AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY 77:367-376 (1988) Distribution of Epidermal Ridge Minutiae DAVID A. STONEY Department of Criminal Justice, University of Illinois at Chicago, Chicago, Illinois 60680 KEY WORDS Pattern, Fingerprints, Dermatoglyphics ABSTRACT The distribution of epidermal ridge minutiae on the distal portion of male human thumbprints has been characterized. For each of 412 thumbprints, a centrally located focal minutia was chosen and neighboring minutiae were sampled. Minutiae were considered to be neighbors if there were no other minutiae appearing in the intervening region defined by the two minutia events and the ridge system. For each neighbor minutia, the total ridge distance between the focal and neighbor minutiae was measured. This distance is the total length of ridges appearing in the intervening region. The number of neighbor minutiae occurring about the focal minutia was found to be normally distributed with a mean of 6.42 (n = 412). The distribution of total ridge distances was not adequately described by the negative exponential distribution, but was well described by a gamma distribution with a shape parameter of 0.193 and a scale parameter of 5.91 mm. This gamma distribution reflects a local overdispersion of minutiae. This study is noteworthy as the first to describe the distribution of epidermal ridge minutiae within the ridge structure. The results support prior work based on quadrat sampling and eliminate two possible sources of error. A possible explanation for the overdispersed distribution lies in the growth stress model for minutia formation. Minutia formation may alleviate local growth stress, thereby removing the impetus for formation of additional minutiae in the immediately surrounding region. Epidermal ridge structures occur on the volar surfaces of all primate species and on some marsupials (Montagna and Parakkal, 1974). Two features of the epidermal ridges are of interest here: 1) the patterns that are formed by groups of ridges and 2) the irregularities and characteristics of the individual ridges, known as minutiae. The present work describes the distribution of epidermal ridge minutiae observed on the distal tips of male thumbs. Locally, epidermal ridges are parallel to one another, but over larger regions the ridges flow in groups and form patterns, known technically as dermatoglyphics, but familiar to most as “fingerprints.” In addition to general convergence and divergence of ridges, two fundamental types of pattern singularities occur: loops, where a group of ridges turns back upon itself, and triradii, where ridges flow outward from one point in three directions. Combinations of these two 0 1985 ALAN R. LISS, INC. pattern singularities create a variety of local pattern environments. Close inspection reveals that individual epidermal ridges do not flow continuously, but often branch or terminate within the confines of the general ridge pattern. These discontinuities in individual ridges are minutiae. Minutiae can take a variety of forms, but the most common types are ridge bifurcations and ridge endings (see Fig. 1). Dermatoglyphics, the epidermal ridge patterns, have been the subject of intensive investigation. Major fields of study have included their comparative anatomy (Cummins and Midlo, 1943; Newell-Morris, 1979), genetics (Holt, 1968; Roberts, 1979; Loesch, 19831, anthropology (Steinberg et al., 1975; Plato et al., 1975; Roberts, 1982), topology Keceived September 14, 1987; accepted April 4, 1988 368 D.A. STONEY (Penrose, 1965,1979; Smith, 1979), utility for medical diagnosis (Schaumann and Alter, 1976), and utility for fingerprint record retrieval (Cowger, 1983). In spite of this extensive and long-standing interest in dermatoglyphics, there has been only minimal study of epidermal ridge minutiae. Systematic studies with a biological focus have been rare (Okajima, 1967a,b, 1970; Loesch, 1973; Dankmeijer et al., 1980; Liu, et al., 1982; Loesch, 1983,1984; Okajima and Usukura, 1984). A number of other studies by forensic scientists and statisticians are also of interest (Stoney and Thornton, 1986a), but the spirit behind these investigations has been justification of fingerprint comparison practices rather than of investigation of minutia variation per se. In all prior work, minutiae are treated a s point occurrences, without reference to the ridge structure in which they are found. The result is that the distribution of minutiae has never been adequately described within the ridge structure. The present work is a n initial effort to describe this distribution. For each print, a centrally located focal minutia is selected and the distribution of total ridge distances to neighboring minutiae is characterized. Total ridge distance is the total length of ridges occurring between the two minutiae, as fully defined below. MATERIALS AND METHODS Sample material Data were collected from inked fingerprints on file at the Contra Costa County Criminalistics Laboratory in Martinez, California. Right and left thumbprints of adult males were examined from sequential fingerprint cards in a file sequenced by date of printing. The cards were screened for clarity of printing of right and left plain thumbprints a t the bottom of the standard fingerprint card (Cowger, 1983). Plain prints approximate a planar projection, as opposed to the cylindrical projection of rolled prints. Sampling was restricted to male thumbprints in order to avoid substantial discrepancies in finger size. A total of 412 thumbprints were sampled: the right and left thumbprints from 206 individuals. The mean age at time of printing was 26.5 years with a range of 18-56 years and a standard deviation of 6.5 years. According to the racial classifications on the cards, there were 147 whites, 56 blacks, and 3 Asians in this sample population. No at- tempt was made to verify these racial classifications. Sample area The distal tips of digits were chosen as the sample area for two reasons. First, the geometry of the ridge flow seen in this region is similar for all dermatoglyphic pattern types. The ridges conform to the digital outline, flowing in a n arch from one side of the finger to the other. Comparable ridge geometry within the sampling region is required in order to use the general ridge flow as a reference for minutia position. The second reason for sampling from the distal tip region was to avoid pattern singularities. These singularities lie proximal to the sampling region on the distal phalange. Minutia density is known to increase significantly about the pattern singularities (Kingston, 1964). Selection of focal minutiae For each print, a centrally located focal minutia was chosen using the following procedure (see Figs. 1, 2). A reticle with a circle of 5-mm radius was placed over the print while viewing the print under a stereobinocular microscope. The reticle was placed so that the outline of the circle coincided with the upper portion of the ridge most closely corresponding to the circle curvature. This ridge was designated the 5-mm ridge. While holding the reticle in place, the center of the circle was marked with a fine pin prick. This point was designated the origin. Typically, the origin was in the area of the print where the pattern singularities occur. Epidermal ridges show their highest degree of curvature as they round the apex of the finger, flattening out as one proceeds laterally. The point of maximum distal extent on the 5-mm ridge was thus easily noted and marked with a second pin prick. A second reticle, marked with angular measure, was placed with the vertex at the origin (see Fig. 2). The search for a focal minutia was conducted beginning on the 5mm ridge. The minutia closest to the point of maximum distal extent was selected if one was within 45". If a minutia was not found within 45", then the search was continued on successively more distal ridges within the total 90" arc until a minutia was encountered. Limiting the search to within the 90" arc ensured that the focal minutia would be on the distal portion of the finger. Searching on sequential ridges introduced some varia- EPIDERMAL R1:DGE MINUTIAE 369 tion in the position of the focal minutia relative to the 5-mm ridge. The intent of selection, however, was not to define a n invariant point within the pattern, but rather to choose a centrally located minutia in a consistent and unbiased manner. ab- Data collection Pattern type. The general pattern type of each print was noted, following Loesch's criteria for arches, radial loops, ulnar loops, and whorls (Loesch, 1983). Ridge density. After selection of the focal Cminutia, ridge density was measured along the direction defined by the origin and the focal minutia. The distance between the ridge located five ridges below the focal minutia and that found five ridges above the %&-+w * . focal minutia were measured (see Fig. 3). Minutia density. Minutia density was Fig. 1. Placcmcnt of the reticle on print to be sammeasured by sampling all minutiae in a cirpled. The 5-mm diameter circular reticle is placed so cular region of 3-mm radius, centered one that the outline of the circle coincides with the upper portion of the ridge most closely corresponding to this ridge directly above the position of maxidegree of curvature. Minutiae labeled (a), (b), and (c) mum distal extent. This position was chosen represent the three fundamental types of ending ridge, because the average focal minutia position fork, and dot, respectively. was not on the 5-mm ridge, but a t 0.91 ridge intervals above it. Sampling of neighbor minutiae. Once a focal minutia was chosen, positions of neighd boring minutiae were sampled using the an\ C gular reticle (see Fig. 4).To be sampled a minutia had to meet three criteria: 1)appear within a 45" angle of the focal minutia, 2) be within a count of six ridges proximally or e nine ridges distally, and 3) be a neighbor f b minutia, as defined below. Minutiae outside the 45" angle, or more distal than nine ridge intervals, approached a the border areas of the print. These areas could not be sampled on all prints because of the variable extent of printing. Minutiae appearing more than six ridges proximal to the focal minutia approached the pattern area where ridges could not be reasonably approximated a s circular about the origin. Ridge counting was performed in the direction perpendicular to the ridge flow, beginFig. 2. Selection of the focal minutia. After placement ning at the focal minutia. Each ridge of the rcticle, the direction showing maximum ridge interval was counted from the focal minutia curvature (d) defines the point of maximum distal extent to the ridge on which the neighbor minutia (fJ.Thc focal munutia is selected from within a 90" sector of this point with vertex at the origin (a).Lines (c) and appeared. Counts to bifurcating ridges were (e) define the sector limits. A minutia is first sought on made to the nearest branch of the bifurcathe fi-mm ridge. If no minutia is found on the 5-mm tion, and counts to ending ridges were made ridge, successively more distal ridges are searched until as if the ending ridge extended past the a focal minutia is found. If two rnunitiae appear on the counting line. ridge, the one closest to the axis (d) is taken. In this Only neighboring minutiae were sampled. example, the minutia at (b) is selected as the focal Minutiae were defined as neighbors when minutia. ' ,>L 1 370 D.A. STONEY examination of fingerprint patterns (Stoney and Thornton, 1986b). Ambiguities in ridge counts arise when minutiae are not neighbors, because of the effect of the intervening minutiae. For each neighbor minutia, the ridge count and the angular distance from the focal b minutia were recorded. Ridge counts ranged from six ridges proximal to the focal minutia to nine ridges distal to the focal minutia. a Ridge counts of zero represent minutiae that appear on the same ridge as the focal minutia. A full discussion of the ridge counting procedure has been previously described (Stoney and Thornton, 1986b). Calculation of total ridge distance. A single measure of distance between the focal and neighbor minutiae was obtained by calculating the total length of ridges appearing in the region between the two minutiae. For Fig. 3. Measurement of ridge density. Ridge density is measured along the direction defined by the origin (a) a ridge count of C, there are C + 1 ridges and the focal minutia (b). The distance (c) is measured appearing between the two minutiae. The between the ridge five ridges below the focal minutia distance along each of these ridges must be and the ridge five ridges above the focal minutia. summed to get the total ridge distance. The primary data for calculating this distance are the ridge count, the angular distance to the neighbor minutia, the radius of ridge curvature at the focal minutia, and the ridge density. Given the ridge density, the radii of curvature Ri for each ridge are calculated, using Equation (l), where Ci is the ridge count (distally) from the focal minutia, Ro is the radius of ridge curvature at the b focal minutia, and B is the ridge density. The total ridge distance is given by sum-a ming Equation (1)over the integral values of Ci, ranging from zero to the ridge count C. (Note that proximal ridge counts give a negative value of Ci since the radius of curvature is decreasing.) ic Ri = Ro + CiB. (1) Statistical methods Standard deviations are given as the Fig. 4. Selection of neighbor minutiae. Neighbor minutiae (circled) are selected from within a 90" sector square root of the adjusted sample variance. centered on the focal minutia (d) with vertex at the Chi-square tests were used to test hypotheses origin (a). Lines (bl and (e) define the sector limits. of homogeneity and goodness of fit. Equality Neighbor minutiae are sampled where the ridge count from the focal minutia is within the range of six ridges of variance was tested using the F-ratio, and the Z statistic was used to test for equality of proximally to nine ridges distally. means. Parameters for the gamma distribution were estimated by the matching mono other minutiae appeared in the region ments method (Hastings and Peacock, 1975), defined by the ridge flow and the perpendic- and expected frequencies for the gamma disular ridge counts from each minutia to the tribution were calculated by linear interpoother. Pairs of neighboring minutiae are the lation from Pearson's tables of the incomplete fundamental unit of variation in the forensic gamma function (Pearson, 1965). EPIDERMAL RIDGE MINUTIAE 371 Distribution of Number of Neighbor Minutiae 100 7 , I 0 2 i 6 i I , 1 0 1 2 1 4 NUMBER OF NEIGHBOR MINUTIAE 5 Fig. 5. The distribution of the number of neighbor minutiae per print. The line follows the expected values for a normal distribution with mean 6.42 and standard deviation 1.76. RESULTS minutiae per square mm for the mean. Given the mean ridge density, the estimated density of minutiae o n the ridges is 0.103 minutiae per mm. The standard deviation, calculated using the individual ridge densities for each print, was 0.0429 minutiae per mm. Variation of density with pattern type is of possible concern. In the present sampling, however, identical mean density values were obtained for the 225 loops and for the 174 whorl patterns (0.222 minutiae per square mm). There was also no significant variation in minutia density between the right and left hands, with mean values of 0.217 and 0.228 minutiae per square mm, respectively (equality of means accepted a t 5% significance with nl and n2 of 206 and a Z value of 0.12, P = ,941. Frequencies of pattern types Although the dermatoglyphic patterns are not the principal focus of this study, the representatives of the sample with respect to pattern type is of interest. This sample of 412 thumbprints contained 223 ulnar loops, 174 whorls, 13 arches, and 2 radial loops. This distribution of pattern types does not differ significantly from the expected distribution, given the racial composition of the sample (Stoney and Thornton, 1987). Ridge density The mean ridge density for the 412 thumbprints was 0.463 mm per ridge. Traditionally, ridge density has been expressed in ridges per cm. This value is 21.6 ridges per cm with a standard deviation of 2.82 ridges per cm. Number of neighbor minutiae per focal minutia Minutia density The distribution of the number of neighbor For each of the 412 thumbprints, the num- minutiae per print appearing in the sample ber of minutiae occurring in the circular region is shown in Figure 5. A total of 2,645 sampling region of 3-mm radius was re- neighbor minutiae were sampled from about corded. The mean number of minutiae was the 412 focal minutiae. The distribution was 6.29, with a standard deviation of 2.61 minu- found to be normal with a mean of 6.42 and tiae. The area of the circular sampling region a standard deviation of 1.76 (normal approxwas 28.3 square mm, giving a density esti- imation accepted a t 5% significance with a mate of 0.223 minutiae per square mm with chi-square value of 4.63 and 6 degrees of a 95% confidence interval of 0.215-0.230 freedom [dfl,P = .59). 372 D.A. STONEY Frequency of Neighbor Minutiae by Total Ridge Distance 200-.- Frequency I . - i ~ ~~ -Gamma Prediction ---- Exponential Prediction 150 0 Observed Frequency 100 ti 0 0 0 5 6 10 15 Total Ridge Distance (mm) 20 25 Fig. 6. Observed TRDs of neighbor minutiae and expected frequencies under the assumptions of 1) the negative exponential distribution and 2) the gamma distribution. Although the two distributions are close for large TRD, the negative exponential distribution pre- dicts unacceptably high frequencies for the shorter distances and is rejected at high significance ( P < < ,0011. The gamma distribution gives a n acceptable fit, demonstrating that minutiae are somewhat overdispersed ( P = .37). Minutia distribution A curve following expected values for the negative exponential distribution, given the mean value of 7.05 mm, is presented in Figure 6. The negative exponential distribution is rejected at high significance (goodness-offit test rejected at 0.1% significance with a chi-square value of 311 and 60 df, P < < .001). A more generalized distribution, known as the gamma distribution, is suitable for describing a wide variety of interevent distributions (Jewell, 1960). Estimates of the gamma parameters were made by the matching moments method (Hastings and Peacock, 1975), using s2/x for the scale parameter b, and (xIsl2 for the shape parameter c, where x is the sample mean and s is the unadjusted sample variance. The estimated scale and shape parameters were 5.91 and 0.193 mm, respectively. Using these parameters, expected frequencies for the gamma distribution were determined using the incomplete gamma function as shown in Equation (3) and tabulated by Pearson (1965). The distribution of the total ridge distance (TRD) for the 2,645 sampled minutiae is shown in Figure 6. The mean TRD was 7.05 mm with a standard deviation of 6.45 mm. The distribution was found to show no significant differences between the right. and left hands and between whorl and loop patterns (see Figs. 7, 8). (Tests for homogeneity accepted at 5% significance. For right and left hands, sample sizes were 206 each, with a chi-square value of 28.3 and 26 df, P = .35. For whorl and loop patterns, sample sizes were 174 and 225, respectively, with a chisquare value of 22.9 and 27 df, P = .69.) Insufficient numbers of arch and radial loop patterns were present in this sample set to extend the test of homogeneity to these pattern types. If minutiae are positioned randomly with respect to TRD, minutia occurrence can be treated as a Poisson process and a negative exponential function should describe the distribution of distances. With a mean value of y, the frequencies of the negative exponential distribution are given by Equation (2). P(X1 < x < X2) = exp(-X2ly) - exp(-Xl/y). (2) 373 EPIDERMAL RIDGE MINUTIAE Comparison of Minutia TRD Frequencies between Right and Left Hands 200 150 Frequency I _______-- E B RIGHT (N=1336) ~ _ I LEFT (N=1309) ~ _ t--- _ _ _ ~ 7 TRD (mm) Fig. 7. Observed TRD frequencies on the right and left hands. The distrlbutions are not statistlcally different ( P = ,351. Comparison of Minutia TRD Frequencies between Loop and Whorl Patterns Frequency F Z WHORLS (N=1139) I LOOPS (N=1414) ~~~ 8 TAD (mm) Fig. 8. Observed TRD frequencies for whorl and ulnar loop patterns. The distributions are not statistically different ( P = .69). A curve following the expected values for the DISCUSSION Ridge density gamma distribution is presented in Figure 6. Observed and predicted values are shown i n The ridge density of 0.463 mm per ridge Table 1. The agreement is very good (goodness-of-fit test accepted at 5% significance obtained in the present sampling is in excelwith a chi-square value of 57.8 and 55 df, lent agreement with prior studies (Cummins et al., 1941; Kingston, 1964) as fully disP = .37). 374 D.A. STONEY TABLE 1. Total ridge distance (TRD) frequencies Interval (mm) Observed frequency 0-0.5 0.5-1.0 1.O- 1.5 1.5-2.0 2.0-2.5 2.5-3.0 3.0-3.5 3.5-4.0 4.0-4.5 4.5-5.0 5.0-5.5 5.5-6.0 6.0-6.5 6.5-7.0 7.0-7.5 7.5-8.0 8.0-8.5 8.5-9.0 9.0-9.5 9.5-10.0 10.0-10.5 10.5-11.0 11.0-11.5 11.5-12.0 12.0-12.5 12.5-13.0 13.0-13.5 13.5-14.0 14.0-14.5 14.5-15.0 15.0-15.5 15.5-16.0 16.0-16.5 16.5-17.0 17.0-17.5 17.5-18.0 18.0-18.5 18.5-19.0 19.0-19.5 19.5-20.0 20.0-20.5 20.5-21.0 21.0-21.5 21.5-22.0 22.0-22.5 22.5-23.0 23.0-23.5 23.5-24.0 24.0-24.5 24.5-25.5 25.5-26.5 26.5-27.5 27.5-28.5 28.5-30.0 30.0-31.5 31.5-33.5 33.5-36.0 36.0 114 132 144 112 147 124 126 154 99 117 95 91 88 82 86 73 70 54 64 55 53 39 45 31 41 33 25 25 29 24 21 27 16 15 Frequency predicted by gamma distribution 125.3 139.3 144.5 143.0 137.7 131.0 124.9 118.6 111.9 103.5 97.5 91.7 86.2 79.4 73.8 69.2 64.7 60.0 55.0 51.4 48.0 44.8 40.8 37.9 35.3 32.9 30.3 27.7 25.9 24.1 22.4 20.3 18.8 17.5 16.3 14.9 13.7 12.7 11.8 11.0 9.9 9.2 8.6 8.0 7.3 6.7 6.2 5.8 5.3 9.3 8.3 6.7 5.8 7.1 5.6 5.6 5.6 9.5 study is comparable to other reported values that range from 0.181 to 0.295 minutiae per square mm (Galton, 1892; Amy, 1946; Kingston, 1964; Osterburg et al., 1977;Dankmeijer et al., 1980). These mean values are not sufficiently close to be regarded as repetitive samples from the same population. The differences are probably attributable to the different sampling regions within the fingerprints and to differences in which fingers and pattern types were sampled. Full discussion of these factors is presented elsewhere (Stoney and Thornton, 1987). Minutia distribution The distribution of TRDs among neighboring minutiae has been shown to be well described by a gamma distribution. The particular distribution parameters will not be valid in other regions of thumbprints, or on other fingers, because the parameters will be affected by variations in minutia density. Acceptable description by the gamma distribution, however, is significant because it demonstrates that, given any particular minutia, there is a decreased probability of encountering neighboring minutiae in the immediate vicinity. Minutiae are thus overdispersed or more regularly spaced than would be expected from a random (Poisson) process. 11 Note that for large TRD both the gamma 22 and the negative exponential distributions 16 8 follow the observed frequencies fairly well. 12 Pielou (1969)has commented on this similar10 ity and on the difficulty that can occur in 5 distinguishing a gamma distribution from a 6 11 random one. In the present data, the large 6 sample size allows sufficiently narrow inter10 vals so that the distinction is quite clear. 2 A complementary demonstration of the 5 4 regularity of minutia spacing results from 3 comparison of the mean TRDs predicted from 11 1) the minutia density estimate and 2) the 6 estimated parameter of the negative expo6 4 nential distribution. Under the hypothesis of 6 a random minutia distribution, these two 6 mean distances are estimates of the same 5 population mean and their values should 7 12 + therefore be within sampling error of one another. In the case of a n overdispersed distribution, the estimated parameter of the cussed elsewhere (Stoney and Thornton, negative exponential distribution will be 1987). lower than the value predicted by the density (Pielou, 1974). Minutia density For the present data, the density gives a The mean minutia density of 0.223 minu- mean distance of 9.70 mm with a variance of tiae per square mm obtained in the present 6.62 mm (n = 412).The negative exponential EPIDERMAL RIDGE MINUTIAE distribution parameter gives a mean distance of 7.05 mm and a variance of 6.46 mm (n = 2,645). The assumption of equal variance is justified, and the hypothesis of equal means is rejected with high significance (equal variance accepted a t 5% significance with a n F value of 1.05 with 411 and 2,644 df, P = .29. The test of equal means is rejected a t 0.1% significance with a n F value of 1.38 with 411 and 2,644 df, P < < .001). The mean distance predicted by the assumption of random spacing is thus significantly lower than the observed mean, indicating a n overdispersed distribution. Two areas of possible bias in the sampling method need to be considered in their relationship to the findings: the method for selection of focal minutiae and the limited extent of the sample region. Focal minutiae were selected after defining the point of maximum distal extent on the ridge with curvature most closely approximating a 5-mm radius. The minutia closest to the point of maximum distal extent on the 5-mm ridge was selected if one was found within 45". If no minutia was found on this ridge, then the search continued on successively more distal ridges within the total 90" arc until a minutia was encountered. In nearest-neighbor studies of the spatial distribution of organisms, Pielou (1974) has noted that selecting the individual nearest to a randomly selected point does not constitute a selection of a random individual. The distinction is of no consequence if the distribution is, in fact, random, but there is a biased selection of comparatively isolated individuals under other circumstances. A similar source of bias in the present work must be considered. Two key features set the present work apart from standard nearest-neighbor organism sampling methods, and both act to reduce the indicated bias. First, the focal minutia was not sought on the basis of minimum TRD, but rather along individual ridges. This means that minutiae present on the other ridges may well show a shorter TRD to the point of maximum distal extent than does the focal minutia. Second, the sampling is not limited to the one nearest-neighbor. All neighbors without intervening minutiae were sampled (an average of 6.7 per focal minutia). This is a much better approximation of a total sampling of interevent distances than is single-distance sampling. In any case, the initial selection of a n event closest to a randomly selected point is a n unbiased random selection under the hypoth- 375 esis of a random distribution, and therefore rejection of this null hypothesis is sound. A second possible source of bias in the sampling is the limited extent of the sampling region. As noted above, the extent of the region was limited in order to ensure that uniform sampling would be possible for all prints and that within the sampling region the course of the ridges could be reasonably approximated by circular arcs. The effect of this limitation is that on each ridge there is an upper bound on the TRD that can be sampled. Minutiae appearing outside the 45" range on either side of the ridge were not sampled. On any given ridge, therefore, distances could not exceed rl4. Thus, based on an average focal radius of 5.42 mm, distances greater than 4.3 mm on the focal ridge fall outside the sampling region. On the focal ridge, this distance is equivalent to the TRD. With a ridge count of +1, assuming the average ridge density, distances of 4.6 mm along the ridge, or 8.9 mm TRD, fall outside the sampling region. Maximum TRDs increase with ridge count, and the probability of encountering neighbor minutiae falling outside the sampling region becomes very small for ridge counts greater than three. Bias introduced by restriction of the sampling region is thus limited to the few ridges about the focal minutia, and the bias is distributed over a relatively wide range of TRDs. The direction of this bias is in favor of shorter TRDs. Kingston's work (1964) is in qualitative agreement with this finding. Kingston ignored the ridge structure itself and performed quadrat sampling of minutiae. His major findings were 1)that there was a regular decrease in the variancelmean ratio with decreasing quadrat size and 2) that among quadrats containing exactly two minutiae the probability of the two minutiae appearing in the same quarter of the quadrat regularly decreased with decreasing quadrat size. Kingston showed that similar results would follow if about each minutia there was a region in which the occurrence of other minutiae was excluded. Kingston suspected two contributing sources for the overdispersal of minutiae. The first was the ridge structure itself. Minutiae appearing on different ridges were necessarily spaced a minimum of one ridge interval apart. Kingston's second suspected source of overdispersion was the way in which minutiae were defined. Kingston had defined certain closely spaced minutia pairs as single, compound events. This practice served in a n 376 D.A. STONEY artificial way to reduce the incidence of closely spaced minutiae. In the present study neither of these two explanations for the observed overdispersion is tenable. The use of ridge distances eliminates the argument that the ridge structure is responsible, and throughout this study closely spaced minutiae were strictly defined as two events. Osterburg et al. (1977) and Sclove (1979, 1980) also studied minutia distribution without reference to the ridge structure, but their model was not based on interminutia distances and cannot be reasonably compared to the present study. ACKNOWLEDGMENTS The author wishes to thank John E. Murdock of the Contra Costa County Criminalistics Laboratory for access to the fingerprint data described in this study. This project was supported in part by grant 82-IJ-CX-0023 from the National Institute of Justice, U.S. Department of Justice. Points of view or opinions stated in this document are those of the authors and do not necessarily represent the official position or policies of the U.S. Department of Justice. 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