A*»3d
C~lSc
1936
B~£
94
74
89
95
92
87
94
84
89
94
85
8©
91
77
87
22 *
1935
1936
A-12
1W*4
86
81
95
85
98
93
95
00
87
85
23*
1933
1936
0-2
B-5
88
88
96
87
94
94
93
90
84
87
24*
1935
1936
C-33b
0
68
43
86
73
88
84
©1
67
81
57
23*
1935
1936
A-9
B-?
66
68
92
79
92
92
83
80
77
72
26#
1935
1936
C-l
B-9
64
73
70
73
74
84
71
77
75
74
27*
1935
1936
A-34b
B-12
76
00
83
78
90
87
83
75
73
65
28*
1935
1936
G-4
8-15
42
30
60
58
84
70
62
55
65
50
29*
1935
1936
0-19
64
73
74
82
70
84
71
80
63
73
15.
16.
17.
18.
19 *
20,
21
,
Itiam
£i
1936
1936
B-22
A-34
@0
1935
1936
C-29
1935
1936
A-36
B~25t>
B—19
a
Test Service
11
127
Table 49 (continued)
Cooperative
Teat Service
Year
£t®»
!i
50.
1955
1936
A—8
8-20
60
62
70
65
84
79
71
69
62
66
31.
1935
1936
B~35e
B-21
48
74
70
81
68
93
62
83
58
77
32*
1938
1936
A-10
B-22
46
64
64
6$
84
80
65
70
55
60
33.
1935
1936
B—3
B-25
50
46
76
78
76
84
67
69
54
62
34.
1935
1936
A—18
B-29
34
53
73
57
78
81
62
64
53
55
55#
1935
1936
A-26
B-38
6
4
22
19
40
51
23
25
34
33
36*
1935
1936
0-23
B-39
10
5
26
12
32
32
23
16
31
20
37*
1933
1936
G-24
3-50
2
0
4
4
6
16
4
7
10
£2
3
138
Table 50
COMPARISON OP DIFFICULTIES OF ITEMS GIVES IK 1956
WHICH INVOLVED THE SAME PROCESS BUT WERE NOT IDENTICAL
TO 1935 ITEMS
e w « M «w iin eM m iiiiiii i m u
<M*w»ilWei m uw.i■■
w m m m w w
rn im rriir nnii*i— w im m ■ » m w im i
uooperstxve
Test Service
XS££
£fe®»
P
11
hk
■£»
IS
1935
■A**12
B~8
0*8
A*3
86
80
86
96
95
80
94
95
98
80
90
96
93
80
92
96
87
70
90
95
1938
B—S
C~2
A***4
82
88
93
84
96
94
92
94
98
86
93
95
rjfty
84
90
3e
1935
1936
B«*5
A**6
08
77
76
91
80
92
75
87
67
82
4*
1933
1936
A*0
0*7
A* 7
60
84
88
92
94
89
92
90
95
83
89
90
77
82
84
5.
1935
1936
8*9
A* 14
44
08
68
79
80
93
64
79
57
73
6.
1935
1936
8-3
A-10
50
88
76
71
76
93
67
72
54
68
7.
1935
1936
A-34'to
A* 34c
A~17
76
82
62
85
78
82
90
90
84
83
83
70
73
81
©8
8.
1935
1936
A* 10
A* 19
46
69
64
75
84
82
85
75
55
69
9.
1935
1936
A-20
A-20
38
59
59
75
76
75
58
70
56
68
10.
1935
A* 3
0*4
A* 21
46
42
41
74
60
60
80
84
78
69
62
60
59
65
53
A*8
8*8
A-22
00
52
51
70
70
58
84
86
69
71
63
59
62
52
52
A—29
B~ll
A* 33
0
80
27
20
48
46
30
64
63
17
47
45
24
44
47
1*
1936
2*
1935
1936
11.
1935
1936
IS.
1935
1936
Total
139
Table 50 (continued)
P
Cooperative
Test Service
Tear
Item
fl
fli
JSL
13*
1933
1936
JW18
4*3 4
34
52
73
S7
78
52
©2
47
53
39
14*
1933
1936
A* 3Tb
A* 38
46
19
50
25
66
43
54
29
49
24
13.
1935
1936
A* 5c
0*1
94
92
96
93
100
99
97
95
93
95
16*
1935
1936
B~S
0*8
B~4
SO
86
81
80
94
85
80
96
93
80
92
86
7©
90
85
17.
1935
1936
A* 2
B~5
84
88
92
87
94
94
90
90
84
87
IS.
1933
1936
e~33&
B*6
m
43
86
73
92
84
82
07
81
57
1©.
1935
1936
A* 54a
B*8
92
68
98
89
96
94
95
84
92
©7
20.
1935
1936
A*1
B~9
76
73
88
73
82
84
82
77
78
74
21.
1935
1936
A* 38a
B*34a
B*ll
54
50
63
78
72
72
66
92
90
6©
73
75
74
70
73
22.
1935
1936
0*32
B-12
40
60
08
78
84
87
66
75
65
65
23.
1935
1936
A* 3a
B~14
70
69
88
80
92
91
83
80
78
72
24*
1935
1936
A* 5
B-*4
0*15
46
38
30
74
70
58
86
78
70
69
62
55
59
58
50
23.
1935
1036
B-0
BU20
52
62
70
65
60
79
63
69
52
©6
26.
1933
B~!2
0*18
B~22
56
52
64
56
64
65
60
86
80
57
67
70
53
60
60
1936
Total
140
TaBle 30 (continued)
27*
28*
29*
Year
Item
F
JL
fli
1935
1836
A—4
0-3
B-25
56
30
46
1935
1936
A«*7
B-27
56
50
1935
A*10
C-18
B~28
1936
30.
1935
1936
31*
1935
1936
Cooperative
Teat Service
B
2
Total
83
58
78
84
70
84
74
53
©9
69
50
62
68
61
6©
79
63
©3
61
54
46
52
52
64
64
84
©5
76
65
67
©4
55
60
54
A~22
B**22
B**80
16
20
55
45
32
57
68
44
81
43
32
©4
44
42
55
A**20
CML1
B-53
0
6
26
20
30
36
40
60
70
17
34
45
24
43
49
86
Keeall that* because of a great number of omissions*
it was deemed necessary in the previous section to eliminate
some items from conaideration*
If this policy were to be
followed here, the following pairs of items would be elimi­
nated from Table 48s
56, 37, 38, 39, 40*
13* 14* 16* 17, 18, 19, 28, 32, 35,
Similarly th® pairs 16, 19, 20, and
37 would disappear from Table 49, and item A-29 would be
dropped from the two pairs labeled 12 and 31 in Table 50#
The effect of this would be to strengthen considerably th©
relationship mentioned#
However, it would mean a loss of
one third of th© identical pairs and hone© all items are
h©pt as given#
In compiling distributions of difficulty indices,
groups of five units each were used#
vals,
Using the same inter-
the Pearson product-moment coefficient of correla­
tion was computed for the paired difficulties as given in
each table by th© Cooperative Test Service*
The coeffi­
cient found for th© identical items was #924 with a
probable error of #016$ for the !falmost identical*1 Items
it was *885 with a probable error of #024$ and for th©
items Involving the same process, th© coefficient was
found to b© #761 with a probable ©rror of *042*
Ho corre­
lation study has been mad© for the Indices derived for the
preparation groups*
These Intervals are undoubtedly too small to tab:©
Car© of all chance fluctuations, and hone© th© correlation
coefficients are probably too small*
Befor® generalising we should make on© point
clear#
The Committee, in constructing the 1936 tests,
mad© a deliberate effort to place the items in a different
setting from that employed in 1935#
At the same time, how­
ever,
theconventional policy of arranging items in th©
order
of difficulty was roughly followed#
In
terns of
these facts, then, there is strong evidence to support the
theory that the difficulty of an Item of a pre-test for
students of first year college mathematics, as computed
for a large heterogeneous population, la Invariant with
respect to its location on a test, with respect to th©
year in which it Is given, and with respect to the specific
population to whom It Is given, provided that population
is similar with respect to preparation to the one for
which the Index was derived#
■i* W l -V#—
hi A
school freshmen*
*1*
nlr.nl|lw
( I# * # ! * #
W W M M w w M M )# **—
—
found for
........................... rrtwn#** * !
.
30# Comparison of difficulty indices
In order to substantiate a belief
that the Invariance described above Is not confined to
the pre-tests nor to college students, th© writer mad©
on© further Investigation*
Th© basis for this study is
furnished by th© 1932, 1934, and 1936 Iowa Every-Pupil Tests
o
in ninth year algebra*
Difficulty Indices for the Items
® These tests constituted, in the years mentioned,
on© part of th© annual Iowa &v©ry-Pupil Testing Program*
This program Is carried on each spring under the direction
of Professor E* F# Lindquist of the State University of
Iowa, and extends to all high schools in the state who wish
to participate#
143
of th© 1932 test* reported for the entire population of
11*572* were presented fey Professor Lindquist at a con­
ference of teachers of mathematics which was helc3 at the
University of Iowa In October* 1935*
Indices for th© 1934
and 1936 tests were taken from two unpublished master’s
theses
%
which were directed by Professor Lindquist*
Th©
1934 study Is based on a sample of 1000 papers and the
1936 study on a sample of 500 papers*
The writer carefully examined the three examina­
tions mentioned and selected all of th© items which were
enough alike to be considered as testing for the same
process*
(Since schools participating in the program have
access to tests of former years* It is very seldom that an
item Is repeated*)
Each test is divided Into three sec­
tions which are labeled ‘’Fundamental Processes*tf
Algebraic Representations and R e l a t i o n s a n d "¥©rfeal
Problems*n
The Items are classified her© In the same manner,
and the number in parenthesis immediately following an item
is Its difficulty index*
In most cases where noticeable
differences occur* It is easy to see th© reason for th©
difference*
S Fridolf Lun&hofom* "Achievement in th© 1934 EveryPupil Test In Algebra*" Thesis (M*A#) Iowa* 1934*
Alfred A* Kaskadden* "A Critical Analysis of th©
1956 Iowa Bvery-Pupll Test In Algebra*" Thesis (M*A«) Iowa,
1936*
144
PffiTt 1*
l t 1934, Item 2t
1936, Item 2t
2 * 1932,
Fundamental Process©®
If a £ - 2 , find the value of (-3 &)^*
(48)
If x »
(5 7 )
2,
what Is the value of Sx'H
Item28 Subtract a2 - 6 a * 7 from 2 a2- 3 *
(50)
item 4s Subtract 2 a2 * a- 3 from 4 a2 - 2 a- 1 #
(67)
3# 1934,
ItemIs Divide
4a2x® by ax2 ,
(62)
1936,
Item9s Divide
6n4x6 by 3n2x2*
(57)
1934,
4# 1934,
1936,
5, 1932,
1936,
6 # 1934,
1934,
Solve for x In terms of
a and b,
item5* a « & *
Item5s Solve for Rx
K s
Item6s Solve for m
s * 2n » k *
Item1 4 •m » ax- b#
j| »
(59)
(60)
(59)
Solve for x in terns
of a, b, and m*
(57)
Item61 If 3A - 2B, what is the ratio of
A to B?
(7)
Item 19s Solve the equation G * ax
ratio l/x*
(18)
7, 1932, Items:
1934, Item 22:
Solve for r:
^j| -
4
/®"*
V *J^
for 1h©
.
What is the value of H?
(5)
( 5)
±4n>
1934, item 9*
Write in factored form the
low®at common denominator that
can be used in adding
.x
and
ay + xy
(35)
Find the algebraic expression that
is the product of th© highest
powers of all factors of the terms
(14)
6 ©xS , 3a^x and 4asxs #
1936, item 105
What is the lowest common
denominator that should be used
In adding three fractions whoa©
denominators are
(57)
Sax, 8 &S , and 12x3?
* 1932, item 10:
1934, item 11:
If 3x - 2y s 11 and 5x+3y m 12
what is the numerical value of x?
(44)
What is the value of x found by
solving simultaneously these
equations?
(36)
x ~ 2y z 10
1936, item 19:
10# 1932, item Hs
and
2x ~ y « 1 1 *
Solve for the unknowns:
2a •* 3b 3 13
♦ 2b « 4
2 m 7
Writ© the expression x ~ 2 ~ r
^'r+~*>
(30)
(19)
as on© fraction having
x + 2 as its denominator*
1934, Item ID: Simplify*
( 9)
1936, item 15s Reduce the following mixed expres­
sion to on© fraction in its
simplest forms
2 _ Sns
x + n - —---=»- ♦
(14)
x -n
XI# 1932, item 15*
If R *
——
write a formula
for 0 in terms of
E, r, and R#
1934, Item 24s
X, m M fe-r, B, # Solve for t In terms
of L, M, and g#
1936, Item 21*
S a ^
(1 2 )
(12)
1
®
* Writ© a formula for
®
Wg in terms of S
and
( 5)
For what negative value of y will
th© expression 5y^+5y**2 he
equal to a©rot
(25)
1934, Item 16*
What are the two roots of the
equation 2 n 2 « 5n * 3f
(2 2 )
1936, item 22*
Find the two roots of the equation
2 b2 - 4b S 18*
(26)
13# 1954, Item 23*
Multiply
( 9)
1936, Item 10s
Simplify
12# 1932, item 19*
2+ &
n
V?
by
- 1] •
/
4n+ 6
.
— ™—
4x - 9
•
(17)
147
?jRjrt II *
Algebraic Representations and Halations
14* 1932 , 1 tern
1936, item
15* 1934, item
7s
1934, Item 11*
16. 1934, item 10*
Write a trinomial whose factors are
(x ~ 0ft) and (2x 4- a)*
(5 4 )
What trinomial has 2x + y
of Its two equal factors?
(3 5 )
as on©
Write the quotient obtained by
dividing a + 1 by a ^ * 1 *
2
(26)
What Is the ratio of x - 9
5x~ 9 ?
(16)
Write th© fraction which is equal
to jj but which has a numerator
(12 )
Of 3$.
1936, item 12*
17, 1934, Item 22*
1936, item 18*
Hi© expression — j^g may be
changed into a fraction whose
denominator is
What Is
the numerator of this fraction?
(28)
For what value of m is the expres­
sion
equal to 3 *
(22)
For what value of x is th© expres*
(20 )
*
18* 1934, item 24*
1936, Item 14*
3
equal to zero?
Suppose H varies inversely as R and
that H Is 6 when H is 2 * What is
the value of N when R Is 3?
A varies Inversely as
16 when B is 3* What
A when B is 8 ?
B,
is
( 3)
and A
is( 3 )
the valueof
148
Problems
If the average speed of a train
should be Increased 4 miles an
hour it would require 30 minutes
less time to travel a distance
of 180 miles* What is the average
speed of the train?
( 1)
1934, item 10*
If the average rate of a certain
train should b© Increased by 4
miles per hour. It would require
30 minutes less time to travel a
distance of 180 miles* What Is
th© average rat© of the train?
( 2)
1934, item
9*
A man worked for a number of days
and earned #63* If he had received
75 cents more per day, he would
have earned the same amount in 2
days less. How many days did he
work?
( 1)
20* 1934, Item
3s
The sum of three numbers is 150*
The first number is twice th©
second, and 20 more than th©
third* What is the third number?
( 9)
1936, Item
2*
Three numbers are so related
that the first Is equal to twice
the second, while th© second
is equal to three times the third.
What is the largest of th© three
numbers If their sum Is 45?
(24)
19* 1932, item 10*
Only one pair of th© entire set of exercises can
b© classified as M e n t l c a l ^ that pair being th© first two
items of th© group numbered 19, and the Indices reported
are practically identical*
The 1934 Item was the last one
on the examination so its index perhaps doesn11 mean much,
but there were five Items following Item 10 in 1932 and
three of them yielded Indices of 10 or more, on© being as
high as 28*
It is quite likely, then, that the true index
of th© item is between 0 and 6*
There are several pairs of nalmost identical”
Items and, with on© exception, the agreement of their In­
dices is very close*
These pairs ares
those items
numbered 5, the last two Items of group 8, th© first two
of group 9, th© last two of group 12, the pair labeled 18,
and th© pairs one and three and two and three of group 19*
Th© two whose Indices differ markedly are those found In
group 8*
Th© unusual terminology of the 1934 Item is un­
questionably responsible for a large part of this difference*
In general, th© paired items taken from the
Every-Pupil Tests behave very much as do those taken from
th© pro*tests*
Th© evidence Is strong that th© invariant
quality of th© difficulty index is characteristic of
mathematics tests at both Instructional levels*
BIBLIOGRAPHY
151
1* Allen, Arthur A*, “Studies of the Retention of Algebra
by High School Graduates* I Algebraic Representa­
tions, II Formal Skills, III Verbal Problems•"
Master1® thesis, State University of Iowa, Iowa
City, August, 1939#
2* Atchison, R* H#, "Variability from School to School
In Difficulty of Items in the 1933 Iowa EveryPupil Test in General Science," Master’s thesis.
State University of Iowa, Iowa City, August, 1934#
3# Bergen, M# C#, "Achievement in College Algebra Com­
pared with the Humber of Semesters of Preparation
in High School," School Science and Mathematics.
XXXVIII (October,^1938), 763-765#
'
~~
4# Bowley, A# L#, Element® of Statistics# 4th editions
Bond on* P • "STJ King and Son, Ltd*, 1920,
5# Branham, Arthur K#, "Knowledge of High School Algebra
Retained by High School Seniors#" Master’s thesis,
State University of Iowa, Iowa City, August, 1938
6# Brumbaugh, Aaron J,, "The Uses of Tests and Examina­
tions in the Selection, Guidance, and Retention
of Students#" Institute for Administrative
Officers of H i g h e r lnstitu tion a , Proceedings-. VIII,
1936, pp# 176-189#
x/t * Burch, Robert L#, "Studies of Retention of Algebra
^
by High School Graduates* I Algebraic Representa­
tion#" Master’s thesis, State University of Iowa,
Iowa City, August, 1939#
8# Chesire, Leone, Milton Saffir, and L* L# Thurstone,
Computing Diagram® for the Tetrachorlo Correlation
ffoefficTeiat» The University of Chicago, 1933*
9# Crawford, A# B# and Paul S* Burnham, "Entrance Exami­
nations and College Achievement," School and
Society. XXXVI (September 10 and 17, 1932), 344352 and 378-384#
10# Douglass, Harl R#, "Permanence of Retention of Learning
In Secondary School Mathematics," The Mathematics
Teacher, XXIX (October, 1936), 287,288*
11# Douglass, Harl R# and Jessie H# Michaelson, "The Rela­
tion of High School Marks and of Other Factors to
College Marks In Mathematics," School Review5
XLIV (October, 1936), 615-619#
12* Garrett, Henry E*, Statistics in Psychology and EducaM o n * 2nd editions Hew York * Longmans, Green
and Company, 1937*
IS# Gilkey, Royal, 11The Helation of Success in Certain
Subjects in High School to Success in the Same
Subjects in College,ff School Review* XXXVII
(October, 1929), 576-588*
14# Hammond, H* P # and George B* Stoddard, A Study of
Placement Examinations, University""of Iowa Studies
in Education, Vol* XV, Ho# 7* Xowa Citys Uni­
versity of Iowa, 1929*
15* Hansen, George W*, 11An Analytical Study of the Iowa
Placement Examinations#1* Master*s thesis, State
University of Iowa, Iowa-City, August, 1928#
16# Hart, William L*, "Some Debatable Facts and Some Sug­
gestions, ” The Mathematics Teacher * XXXI (Decem­
ber, 1938), 355-362*
17* Hawkes, H* E*, E* P* Lindquist, and C* R* Mann, The
Construction and Use of Achievement Examinations*
Hew Yorks Houghton, Mifflin Company, 1936*
18* Kaskadden, Alfred A#, ”A Critical Analysis of the 1936
Iowa Every-Pupli Test in Algebra**1 Master1s thesis,
State University of Iowa, Iowa City, August, 1936*
19* Kenney, J* P*, "Some Topics in Mathematical Statistics,”
The American Mathematical Monthly % XL VI (February,
19391, Ho* 2, 59-74*
20* Lindquist, E* F*, "Cooperative Achievement Testing,”
Journal of Educational Research, XXVIII (March,
1935), Ho* 7, 511-520*
21# Lindquist, E* P*, Statistical Analysisin Educational
Research* Hew York: Houghton, Mifflin Company,
1940*
22* Lundholm, Frldolf, "Achievement In the 1934 Every-Pupil
Test In Algebra•” Master*s thesis, State Uni­
versity of Iowa, Iowa City, August, 1934*
23* Miller, Laurence William, An Experimental Study of the
Iowa Placement Examinations * University of Iowa
Studies in Education, Vol* V, Ho* 6* Iowa City:
University of Iowa, 1930*
153
24* Olson, Owen L #, "Studies of Retention of Algebra by
High School Graduatest
I Algebraic Representa­
tion, IX Formal Skills, III Verbal Problems*"
Master1s thesis, State University of Iowa, Iowa
City, August, 1939.
25* Hemmers, H* H. 4 Study of Freshmen Placement Testa
at Purdue university. 1926-29* Purdue University
Bulletin, XXIX,No* 13, Lafayette, Indiana, 1929*
26* "Report of the Committee on Tests," The American
Ma them a 11ca! Monthly. XL VII (May, 194QT, Ho* 5*
27* Robb, Eva V., "A Study of the Variability In Item Dif­
ficulty from School to School for the 1933 Iowa
Every-Pupil Teat in Ninth Grade Algebra." Master's
thesis, State University of Iowa, Iowa City,
August, 1934*
28* Schoonmaker, Ha&el E*, "The Value of the Hotz Algebra
Scales In Sectioning College Classes In Freshman
Mathematics." School Science and Mathematics
XVIII (November, 1928),880-884*
29* Segel, David, Prediction of Success in College*
Bulletin 1934, Ho* 15* Washington, D* G*: United
States Department of the Interior, 1934*
50* Stoddard, George D., Iowa Placement Examinations*
University of Iowa Studies in Education, Vol. Ill,
Ho* 2* Iowa City: University of Iowa, 1925.
31* Stoddard, Georg© D., "Iowa placement Examinations,"
School and Society* XXIV (August 14, 1926), Ho.
607, 212-216*
32* Tippett, L. H* C*, Random Sampling Numbers * Tracts
for Computers» S o * XV, Cambridge University Press,
1927*
33* Turner, Nura, "A Study of Certain Pre-Tests for First
Year College Mathematics*" Master's thesis, State
University of Iowa, Iowa City, August, 1936*
34. Yule, G* Udny and M* G. Kendall, An Introduction to
the Theory of Statistics*
11th edition®
Charles Griffin and Company, 1937.
London:
its4
APPENDIX
Table 51
DIFFICULTY AND VALIDITY INDICES FOR EACH ITEM
OF FORM A, 1935, AS GIVEN BY THE COOPERATIVE TEST SERVICE
Item
1
2
5a
b
c
d
4
5
8
7
8
9
10
11
12
15
14
15
16
17
18
19
30
21
22
23
24
25
26
27a
b
28
29
30
31
32
33
34a
b
c
d
35a
b
c
Difficulty
Validity
78
84
78
95
93
91
69
59
86
61
62
77
55
55
87
70
52
42
35
52
53
42
56
56
44
34
51
47
34
45
50
35
24
3
28
18
7
92
73
81
83
3
6
6
4
6
4
8
7
3
3
5
5
5
10
7
8
8
9
11
7
7
8
6
9
11
11
11
12
12
10
10
4
9
4
11
9
6
3
4
5
6
mm
mm
-
1^6
Table 61 (continued)
Item
66
37a
b
58a
b
c
59a
b
e
40
41
42
45
44
Difficulty
Validity
24
64
49
74
68
64
8
5
6
4
•7
4*
m
*»
m
81
24
17
51
4
12
6
4
8
4
5
4
Table 52
AMD VALIDITY IHDIOES FOR EACH
FORM B* 1956
Item
Difficulty
1
8
5
4
5
0
7
a
9
10
11
12
15
14
16
16
17
18
19
20
'74
64
58
67
52
78
76
57
50
44
53
60
62
72
85
54
45
26
53
Validity
4
5
10
7
6
6
5
5
-8
8
5
7
10
5
5
4
7
6
10
12
157
Table 52 (continued)
Item
Difficulty
21
22
23
24
25a
b
26
27
28
29
30
81
32a
%
a
35
54a
b
0
58a
b
0
m
57
38
39
40
41
42
37
42
41
39
49
58
26
32
15
25
17
78
Validity
8
12
12
3
11
10
9
3
9
9
5
«•»
■m
34
70
52
25
**
m
58
22
25
14
86
4
IS
9
7
8
7
7
**
9
8
7
6
5
5
6
6
Table S3
DIFFICULTY AMD VALIDITY INDICES BOR EACH ITEM OF
FORM C, 1938
Item
1
2
3
4
5
6
Difficulty
Validity
75
84
50
65
68
58
82
4
4
6
5
7
7
4
Table 555 (continued)
Item
Difficulty
a
9
10
11
12
13
14
15a
b
6
16
J7
IB
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33a
b
e
&
34
35
56a
b
37
38
39
40
41
42
43
44
90
95
71
43
39
71
76
87
88
77
68
70
•60
68
46
40
24
31
3
11
47
14
29
Validity
81
ai
78
77
6
4
6
8
10
7
9
8
7
4
5
4
6
6
14
9
a
9
4
6
6
8
9
9
8
8
6
7
9
8
7
82
8
85
17
27
65
38
m
49
22
85
12
59
23
27
19
6
5
m
7
5
6
7
11
10
10
7
6
Table 54
DIFFICULTY AMD VALIDITY INDICES FOR EACH 1936 ITEM
AS GIVES BY THE COOPERATIVE TEST SERVICE
Form A* 1936
Item
Difficulty
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
13
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
92
74
95
90
66
82
84
78
47
72
75
75
71
73
77
68
68
77
69
68
53
52
78
63
70
30
54
75
73
51
48
63
47
39
45
59
46
24
28
43
Validity
1
7
4
6
4
5
6
7
3
6
7
4
3
6
6
9
4
10
7
5
7
4
4
6
6
11
7
10
6
11
14
10
7
5
8
9
Y
9
Y
8
Form
Difficulty
95
87
91
85
87
57
72
67
74
31
73
65
64
72
50
84
68
55
73
66
77
60
24
45
62
83
54
54
55
66
57
19
49
35
44
15
29
33
20
36
1936
Validity
5
1
3
4
4
9
6
5
3
8
7
7
7
3
6
5
5
8
4
6
6
5
7
6
7
6
7
5
8
7
6
7
7
9
10
8
6
9
8
10
•]*
‘Table 84 (continued)
Form A m 1956
Item
Difficulty
41
43
43
44
45
46
47
48
49
50
50
20
42
34
15
18
21
14
10
28
Validity
10
6
8
6
7
9
9
8
7
3
Form B, 1936
Difficulty
Validity
27
25
18
18
18
36
15
16
4
10
9
9
8
9
5
3
8
6
5
6
* f;
ii *>-•* M
Tabl® 55
NUMBER OF CORRECT RESPONSES (R), INCORRECT RESPONSES (W),
AND OMISSIONS (0), ARISINQ FROM EACH SAMPLE,
FOR EACH ITEM OF FORM A, 1935
1 Year
1& Years
E Years
Item
E
W
0
R
mtm*
W
0
H
W
0
1
2
3*
b
e
d
4
5
6
7
8
9
10
11
12
13
14
18
16
17
18
19
20
21
22
23
24
23
26
27a
b
28
29
30
31
32
33
34a
38
42
38
46
47
47
28
28
48
28
30
33
23
18
43
53
18
8
8
24
17
15
19
10
8
8
8
3
11
7
14
4
3
2
21
21
7
21
16
14
19
26
6
11
16
20
22
19
19
18
15
21
24
20
11
10
5
6
5
6
6
4
3
1
2
2
10
7
7
1
1
1
O
0
1
1
6
0
1
4
3
8
6
1
6
16
22
23
7
14
17
16
19
18
22
31
37
42
36
40
40
44
46
45
49
48
2
2
2
3
88
92
88
98
96
95
83
74
90
68
70
92
64
55
95
80
75
45
28
60
73
58
59
33
45
32
24
47
22
44
47
25
20
0
26
16
6
98
83
78
91
11
8
11
2
3
3
15
23
9
31
27
6
30
37
5
19
21
48
43
30
20
33
20
54
37
45
38
13
15
13
10
18
15
24
5
S
10
2
15
20
8
1
0
1
0
1
2
2
3
1
1
3
2
6
8
0
1
4
7
29
10
7
9
21
13
18
23
38
40
63
43
43
57
65
76
69
79
84
0
2
2
1
41
47
46
49
50
47
42
43
49
33
42
46
42
39
49
43
35
27
17
38
39
35
38
26
34
22
21
38
20
36
38
24
15
2
15
13
3
48
45
45
45
8
3
4
1
0
3
8
3
1
16
8
4
7
9
0
7
13
17
18
11
10
12
6
19
12
23
15
3
11
3
2
10
14
20
4
6
8
2
4
4
3
1
0
0
0
0
0
0
4
0
1
0
0
1
2
1
0
2
6
15
1
1
3
6
5
4
5
14
9
19
11
10
16
21
28
31
31
39
0
1
1
2
b
c
a
3
8
7
4
0
0
2
0
0
46
58
41
40
±S2
Table 58 (continued)
1 Year
Item
35a
b
0
36
37 a
b
38a
b
e
39a
b
e
40
41
42
43
44
14 Years
H
W
0
**
mm
e*
m¥
mm-
a#
4m-
6
27
23
27
24
10
26
22
22
18
9
18
18
1
5
a
17
25
22
6S
50
78
72
29
1
W
2 Years
MM*
«MWM
0
E
w
0
#*
4m
mm
mm
*
*»
ee
M
mm
58
32
42
16
14
34
20
3
8
6
14
57
14
41
33
33
40
20
23
9
12
14
2
16
13
0
5
5
a
14
m
mm
ae
mm
flW-
m*
mm
**
mm
m-
4m
aw
eft-
«
4m
mm
**
m
20
10
7
3
1
9
13
12
3
3
2
3
17
28
35
39
47
39
56
19
85
28
2
11
25
48
59
6©
©7
80
40
20
13
10
4
8
19
33
16
©
11
9
5
14
9
2
4
7
5
16
28
30
42
36
Table 86
NUMBER OF CORRECT RESPONSES (R), INCORRECT RESPONSES (W),
AND OMISSIONS (0), ARISING FROM EACH SAMPLE,
FOR EACH ITEM OF FORM B, 1935
l£ Years
1 Year
2 Years
■ M A a r t M M M M i M W W M *
Jin nnrm-. Jinn NTT
Item
E
W
0
H
W
£
E
W
0
1
2
5
4
5
6
7
8
9
10
11
37
41
25
19
34
26
41
40
22
20
15
13
9
18
17
15
24
8
8
21
17
19
0
0
7
14
1
0
1
2
7
13
16
36
42
38
35
38
35
40
40
34
36
24
13
8
12
8
12
15
9
9
15
10
18
i
0
0
7
0
0
1
1
1
4
8
55
4©
58
39
40
33
47
40
40
55
52
15
4
10
6
9
17
5
9
10
14
16
0
0
2
5
1
0
0
1
0
1
2
fable 56 (continued)
1 Year
Item
1
IB
13
14
15
16
17
18
19
20
21
22
23
24
25a
b
26
87
28
29
30
31
52a
b
c
33
34a
b
c
35 a
b
c
36
37
38
39
40
41
42
28
25
34
38
43
26
15
0
4
- h, 8
10
9
4
11
8
3
7
1
2
3
33
*•.
li Years
«•»#
w
0
20
16
11
9
4
19
26
25
16
13
13
12
19
14
@
4
4
2
2
8
9
#»
2 Years
m
W
0
&
W
0
2
9
6
5
3
5
11
25
30
24
27
29
27
25
34
43
39
47
46
39
8
28
32
34
40
49
38
22
8
9
19
16
20
19
24
11
4
13
1
3
22
JLi^r
12
9
1
18
19
29
25
0
5
4
1
0
0
9
13
16
16
21
20
21
23
24
31
32
36
41
36
1
30
40
40
41
47
35
24
19
1
3
4
2
0
2
7
13
11
14
14
14
16
14
17
24
25
34
31
34
1
■put
6
44
IS
13
10
10
5
15
15
5
13
6
8
5
a
12
18
22
25
81
30
21
9
15
5
14
7
49
«■»
«»
**
*»
mm
«*
urn
«■*
mm
mi
mm
mi
«*
am
m
m
tm
8
7
17
19
21
56
33
9
m
14
28
14
1
28
15
19
30
23
12
11
32
m
mm*
«►
!*»
24
3
4
1
11
0
2
2
5
6
17
8
4
6
3
4
21
41
29
41
35
44
45
44
35
7
12
6
18
2
8
2
6
2
6
9
-
mm
mm
4
11
15
7
3
8
4
1
11
32
23
37
29
40
38
47
23
46
35
13
y
6
7
3
13
19
29
27
18
14
11
13
6
12
17
10
11
5
9
0
-
mm
m
34
9
11
7
25
2
9
2
as
20
3
8
28
7
1
7
9
an*
mm
7
13
20
10
5
3
5
9
28
19
33
20
45
36
42
mm
mm
—
m
6
Table 57
NUMBER OF CORRECT RESPONSES (R), INCORRECT RESPONSES (W),
AND OMISSIONS (0), ARISING FROM EACH SAMPLE,
FOR EACH ITEM OF FORM C, 1935
I Tear
Item
E
PM*
1
2
3
4
5
6
7
8
9
10
11
12
IS
14
15 a
b
c
16
17
18
19
20
21
22
25
24
25
26
27
28
29
30
31
32
33a
b
C
a
32
44
15
21
51
32
42
43
46
29
3
9
33
35
39
44
37
32
34
26
52
16
10
3
5
1
2
15
2
1
6
0
4
23
34
34
28
32
W
18
6
27
14
19
14
@
7
4
18
31
34
15
12
10
2
10
16
14
19
11
15
19
6
18
13
7
6
12
10
11
9
8
25
7
5
7
8
lit Years
0
0
0
3
15
0
4
0
0
0
3
16
7
2
3
1
4
3
2
2
5
7
19
21
41
27
56
41
29
36
39
33
41
38
4
9
11
15
10
R
W
38
43
29
30
36
28
47
47
50
36
18
13
41
47
50
45
46
40
38
32
37
24
26
10
13
2
3
20
5
8
12
6
7
34
43
45
36
46
12
2
20
13
14
21
2
3
0
12
25
28
9
2
0
4
3
7
11
15
5
17
13
15
18
19
12
8
14
9
7
8
9
14
6
6
12
3
2 Years
0
R
W
0
0
1
7
0
1
1
0
0
2
tp
37
47
35
42
33
38
45
48
46
41
30
27
45
47
49
47
42
41
38
43
38
38
30
13
16
3
6
28
10
21
21
13
14
42
46
44
41
48
12
3
15
5
17
11
4
2
4
9
17
21
5
3
1
3
8
9
11
7
7
8
14
27
21
28
10
8
20
9
8
5
9
8
3
5
8
2
9
0
1
0
1
1
3
1
3
8
9
11
25
19
29
35
22
31
33
31
36
34
2
1
1
2
1
0
PPM-
1
0
0
3
0
1
1
0
0
0
3
2
0
0
0
0
0
0
1
0
5
4
6
10
13
19
34
14
20
20
21
32
27
0
1
1
1
0
105
Table 67 (continued)
1 Year
Item
R
w
M M
34
35
36a
b
37
38
39
40
41
42
43
44
6
9
25
24
20
7
7
1
10
1
4
2
0
12
18
14
12
18
6
O
3
7
ijfr Years
*0"
19
17
«•>
18
25
29
37
22
43
46
45
43
2 Years
R
W
0
R
W
0
14
25
25
22
11
5
22
26
22
16
6
8
mm
mm
26
9
20
5
19
8
6
5
1
15
27
14
13
12
14
6
2
11
mm
9
14
16
32
19
28
38
43
38
52
12
29
9
23
13
14
15
5
mm
13
17
8
©
9
14
6
3
5
5
21
15
35
18
23
30
32
42
fable 58
NUMBER OF CORRECT RESPONSES (R), INCORRECT RESPONSES (W),
AND OMISSIONS (O), ARISING FROM EACH SAMPLE,
FOR EACH ITEM OF FORM A, 1936
1 Year
l| Years
2 Years
Item
K
W
0
R
W
0
R
W
0
1
2
5
4
5
6
7
8
9
10
11
12
13
14
15
1©
88
56
96
93
61
12
43
4
7
22
22
15
17
28
23
26
26
32
24
19
36
0
1
0
0
17
1
0
2
20
6
4
3
2
11
10
11
93
78
95
94
87
91
89
84
64
85
80
81
72
79
78
71
7
21
3
6
10
9
11
15
30
15
19
IB
28
18
18
28
0
95
82
96
98
88
92
95
91
84
82
85
88
89
93
85
93
5
16
4
2
11
8
5
8
15
18
14
12
11
6
14
7
0
2
0
0
1
0
0
1
1
0
1
0
0
1
1
0
77
85
81
52
71
70
71
6©
65
71
53
2
0
3
0
0
1
6
0
1
1
0
3
4
1
Table 58 (continued)
1 Tear
lj» Tears
2 Years
Item
R
W
0
R
W
0
R
W
0
17
IB
19
20
62
92
69
59
41
51
77
38
52
8
45
54
60
22
21
32
27
32
21
39
26
19
11
16
7
3
22
3
1
2
1
0
1
17
26
8
28
31
30
40
21
45
19
59
42
20
35
43
24
33
27
36
27
40
33
34
38
26
16
23
6
21
23
21
11
21
2
10
12
0
3
10
29
9
2
17
29
33
IS
26
7
35
55
35
48
32
52
21
41
47
51
58
77
74
72
76
76
*77
88
79
97
73
82
99
75
75
60
58
89
70
80
31
61
86
82
65
60
75
46
57
36
64
43
26
26
34
27
18
53
27
10
9
14
6
9
31
13
0
22
10
27
39
8
24
8
@4
32
4
15
27
25
22
31
26
42
29
43
37
40
39
34
20
13
22
32
28
21
18
8
5
5
1
3
15
13
3
3
6
12
15
7
10
3
8
15
3
23
17
22
7
14
38
34
27
39
62
34
51
58
63
65
76
83
64
84
97
82
75
78
69
83
80
85
48
82
89
87
77
68
94
63
52
57
79
66
43
46
68
51
24
73
46
25
19
29
18
12
SB
14
2
16
13
16
31
15
16
9
46
18
8
12
18
24
5
51
40
59
20
31
39
48
26
29
25
9
23
40
43
20
16
10
3
2
1
2
12
6
0
2
4
6
6
0
3
1
5
8
1
6
8
4
1
3
18
6
6
20
51
18
31
37
38
51
66
78
59
21
22
23
24
25
26
27
26
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
48
47
48
49
50
I—j
JL*5 J
Table 59
NUMBER OF CORRECT RESPONSES (R), INCORRECT RESPONSES <W),
AND OMISSIONS (0), ARISING FROM EACH SAMPLE,
FOR EACH ITEM OF FORM B, 1936
1 Year
Item
R
Wmm*
m
1
2
3
4
5
6
92
89
92
81
88
43
68
68
73
27
63
60
55
69
30
85
60
35
73
62
74
64
8
36
46
80
50
52
53
41
46
10
26
15
23
8
24
4
8
3
7
18
12
35
30
8
83
64
30
24
24
28
38
9
33
47
12
33
6
31
82
37
39
9
39
42
32
15
33
64
27
44
45
39
42
32
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
23
29
30
31
32
33
34
35
36
37
38
li Years
2 Years
0
£
W
0
R
W
0
0
3
1
1
0
22
2
24
4
9
7
16
21
3
38
6
*7
18
15
5
21
5
10
27
15
11
11
6
15
44
21
26
47
41
32
53
34
64
93
87
97
85
87
73
79
89
73
42
72
78
76
80
58
89
75
67
82
65
31
65
15
51
78
79
61
65
57
71
58
11
40
26
36
18
35
19
5
10
3
13
12
20
20
7
24
54
28
19
19
19
25
9
25
24
8
33
10
32
77
34
21
10
36
29
26
18
22
72
30
42
45
44
37
35
2
3
0
2
1
7
1
4
3
4
0
3
5
1
17
2
0
9
10
2
9
3
8
15
1
11
3
8
17
11
20
17
30
52
19
38
28
46
99
89
98
93
94
84
92
94
84
59
90
87
88
91
76
97
92
82
84
79
93
80
39
69
84
89
79
76
SI
93
80
32
70
51
67
47
61
51
1
11
2
6
6
12
8
6
16
38
10
13
10
9
15
3
8
14
8
19
3
20
56
29
15
5
19
23
14
5
17
61
24
41
31
39
28
50
0
0
0
1
0
4
0
0
0
3
0
0
2
0
9
0
0
4
8
2
4
0
5
2
1
6
2
1
5
2
3
7
6
8
2
14
11
19
±68
Table 59 (continued)
1 Year
ijr Years
2 Years
Iteia
B
W
0
a
W
0
R
W
0
39
40
41
42
43
44
45
46
47
48
49
50
5
8
5
4
5
3
6
23
4
2
0
0
46
15
7
28
5
5
4
8
10
6
4
16
49
77
88
68
92
92
90
69
86
92
96
84
12
21
8
10
5
11
5
23
2
8
1
4
51
17
22
26
3
3
12
2
11
1
7
16
37
62
70
64
92
86
83
75
87
91
92
80
32
62
43
36
27
35
51
41
20
15
8
16
51
15
29
36
9
9
17
7
14
6
14
16
17
23
28
28
64
56
52
52
66
79
78
68
1G9
Table 60
V npq
u p * 10*000 /
FORM A, 1935
Item
1
2
3a
b
o
&
4
3
6
7
a
9
10
11
la
13
14
15
16
17
1©
19
20
21
22
23
24
25
26
27a
b
28
29
30
31
52
33
34a
b
c
a
31*064 '
25*360
30*976
14.100
9*768
9*232
46*040
44*108
21*668
42*924
39*444
42*260
43*272
44* 924
23*436
39*592
41*308
24.988
16.176
42.272
38.828
36.048
36.732
27.936
25.064
25*596
20.396
9,544
9, 544
20*436
17.176
6.692
O
O
5.852
O
0
14.176
34.372
22.900
29.900
9*913
7,146
10*046
1.913
5.753
4*653
13.933
19.013
8.966
21.086
19.986
7.113
21.346
23.746
4.553
15*446
18.32
23.833
18*713
23.48
19.146
23.613
22.866
19.886
24.133
20.453
17*086
21.113
16.153
21.533
22.753
18.466
14.553
0
17.286
12,253
5.133
1.913
13.113
16.546
7,946
28,172
10,668
14.000
3.852
O
10.740
25.504
20*304
3.852
42,804
23.212
14.376
24*072
30*376
3*852
22.596
30.000
43.472
33.204
30.000
31.508
39.244
34 «156
38.036
40.396
41.508
42 *804
32.156
35.364
35,284
33.880
36,060
34.348
10*156
38.244
36*848
13,712
7.408
17.636
17*212
17.212
Table 60 (continued)
Item
35a
13
c
36
37a
b
38a
b
o
39a
b
0
40
41
42
43
44
£1
h i
m
«»
*»
19.938
43.984
46.744
48.076
38.844
88.438
15.753
21.846
83.113
16.313
19.213
19.433
m*
a*
46.260
28.936
81.640
84.436
3 *852
28*064
«*
*
34.396
85.932
38.544
44.036
88*840
44.176
«*
m
**
28.68
15*158
18*613
19.646
1.92
9*453
28*520
42*908
35*064
42*316
13.712
23.516
Table 61
CT| ?
4 ^ npt|
FOBM 8 # 1935
Item
1
2
3
4
5
6
7
8
9
10
11
12
15
14
15
16
fi
35.472
28*596
48.132
44*448
42.976
48*696
29*376
30.976
43.584
42.744
35.796
44.000
46.428
41.376
34*464
20*316
Pl§
39*296
22*228
32*916
39*524
33.524
37.296
31.828
28.096
41.423
37.924
48.384
46*668
40*896
39.200
27.028
3.716
I*
36.376
14.236
55.004
27.676
28.036
41.588
10.760
30.396
29.840
39.500
44.856
46.828
29.748
30.968
29 *444
9.600
tTable 61 (continued)
Item
17
18
19
20
21
22
23
24
25a
b
26
27
28
29
30
31
32a
b
G
33
54a
b
c
35a
b
c
36
37
38
39
40
41
42
47*096
52.272
O
10.780
23.544
28.464
22.448
10.780
30.064
23.900
7.228
20.848
O
3.896
7.580
40.384
m
36.096
41.584
25.524
25.184
46.584
41.296
44.000
42.868
47.984
29.924
14.516
36*856
3.2
10.784
20.856
20.784
39.272
41.728
85.784
55.708
41.472
40.500
45.856
46.272
40.640
46.804
27.340
41.516
17.064
58.140
21.748
3.732
#*-
35.604
46.152
34.272
3.896
47.316
56.628
41.656
29.256
46.456
14.836,
58*204
36.468
37.896
10.408
14.384
3.896
32.804
0
7.580
7.580
41.256
22.916
33.584
20.724
40.384
6.856
23.296
7.584
39.804
28.300
53.588
23.348
47.596
7.304
28*100
7.304
Table 62
Op* S
4 £ npq
FORM C» 1936
Item
1
2
3
4
5
6
7
8
9
10
XI
12
13
14
15a
b
c
16
17
18
19
20
21
22
23
24
25
2©
27
28
29
30
31
32
53a
b
c
a
34
35
36a
P1
40*076
19*900
41*268
45*204
42*020
44*860
26*416
23*820
14*604
46 *496
9*636
23*820
43*91©
41*212
33*624
19*720
57*784
45*568
42* 504
49*648
44*308
36*29©
31*240
10*840
11*248
2.000
7*484
36.980
7.636
2*000
14 *604
0
13.248
46.360
42*624
42*020
43*916
44*060
19*900
29.240
mm
*1*
56*000
7*600
45*900
45*600
39*500
49*100
9.400
9.900
0
59*600
45*500
35*600
25*500
11*100
0
17*100
14.400
31*400
35*500
45*500
37*500
49*900
49.500
31.100
57*400
7*600
9*400
47.100
17*500
25.500
55*500
19*500
25.000
41*900
23.500
23 *500
39.600
14.600
39.400
45.900
mm
*8
37.656
11*223
41*428
24*228
43 *916
34*628
17.628
5*800
14.628
28.200
44*916
47*056
15.716
11*05©
3.800
11.228
26.516
29.056
35*656
22.228
35.228
33*916
46*428
37*85©
41.716
11.228
20.228
39*65©
30.056
44.656
44.856
33.916
39.428
25.716
11.228
20*916
28.716
7.656
46.516
49.856
mm
Table 62 (continued)
Item
36b
37
38
39
40
41
42
43
44
Ei
h k
44*860
23*820
22*540
3*764
26*172
2*000
12*932
5*764
0
49*500
29.100
46.600
16.600
41.600
99.400
19.100
17*900
S.500
Ha
44*428
36*000
45.516
29*056
43*716
37*228
36*916
36.628
10.716
Table 63
<r/ = H
[
npq
FOES A, 1936
Item
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
El
10*509
21*271
3*797
6*444
21.171
17.530
12.394
14.430
22.469
20.064
20.418
19.984
21*976
21*438
19.543
25*097
20*571
7.190
19.451
22*894
21.418
21.970
17.193
19.348
M
Er
6*380
15*564
4*588
5*535
10*960
7*819
9*501
12.789
21.313
12.627
14.593
14.914
19.594
16*322
16.326
19.529
14*141
980
18.280
17.106
22*335
22.268
9.378
19.329
4*674
13.197
3.659
1.924
9.832
6.797
4.600
7.879
12.449
14* 015
12.183
10.394
9.226
5.668
12.628
6.068
13.000
2.847
13.^92
18.226
17.038
20.396
13.805
15*583
•
f,ryA
UXm *+
Table 63 (continued)
Item
25
26
27
28
29
50
31
32
35
34
35
36
37
58
39
40
41
42
45
44
45
46
47
48
49
50
23.597
S.261
22.110
22.771
22.469
14.836
11.575
17.037
17.159
20.184
13.568
22.218
17.317
10.250
3 *084
8*284
4.849
2.762
15.461
2.275
0
1.5
.5
0
.875
13.777
15.538
20.770
20.823
11.817
14.509
19.396
20.437
16.670
22.866
22.566
20.017
20.591
22.431
18.340
16.830
20.135
18.817
14.318
23.348
19.359
8.860
7.747
11.672
5.514
7.809
20.250
12.571
22.913
14.112
9.350
10.894
17.318
19.610
5.150
21*619
23.134
21.823
15.866
19.834
22.776
24.429
19.692
23.198
18.220
18.967
25.949
16.029
15.291
18.918
14.288
9.698
22.137
Table 64
C~p2 » 21
FORM B, 1936
11
1
2
3
4
5
6
7
8
9
10
11
12
15
14
15
16
17
18
19
00
21
22
25
04
25
26
27
28
29
30
31
32
33
34
36
36
37
38
39
40
41
7 b198
9b681
7 b198
14b603
10w420
22•718
20*093
20*971
18*579
17*818
22.521
22*820
24.195
20.862
17.316
12.420
21.745
21.181
19*387
23.179
18*720
22.554
6.795
19.454
23.526
14.806
25.987
23.798
24*256
00.391
23.141
8.862
14.471
9.281
15.993
5.318
17.691
2.333
4.479
S. 396
3.776
pi&
6*238
10*786
2*689
12*417
10b146
17.884
15.076
9b 451
19.418
22.966
19.678
15.551
17.687
15.543
21.44©
9.586
14.700
21.684
14.307
21.957
14.700
21.613
11.240
23.235
16.551
15.528
23.479
22.413
22.735
17.518
22*849
9.197
17.203
18.543
19.484
12.653
22.165
13.261
9 .494
13.219
6.952
f®
.8
9.211
1.918
6.426
5.502
12 .953
7.009
5.486
12.79©
23.086
8.729
11.169
10.362
7.559
16.203
2.875
6.900
14.412
13.323
15.572
6.091
13.552
22.918
19.498
12.453
8*990
16.547
17.714
14.964
5.395
15.571
21.101
19.872
24.217
21.267
24.274
22.407
23.348
20.229
21.512
22.524
Table ©4 (continued)
Item
42
43
44
45
4©
47
48
49
50
II
3*526
2*583
2*541
4*712
15.816
2.688
.985
0
0
fit
8.338
4.633
9.310
4.634
17*243
1.884
7.141
.934
3.084
fs
18.457
18.525
20.009
19.845
20.540
14.614
11.336
7.041
12.600
-r
J
i5
- t* w
Tabid 65
COXFARXSOB OF OBSERVED DIFFERENCES BETWEEN GROUPS P^ and P-ji
IN PERCENTAGE OF CORRECT RESPONSES TO EACH ITEM
IB FORM A, 1935, AMD THE STANDARD ERRORS OF THE DIFFERENCES
Item
1
a
3a
b
e
d
4
3
6
7
a
9
10
11
12
13
14
15
16
17
1©
19
20
21
22
23
24
25
26
27a
b
28
29
30
31
32
33
34a
b
o
a
Observed Difference
12
8
18
6
2
1
27
28
4
12
10
26
18
19
9
14
39
29
18
12
39
28
21
13
29
16
8
41
1©
28
33
17
20
0
22
16
6
6
7
4
11
5*12# (diff * )
6*4
5*7
6*4
4*0
3*7
3*7
7*7
7*9
5*5
8*0
7*7
7*0
8*0
8*3
3*3
7*4
7*7
7*0
5*9
8*1
7*6
7*7
7*7
6*9
7*0
6*8
6*1
5*5
5*1
6*6
6*3
5.0
3*8
0
4*8
3*5
2*3
4*0
6*9
6*5
6.2
S.B.Td:
1*9
1*4
2*8
1*5
*54
*27
3*5
3*5
*73
1*5
1*3
3*7
2*3
2*3
1.7
1.9
5.1
4.1
3.1
1*6
5*1
3*6
2*7
1.9
4*1
2*4
1*3
7.5
3*1
4*5
5.2
3* 4
5.8
4*6
4*6
2*6
1.5
1*0
*63
1.8
178
Tab!a 66 (continued)
Item
85a
b
c
36
37a
b
38a
b
c
39a
b
a
40
41
42
43
44
Observed Difference
m
m
S*E*(diff*)
■m
m
m
10
11
4
24
24
9
6*0
8*1
8 #4
7*6
7*6
6*9
e*»
mm
a*
Difference
§*E*(diff*)
mm-
1*7
1.4
*48
3*2
3*2
1.3
«*
a t
16
1
11
12
0
7
8*3
6*6
6*3
6*6
2*4
6*0
1.9
.15
1*7
1*8
0
1*2
Table 66
COMPARISON OP OBSERVED DIFFERENCES BETWEEN GROUPS Pji and P,
IN PERCENTAGE OF CORRECT RESPONSES TO EACH ITEM
IN FORM A, 1935* AND THE STANDARD ERRORS OF THE DIFFERENCES
Item
Observed Difference
1
2
3a
b
0
d
4
5
6
7
8
9
10
11
12
13
6
2
4
0
4
1
1
12
8
2
14
0
20
23
3
6
S*B.(diff.)
6*2
4*2
4*9
2*4
1*9
3*9
6.3
6.3
3*6
8*0
6*6
4*6
6.7
7*4
2 *9
6*2
S*E*(d;
*97
*48
*82
0
2.1
*26
*16
1.9
2.2
*25
2*1
0
3.0
3*1
1*0
.97
-f. S ^ Q
m- w
Table 66 (continued)
item
Observed Difference
14
15
16
17
18
19
20
21
22
23
24
25
26
27a
b
28
5
9
6
16
5
12
17
19
23
12
18
29
18
28
29
23
10
4
4
10
0
2
7
12
1
30
31
32
33
34a
b
e
a
35a
b
e
36
37a
b
38a
b
e
39a
b
c
40
41
42
43
44
*»
aa
4*
6
17
16
12
8
11
*»
24
21
1
8
6
5
S*E*(aif£)
7*0
8*2
7*2
7*5
7.1
7 *9
7*6
7 *6
8.0
7.9
7*7
7.5
7.2
7.6
7.5
7.4
7.0
5.2
7*5
7.0
4.3
3.1
5.5
3.8
5.0
£~£S§|§g®
S»£«vdirf•/
.71
1*1
*83
8.2
*70
1*5
2.2
2.5
2.9
1.5
2*3
4.0
2.5
3*7
3*9
3.1
1.4
1.3
.55
1.4
0
.65
1*3
2.1
.20
<*
■w*
m
7*1
6*9
7.9
7.8
6.9
8.0
**
7.2
7.6
7.3
7.9
4*0
5.7
**
.85
2.5
2.0
1.5
1.2
1.4
e*
4*
3.3
2.8
*14
1.0
1.5
.88
180
Table 67
COMPARISON OP OBSERVED DIFFERENCES BETWEEN GROUPS P^ and Pg
IN PERCENTAGE OF CORRECT RESPONSES TO EACH ITEM
IN FORM A, 1935, AND THE STANDARD ERROR OF THE' DIFFERENCES
Item
I
®
3a
to
6
0
4
8
€
7
8
9
10
11
18
13
14
15
16
17
19
80
21
22
23
24
25
26
27a
b
28
29
30
31
32
34a
b
q
d
Observed Difference
6
10
28
0
6
0
28
40
12
10
24
26
38
42
12
80
34
38
24
28
44
40
38
32
52
28
26
70
34
56
02
40
30
4
26
26
6
4
14
8
10
S.E.{diff.)
7.7
6.0
6.7
4,2
3,1
4.8
8,5
8,0
5.1
9.3
7.9
7.5
8.2
8.7
5.2
7.9
8.4
8.3
7.0
8.5
3.4
8.7
8.4
8.1
8.1
8.2
7.9
6.5
6.7
7.5
7.1
6.5
5.9
3.2
6.6
6.1
3.7
4.6
7.2
6.3
6.9
.78
1.7
3.1
1,4
1.9
0
3.3
5.0
2.4
1,1
3.0
3.5
4.'S
4.8
2.3
2.5
4*0
4.6
3.4
3.3
5.2
4«'6
4.5
4.0
6*4
3.4
3.3
10.8
5.1
7.5
8.7
6.2
5,1
1.3
3.9
4.3
1.'6
.87
1.9
1.3
1.4
131
Table ©7 {continued)
X test
Difference
S.B.(dlff*)
Difference
8*E*{diff.)
b
e
56
57 a
b
50a
b
c
b
e
46
41
42
45
44
16
28
20
12
20
6
2
f.
2*2
8*4
9.2
9*5
8 #2
©•6
5*5
2.2
1*3
3.9
2*4
8*6
8*5
7*5
8*2
4*2
7*2
4.7
2.4
1*6
2*4
1.4
.28
Table 68
COMPARISON OP OBSERVED DIFFERENCES BETWEEN GROUPS Px and Px j
IN PERCENTAGE OF CORRECT RESPONSES TO EACH ITEM
IN FORM B* 1935, AND THE STANDARD ERRORS OF THE DIFFERENCES
1 teia
Observed Difference
1
2
3
4
5
6
2
2
03
32
8
18
2
0
24
32
18
0
14
0
4
12
8
9
10
11
12
13
14
15
1©
S.E.{dlff*)
8*6
7.1
9.0
9.2
8.7
9.3
7.8
7.7
9.2
9.0
9 #1
9.5
9.3
9.0
7.8
4.9
uxx x or1
SZSTTHi
.23
.28
2 .9
3.5
.92
1.9
•26
0
2.6
3.6
2.0
0
1.5
0
.51
2.4
S .S S
Table 68 (continued)
Item
17
18
19
SO
21
22
23
24
25a
b
26
27
28
29
30
31
32a
b
o
33
34a
b
c
38a
b
c
36
37
38
39
40
41
42
Observed Difference
24
18
16
10
22
12
22
30
26
6
2
12
0
8
6
22
**
m
14
16
38
16
**
22
8
16
10
14
4
12
0
S*E.(diff*)
9*1
8*6
5*1
6.0
8*4
8*4
8*2
7*3
8*8
7*3
4*7
7.6
1*8
3*8
5*3
7*8
PMtwmse
SeETTWIfFr)
2*6
2.1
3*1
1*7
2*6
1*4
2*7
4*1
3*0
.82
.43
1.6
0
.53
1.1
2.8
*•*
dr
#*
d»
«*»
*»
9.1
9 .1
8.7
5.8
1.5
1*8
4*4
2.8
CM
dr
dr
8*9
5*8
6*9
5.0
8.6
2*6
5*6
3.9
m
2*5
1.4
2.3
2*0
1*6
1*5
2.1
0
133
Table 69
COMPARISON OF OBSERVED DIFFERENCES BETWEEN GROUPS P-ji and Pg
IN PERCENTAGE OF CORRECT RESPONSES TO EACH ITEM
IN FORM B, 1935, AND THE STANDARD ERRORS OF THE DIFFERENCES
Item
1
2
3
4
5
6
7
8
9
10
11
12
13
14
IS
16
17
18
19
20
21
22
23
24
23a
b
26
27
28
29
30
31
32a
b
c
33
34a
b
0
35a
b
c
36
Observed Difference
2
3
0
a
4
4
14
0
12
2
16
4
16
12
2
4
6
4
0
6
2
12
10
4
12
20
10
4
8
22
2
10
*■
m
mm
4
20
4
8
3 «E.(d
8.7
6.0
8.2
8.2
7.8
8*9
6.5
7.0
8.4
8.8
9.7
9.7
8.4
8*4
7.5
3.6
8.7
9.1
7.2
7.8
9.4
9.0
9.5
9.4
9.4
8.8
6.5
8.9
4.5
7.0
6.5
4.9
p 1
Difference
•23
1*3
0
•98
•51
•45
£•2
0
1*4
.23
1.6
•41
1.9
1.4
.27
1.1
•69
•44
0
•77
•21
1.3
1.1
.43
1.3
2*3
1.5
*45
1.8
3.1
•31
2.0
m
mm
mm
9.7
7.1
8.9
8.1
4*
mm
m
mm
2
4
9.0
7.2
.41
2.8
.45
*99
.22
.36
134
Table 69 (continued}
Item
3?
38
39
40
41
42
Observed Difference
S.E.(dlff.)
a
2
14
O
2
0
8.2
6.6
9.4
3.8
7.2
3.9
.24
.30
1.5
0
.28
O
Table 70
COMPARISON OP OBSERVED DIFFERENCES BETWEEN GROUPS.
and Pg
IN PERCENTAGE OF CORRECT RESPONSES TO EACH ITEM
IN FORM B, 1935, AND THE STANDARD ERRORS OF THE DIFFERENCES
Item
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25&
b
26
27
Observed Difference
4
10
26
40
12
14
12
0
36
30
34
4
30
12
6
8
18
22
16
16
20
24
32
34
38
26
12
16
s*E*{diff•)
8*5
6*5
9*1
8*5
8.4
9*5
6*3
7*8
8*6
9*1
8.9
9*5
8«?7
8*5
8*0
5.5
9*3
8*6
5*1
6*8
8.1
8*3
8*3
7*6
8*4
8*4
5.9
7.9
Difference
S.E*(dI:
*47
1 *5
2.9
4*7
1*4
1*5
1.9
0
4*2
5.3
3*8
*42
3.4
1.4
*75
1*5
1.9
2*6
3*1
2*4
2*5
2*9
3 .9
4*5
4*5
3 *1
2.0
2.0
£85
3?&ble 70 (continued)
s,E.(aiff.)
Item
Observed Difference
28
29
30
31
32a
b
0
33
34a
b
c
35a
b
e
36
37
38
39
40
41
42
8
24
8
32
4*1
6*5
S.4
6.6
2*0
3*7
1*5
4*8
-
to*
w
18
56
42
24
*
to*
20
18
14
12
28
4
14
0
«*
to*
to«
to*
9*1
7*8
8.5
6*4
2.0
4*6
4.9
3*8
to*
to*
8.8
6*8
6*9
5.2
9.0
2*7
6*0
3*9
to*
2.3
1.9
2*0
2*3
3*1
1*5
2*3
0
Table 71
COMPARISON OF OBSERVED DIFFERENCES BETWEEN GROOTS P-j_ and Pj|
IN PERCENTAGE OF CORRECT RESPONSES TO EACH ITEM
IN FORM C, 1935, AND THE STANDARD ERRORS OF THE DIFFERENCES
Item
Observed Difference
1
2
3
4
5
6
7
8
9
10
11
12
8
20
IS
10
8
10
8
8
14
30
s*E*(aiff.)
8.7
5*2
9*3
9*5
9*0
9*7
6*0
5.8
3.8
9*3
7.4
Difference
1*4
1*5
3*0
1*9
1.1
.82
1*7
1*4
2.1
1*5
4.1
1LSS
lable 71 (continued)
Item
12
13
14
15a
t>
e
16
17
18
19
20
21
22
23
24
25
m
27
28
29
30
31
32
33a
b
c
a
34
35
36a
b
37
38
39
40
41
42
43
44
Observed Difference
S.H.(dif£.)
PM£er|R.go
8
16
24
22
2
18
16
8
12
10
16
32
14
16
2
8
10
6
14
12
18
6
22
IB
IB
16
28
16
28
7*7
8*3
7*2
5*8
6.1
7*8
8*8
3*8
9*3
9*0
9*3
9*0
8*5
7*0
3.1
4*1
9*2
5*0
5*2
7*1
4* 4
8*0
9*4
3*1
8*1
9.1
7.7
7.7
8*7
m
•»
im
9.7
7*3
8*3
4*6
8.2
4*9
5.7
4*9
1.9
1.2
12
4
26
S
IB
14
4
6
2
1*0
1*9
3.3
3.8
.35
2*5
1*8
*91
1.2
1.1
1*7
3.6
2.2
2.3
.65
.49
1.1
1.2
2.7
1*7
2*7
1.0
2.2
2.2
2.2
1*8
3.6
2.1
3.2
esq
.<oO
3.1
1*8
2.2
2.9
*70
1*2
1*1
137
Table 72
COMPARISON OF OBSERVED DIFFERENCES BETWEEN GROUPS Pji and Pg
IN PERCENTAGE OF CORRECT RESPONSES TO EACH ITEM
IN FORM C, 1935, AND THE STANDARD ERRORS OF THE DIFFERENCES
Item
1
2
5
4
5
6
7
3
9
10
11
12
13
11
15a
"cb
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
53a
b
cs
a
34
35
Observed Difference
2
2
12
24
6
20
4
2
a
10
24
28
8
0
2
4
82
0
22
2
28
a6
6
2
6
16
10
26
IS
14
14
166
2
10
4
16
6
S.2.(diff.)
8*6
4*3
9.3
8.4
9*1
9*2
5*2
4.0
5*8
9*5
7*4
7*7
6.4
4*7
1.9
5*3
07*8«4
8.4
8*2
8.5
9.2
9*8
8*3
8*9
4.3
5.4
9*3
6 *9
8*4
9.0
7*3
7 *9
8*2
5*9
6*7
8.3
4*7
9*3
9*8
^ § " ~fdfe?^')
*23
,47
1,3
2.9
.66
2.2
,77
,50
2*1
1*1
3*2
3*6
1*3
0
1*1
*75
1*3
*26
0
2*7
*24
3*0
.82
*72
*67
.47
1*1
1*7
1*4
3*1
2.0
1.9
1*8
2.0
1*0
.30
1*2
.85
1.7
.61
138
Table 72 (continued)
Item
36a
to
37
38
39
40
41
42
43
44
Observed Difference
12
6
18
8
8
10
16
20
4
S.E*(diff*)
9*7
8*1
9*6
6*8
9*2
7*7
7*5
7*4
3*8
Difference
s.S.fdiff*;)
1*2
*74
1*9
1*2
.87
1*3
2*1
2*7
1*1
Table 73
COMPARISON OP OBSERVED DIFFERENCES BETWEEN GROUPS Pj_ and Pg
IN PERCENTAGE OF CORRECT RESPONSES TO EACH ITEM
IK FORM C, 1935, AND THE STANDARD ERRORS OF THE DIFFERENCES
Item
1
2
3
4
3
6
7
8
9
10
11
12
13
14
15a
to
c
16
17
18
19
20
Observed Difference
10
6
40
42
4
12
6
10
0
24
54
36
24
24
20
6
10
18
8
34
12
44
3*K*(diff*)
8*8
5*6
9*1
8*3
9*3
8*9
6*6
5*4
5*4
8*6
7*4
8*4
7*7
7*2
6*1
5*6
8*0
8*6
8*8
8.5
8*9
8*4
iiirrerei
&# & •(dl:
1#1
1 *1
4* 4
5*1
*43
1*5
*91
1*9
0
2*8
7*3
4*3
5*1
5*3
3.5
1.1
1 *e
2 *1
*91
4*0
1 *3
5 *2
139
T a b le
Item
21
22
23
24
25
26
27
28
29
30
31
32
33a
b
©
&
34
35
36a
b
37
38
39
40
41
42
43
44
75
( c o n tin u e d )
Observed Difference
40
20
22
4
5
26
16
40
30
26
20
38
24
20
20
32
32
34
s,E.(diff.)
8,8
7.0
7.3
3.6
5.3
8,8
6.1
6,8
7,7
5,8
7,3
8,5
7,3
7,9
8,5
7,2
8.1
8,9
m*
24
10
44
10
26
24
20
26
0
§-£--f£vB£9o • \fixfr•/\
4.5
2.9
3,0
1,1
1.5
3.0
2,6
5.9
3,9
4,5
2.7
4.5
3.3
2.5
3.1
4.4
4.0
3.8
*»
9*4
^*7
8.2
5,7
8.4
6.3
7.1
6.5
3,3
2*6
1*3
5.4
2.8
3.1
3.8
2.8
4.0
1.8
190
Table 74
COMPARISON OP OBSERVED DXPFEE.BKCES BETWEEN GROUPS
and
IH PEHCKNTAOE OF CORRECT RESPONSES TO EACH ITEM
IB FORM A, 1936, Af W THE STANDARD ERRORS OF THE DIFFERENCES
Item
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
26
27
28
29
30
31
32
33
34
35
30
37
58
40
41
Obaerred Difference
5
22
1
1
26
14
4
5
12
14
10
10
6
14
7
18
20
7
0
16
19
7
12
52
28
23
16
32
22
43
39
43
19
25
15
25
17
6
15
18
20
S.E*(diff*)
4*1
6*1
2*9
3*5
5*7
5*0
4*7
5*2
6*6
5*7
5*9
5,9
0,4
6,1
6*0
6*5
5*9
2*9
6,1
6*3
6*6
6*7
5*2
8*2
6*3
5*1
6*6
5*9
6*1
5*9
5*7
5*8
6*3
6*5
6*8
6*5
6*3
b *3
5*0
5*3
4*9
Difference
S.E.fdIff*)
1*2
3*0
.34
*29
4*6
2*8
*85
*58
1.8
2,5
1.7
1.7
,94
2.3
1*2
2.8
3*4
2.4
.98
2.8
2*9
1*0
2*8
5*2
4.4
4*8
2*4
5*4
3*6
7*3
6.8
7*4
3*0
3*8
2.6
3*8
2*7
1.1
3*0
3.4
4.1
19:1
T&ble 74 (contlnuod)
Item
Observed Difference
42
43
44
45
46
47
48
49
50
15
31
24
9
7
18
6
8
14
S.E,(diff*)
4.1
6*8
4.7
3*0
3*0
3.5
2*5
2*9
5.8
Difference
o *hi*{(3if f .)
3.7
4*9
5*1
3.0
2.3
3.7
2*6
2*8
2,4
Table 75
COMPARISON OF OBSERVED DIFFERENCES BETWEEN GROUPS P^.3, and Pg
IN PERCENTAGE OF CORRECT RESPONSES TO EACH ITEM
IN FORM A, 1930, AND THE STANDARD ERRORS OF THE DIFFERENCES
Item
Observed Difference
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
2
4
1
4
1
1
6
7
20
5
5
7
17
14
7
22
2
2
7
0
18
11
6
S.E.(diff,)
3.3
5*4
2.9
2.7
4.6
3*8
3.8
4, 5
5.8
5*2
5,2
5,0
5.4
4,7
5*4
5.1
5,2
2,0
5.7
5,9
6,3
6,6
4,8
-uirrere*
S.E* {dlj
.61
.74
*34
1*5
*22
,26
1,6
1,6
3*4
*53
.96
1.4
3*1
3.0
1.3
4*3
♦38
1.0
1*2
0
2.9
1.7
1 *3
192
Table 75 (continued)
Item
04
25
2$
27
28
88
S0
31
m
33
34
33
3©
37
m
39
40
4%
42
43
44
45
46
47
48
40
50
Observed Difference
10
5
17
01
3
5
10
8
19
17
5
01
15
23
18
20
34
24
©
00
It
13
10
13
12
5
7
S*E*(diff.)
3*9
5*3
©*©
5*9
4*6
3*0
6*1
6*3
4*7
6*7
6.8
6*3
6*0
6*5
6*4
6*4
6*3
©*S
5*7
8*5
©*©
5*0
4*8
5*5
4.4
4*2
6.5
^flCSXSl
S.E,{dt:
1.7
*94
2*6
3*6
*63
1*0
9*0
1*3
4*0
0*5
*74
3*2
2.5
3*5
2.8
3.1
5.4
3*7
1*1
3*1
2.9
2.6
2.1
2.7
2.7
*71
1*1
Table 76
COMPARISON OF OBSERVED DIFFERENCES BETWEEN GROUPS
and Pg
IS PERCENTAGE OF CORRECT RESPONSES TO EACH ITEM
IN FORM A* 1936, AND THE STANDARD ERRORS OF THE DIFFERENCES
Item
Observe© Difference
1
9
7
96
0
8
4
8
©
3
27
15
s.K„(diff.)
3,9
5.9
2.7
2*9
5*6
4.0
Difference
1*8
4.4
0
1*7
4*8
3*1
193
Table 70 (continued)
Item
7
a
9
10
11
12
IB
14
IB
16
17
18
19
20
2%
22
2B
24
25
20
27
28
29
50
31
32
33
34
50
38
37
38
39
40
41
42
43
44
4S
40
47
48
49
50
Observed Difference
10
10
32
11
15
17
23
28
14
40
22
5
13
16
37
18
6
42
S3
40
37
35
27
55
47
62
36
20
36
40
40
24
38
52
44
21
81
43
22
17
28
18
11
21
$*B.(aiff.)
4.1
4*7
5*9
8*8
5*7
5.5
5.6
8*8
5*7
5*4
8*8
3.2
5*8
6*4
6*2
6*5
5*6
5*9
6*0
5*3
6.0
5*7
5*8
k *f
»
O.
5*6
4*7
6*2
6*6
8*9
6.2
'6.1
5*7
8*7
5.8
5*3
4*6
8*9
5*1
4.0
4.1
4*4
3*8
3.3
6.0
Difference
STKfdiff*)
2*4
2*1
8*4
1*9
8*8
3*1
4*1
6*4
2*5
7*4
3*8
1*6
2*8
2*5
6*0
8*8
1*1
7*1
5*0
7*5
6.2
6*1
4*7
9*6
8*4
13*2
5*8
3*0
6*1
0*5
0*6
4*2
0*1
9*8
8*3
4*0
8*0
8*4
5*5
4.1
6*4
4*7
3*3
3*5
194
Table 77
COMPARISON OF OBSERVED DIFFERENCES BETWEEN GROUPS P^ and P ^
IN PERCENTAGE OF CORRECT RESPONSES TO EACH ITEM
IN FORM B* 1936* AND THE STANDARD ERRORS OF THE DIFFERENCES
Item
Observed Difference
%
1
2
3
4
B
6
7
5
4
1
30
a9
2
11
21
10
11
12
13
14
15
16
17
ia
19
20
21
22
23
24
25
26
27
28
29
30
31
32
35
54
35
36
87
38
59
40
9
18
21
11
28
4
15
82
8
5
7
1
7
IS
1
11
13
4
30
12
1
11
15
10
11
IS
7
13
•*b *{diff*)
3 #7
4*5
5*1
5*2
4*5
6*4
5*9
5*5
6*2
6*4
6.5
6*2
6*5
6*0
6*2
4*7
6*0
6*5
5*8
6*7
5.8
6.6
4*2
6*5
6*3
5*5
6.9
6*8
6*9
6*2
6*8
4*2
5*6
5*3
6*0
4*2
6*3
5*9
3*7
4*5
>61100
S.E.tdirn)
*27
*44
1*6
•77
•22
4*7
1*9
8*8
0
2*3
1*4
2*9
3.2
1.8
4*5
*85
2*5
4*9
1*6
*43
1*2
*15
1*7
2*3
5*1
*18
1*6
1*9
*58
4*8
1*8
*24
2*5
2*1
2*2
2*4
1*7
5*8
1*9
3*0
195
Table 77 (continued)
Item
41
42
43
44
45
46
47
48
49
50
Difference
s. hi.(diff.)
Observed. Difference
3
6
2
8
X
0
2
6
1
4
*91
1*8
*74
3*3
3*4
2.7
3*4
3*1
5*7
2*1
2*9
1*0
1*8
*32
0
*95
2*1
1*0
2*2
Table 78
COMPARISON OP OBSERVED EtPFERE/X-ES BE W E E N GROUPS Px£ and Pg
IN PERCENTAGE OF CORRECT ilSSPOHSES TO EACH ITEM
IN FORM B, 1936, AND THE STANDARD ERRORS OP THE DIFFEKEtfCES
Item
12
3
4
a6
7
8
9
10
11
12
13
14
15
16
17
18
10
20
21
22
Observed Difference
62
18
7
11
13
5
11
17
18
9
12
11
18
8
17
15
2
14
12
15
S.E*(dlff
2*7
4.6
2*1
4.3
3*9
5*6
4*7
3*9
5*7
6*8
5*3
5*2
5*3
4 *0
6,1
3*5
4*6
6*0
5,3
6*1
4*6
5,9
*J
Differs
TiiTm
2*2
*44
*48
1*9
1*8
2,0
2*8
1*3
1*9
2,5
3*4
1*7
2,5
2.3
3*0
2*3
3*7
2*5
*38
2*3
2*6
2*5
-a
Table 79 {continued}
Item
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
£7
28
29
30
31
32
33
34
35
36
37
38
89
40
41
42
43
44
45
46
47
48
49
50
Observed lifference
41
2 4
26
11
82
27
rvr/
rv
5 t>
22
46
12
32
47
11
17
19
16
51
«.>o
38
9
29
24
28
52
84
22
4-4
36
44
89
37
47
27
54
58
52
24
52
25
18
16
15
8
16
S.lMdtff.)
6 ,0
5 *2
5*1
5*6
6 *4
5 *6
5 *8
5 *9
5*3
5*8
8*9
5*4
6 *0
5*7
6*2
5 *0
6*0
5*5
6 .2
6*0
4 .9
6 .4
6*4
6*3
5*1
6 .2
5*5
5*9
■5*8
6 ,1
5 *4
6*3
5 .1
5*0
5 .2
5*1
4 *7
4 *6
4 .7
5*0
8*0
4*2
3 *5
2*7
3*5
Mlgerence
3 * Aii* (d 12i?V)
6*8
4 .6
b « J2 .0
5 ,0
4*8
4 .7
5 ,6
4 .2
7 .9
3 ,1
5 .9
7 .8
1 .9
2 .7
3 .8
2 .7
5 *6
5 .3
8 *5
1*8
4 ,5
3 .8
4 ,4
1 0 .2
5 ,5
4, 0
7 ,5
6 .2
7 ,2
7 ,2
5 ,8
9 .2
5 *4
1 0 .4
7 .5
6 ,3
5 .2
6 ,3
5 ,0
3 .0
3 *8
3 .7
3*0
4 *6
±96
Table 78 (continued)
Ite m
Differen.ee
O b s e rv e d
23
24
05
26
27
28
29
30
31
32
55
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
24
18
6
10
IS
11
24
02
02
21
30
25
31
29
26
32
00
41
55
26
22
24
20
18
18
7
7
12
C O M P A R IS O N
IN
FORM
Ite m
1
2
3
OF
OBSERVED
PERCENTAGE
B,
1936,
OF
AND
O b s e rv e d
t*
**>*> \
tauf.,
Difference
5
- [ Z
5 *8
6*5
5 *4
5 *0
6*3
6*3
6*1
4 *8
6 .2
5*5
6 ,1
6 ,5
6 ,4
6 .1
6 .7
6 ,1
5 ,5
5 *9
5*4
6*2
4*8
5*4
4*9
6 *1
4*1
4*3
2 ,8
4*0
T a b le
IB
S .E *
T IE -
D iffe re n c e
7
O
6
_
. )
4 .1
2,8
1*1
2*0
2*9
1*7
3*9
4*6
3*5
3.8
4*9
3.8
4.8
4.8
3.9
5 .2
3.6
6*9
6.5
5*0
4*6
4*4
5.3
3*0
4 .4
1*6
2*5
3.0
BETWDDN G R O U PS
RESPONSES
STANDARD
_
79
D li< F E K K K C S S
CORRECT
- T _
ERRORS
TO
EACH
OF
THE
and
Pg
IT E M
D IF F E R E N C E S
S * E * (d iff* )
)
2 .8
4*3
3*0
2 .5
0
2*0
4
12
4*6
2.6
5
6
4 .0
1*5
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