Chapter 6 Part 3 X-bar and R Control Charts Attribute Data Data that is discrete п‚· Discrete data is based on вЂњcounts.вЂќ п‚· Assumes integer values п‚· пЃ‡ Number of defective units пЃ‡ Number of customers who are вЂњvery satisfiedвЂќ пЃ‡ Number of defects Variables Data X-bar and R chart is used to monitor mean and variance of a process when quality characteristic is continuous. п‚· Continuous values (variables data) can theoretically assume an infinite number of values in some interval. п‚· Time п‚· Weight п‚· Ounces п‚· Diameter п‚· X-bar and R Chart X-bar chart monitors the process mean by using the means of small samples taken frequently п‚· R chart monitors the process variation by using the sample ranges as the measure of variability п‚· п‚· Range = Maximum value вЂ“ Minimum value Notation X пЂЅ quality characteri stic n пЂЅ sample size k пЂЅ number of samples X пЂЅ sample mean ( X - bar) пЂЅ пѓҐ n X Notation X пЂЅ sample mean of X ( X - bar) X пЂЅ mean of the X s X is also called " X double bar, " the estimated the overall process mean mean, or Notation R пЂЅ Sample Range пЂЅ Maximum value - Minimum R пЂЅ Mean of ranges value R пѓҐ пЂЅ k пЃіЛ† пЂЅ estimated пЃіЛ† n пЂЅ estimated standard deviation of X standard error of X Example of Notation A company monitors the time (in minutes) it takes to assemble a product. п‚· The company decides to sample 3 units of the product at three different times tomorrow: п‚· 9 AM п‚· 12 Noon п‚· 3 PM п‚· What is the sample size, n? п‚· What is k, the number of samples? п‚· п‚· Suppose the following data are obtained. Hour п‚· Assembly Time (minutes) X1 X2 X3 9:00 AM 5 12 4 12 Noon 3:00 PM 6 9 8 4 10 2 How would you compute п‚· X-bar? п‚· R п‚· R-bar п‚· X double bar Example of Notation Hour Assembly Time (minutes) X1 X2 X3 Sample Mean, X Sample Range, R 9:00 AM 5 12 4 21/3 = 7 12 вЂ“ 4 = 8 12 Noon 6 8 10 24/3 = 8 10 вЂ“ 6 = 4 3:00 PM 9 4 2 15/3 = 5 9вЂ“2=7 X = 20/3 = 6.7 R =19/3 =6.3 Sample Means X пЂЅ пѓҐ X n First Sample X пЂЅ пѓҐ n X пЂЅ 5 пЂ« 12 пЂ« 4 3 пЂЅ 21 3 пЂЅ 7 Sample Means Second Sample X пЂЅ пѓҐX пЂЅ n 6 пЂ« 8 пЂ« 10 пЂЅ 3 24 пЂЅ8 3 Third Sample X пЂЅ пѓҐX n пЂЅ 9пЂ«4пЂ«2 3 пЂЅ 15 3 пЂЅ5 Estimated Process Mean X пЂЅ пѓҐ X k пЂЅ 7пЂ«8пЂ«5 3 пЂЅ 20 3 пЂЅ 6 .7 Sample Ranges R пЂЅ Maximum First Sample : R пЂЅ 12 пЂ 4 пЂЅ 8 Second Sample : R пЂЅ 10 - 6 пЂЅ 4 Third Sample : R пЂЅ 9-2 пЂЅ 7 Value - Minimum Value Mean of R R пЂЅ пѓҐR k пЂЅ 8пЂ«4пЂ«7 3 пЂЅ 19 3 пЂЅ 6 .3 Underlying Distributions п‚· When constructing an X-bar chart, we actually have two distributions to consider: п‚· The distribution of the sample means X , and п‚· The process distribution, the distribution of the quality characteristic itself, X. п‚· The distribution of averages. п‚· The distribution of X is a distribution of ??? X is a distribution of Underlying Distributions п‚· These distributions have the same mean Mean of X пЂЅ Mean of X Their variances (or standard deviations) are different. п‚· Which distribution has the bigger variance? п‚· п‚· п‚· Would you expect more variability among averages or among individual values? The variability among the individual values is ??? Underlying Distributions п‚· The standard deviation among the sample means is smaller by a factor of 1 n п‚· Therefore, Std of X пЂЅ пЃі пѓ¦ 1 пѓ¶ пЃіЛ† Std of X пЂЅ пЃіЛ† пѓ§пѓ§ пѓ·пѓ· пЂЅ n пѓЁ n пѓё Underlying Distributions Sampling distribution of X Distribution of X X Distribution of X пЃіЛ† M M m LCL X M UCL M M M X m m The distribution of X is assumed to be normal. This assumption needs to be tested in practice. Distribution of X-bar пЃіЛ† n M LCL M X m M UCL M M M X m m If the distribution of X is normal, the distribution of X-bar will be normal for any sample size. Control Limits for X-bar Chart п‚· Since we are plotting sample means on the Xbar chart, the control limits are based on the distribution of the sample means. п‚· The control limits are therefore UCL пЂЅ X пЂ« 3 пЃіЛ† n LCL пЂЅ X пЂ 3 пЃіЛ† n Control Limits for X-bar Chart Distribution of X LCL X пЂ3 UCL пЃіЛ† n X X пЂ«3 пЃіЛ† n Control Limits for X-bar Chart LCL пЂЅ X пЂ 3 пЃіЛ† n UCL пЂЅ X пЂ« 3 пЃіЛ† n LCL пЂЅ X пЂ A 2 R UCL пЂЅ X пЂ« A 2 R Control Limits for X-bar Chart 3 пЃіЛ† n пЂЅ A2 R A2 is a factor that depends on the n, the sample size, and will be given in a table. Example of X-bar Chart п‚· п‚· п‚· п‚· п‚· A company that makes soft drinks wants to monitor the sugar content of its drinks. The sugar content (X) is normally distributed, but the means and variances are unknown. The target sugar level for one of its drinks is 15 grams. The lower spec limit is 10 grams. The upper spec limit is 20 grams. Example of X-bar Chart The company wants to know how much sugar on average is being put into this soft drink and how much variability there is in the sugar content in each bottle. п‚· The company also wants to know if the mean sugar content is on target. п‚· Lastly, the company wants to know the percentage of drinks that are too sweet and the percentage that are not sweet enough. (Next section) п‚· Example of X-bar Chart п‚· To obtain this information, the company decides to sample 3 bottles of the soft drink at 3 different time each day: 10 A.M, п‚· 1:00 P.M. and п‚· 4:00 P.M. п‚· The company will use this data to construct an X-bar and R chart. (In practice, you need 2530 samples to construct the control limits.) п‚· For the past two days, the following data were collected: п‚· Example of X-bar Chart Day Hour X1 X2 X3 1 10 am 17 13 6 2 1 pm 15 12 24 4 pm 12 21 15 10 am 13 12 17 1 pm 18 21 15 4 pm 10 18 17 What is n? What is the k? What is the next step? Example of X-bar Chart Day Hour 1 10 am 2 X1 17 X2 13 X X3 6 36/3 =12 R 11 1 pm 15 12 24 51/3 =17 12 4 pm 12 21 15 48/3 =16 9 10 am 13 12 17 42/3 =14 5 1 pm 18 21 15 54/3 =18 6 4 pm 10 18 17 45/3 =15 8 X = 92/6 R = 51/6 = 15.33 = 8.5 X-bar Chart Control Limits LCL пЂЅ X пЂ A 2 R UCL пЂЅ X пЂ« A 2 R Table A: X-bar Chart Factor, A2 n A2 2 1.88 3 1.02 4 0.73 5 0.58 X-bar Chart Control Limits X пЂЅ 15 . 33 R пЂЅ 8 .5 From Table A in notes or Table 6 - 1, p. 182 of text n пЂЅ3 A 2 пЂЅ 1 . 02 X-bar Chart Control Limits LCL пЂЅ X пЂ A 2 R пЂЅ 15 . 33 пЂ 1 . 02 ( 8 . 5 ) пЂЅ 6 . 66 UCL пЂЅ X пЂ« A 2 R пЂЅ 15 . 33 пЂ« 1 . 02 ( 8 . 5 ) пЂЅ 24 . 0 X -b a r C h a rt fo r S u g a r C o n te n t 30.00 25.00 20.00 15.00 10.00 5.00 0.00 10 1 4 10 1 H our H our 1 2 D ay 4 Interpretation of X-bar Chart п‚· п‚· п‚· The X-bar chart is in control because ???? This means that the only source of variation among the sample mean is due to random causes. The process mean is therefore stable and predictable and, consequently, we can estimate it. Interpretation of X-bar Chart п‚· п‚· Our best estimate of the mean is the center line on the control chart, which is the overall mean (X-double bar) of 15.33 grams. If the process remains in control, the company can predict that all bottles of this soft drink produced in the future will have a sugar content of, on average, 15.33 grams. Interpretation of X-bar Chart This prediction, however, indicates that there is a problem with the location of the mean. п‚· The process mean is off target by 0.33 grams (15.33 -15.00). п‚· The process mean, although stable and predictable, is at the wrong level and should be corrected to the target. п‚· Interpretation of X-bar Chart Since the process mean is in control, there are no special causes of variation that may be responsible for the mean being off target. п‚· Since the operators are responsible for correcting problems due to special causes and management is responsible for correcting problems due to random causes of variation, management action is required to fix this problem. п‚· Interpretation of X-bar Chart The reason is that, because the process is in control, the filling machines require more than a simple adjustments (typically due to special causes) which can be made by the operators. п‚· The machines may require п‚· new parts, п‚· a complete overhaul, or п‚· they may simply not be capable of operating on target, in which case a new machine is required. п‚· Interpretation of X-bar Chart Expecting the operators to adjust the mean to the target when the process is in control is analogous to requiring that you produce zero heads (head = defective unit) if you are hired to toss a fair coin 100 times each day. п‚· Why? п‚· R Chart Monitors the process variability (the variability of X) п‚· Tells us when the process variability has changed or is about to change. п‚· R chart must be in control before we can use the X-bar chart. п‚· R Chart п‚· Rules for detecting changes in variance: If at least one sample range falls above the upper control limit, or there is an upward trend within the control limits, process variability has increased. п‚· If at least one sample range falls on or below the lower control limit, or there is a downward trend within the control limits, process variability has decreased. п‚· R Chart Control Limits LCL пЂЅ D 3 R UCL пЂЅ D 4 R Table B: Factors for R Chart n D3 D4 2 3 4 5 0 0 0 0 3.27 2.57 2.28 2.11 R Chart Control Limits n пЂЅ3 D3 пЂЅ 0 D 4 пЂЅ 2 . 57 LCL пЂЅ 0 ( 8 . 5 ) пЂЅ 0 UCL пЂЅ 2 . 57 ( 8 . 5 ) пЂЅ 21 . 85 R C h a rt fo r W e ig h t 25 20 R 15 LCL 10 UCL R -b a r 5 0 10 1 4 10 1 Hour Hour 1 2 Day 4 Interpretation of R Chart Since all of the sample ranges fall within the control limits, the R chart is in control. п‚· The standard deviation is stable and predictable and can be estimatedвЂ”done in next section. п‚· This does not necessarily mean that the amount of variation in the process is acceptable. п‚· Interpretation of R Chart Continuous improvement means the company should continuously reduce the variance. п‚· Since the process variation is in control, management action is required to reduce the variation. п‚· Expected Pattern in a Stable Process X-bar Chart UCL LCL Time Expected pattern is a normal distribution How Non-Random Patterns Show Up Sampling Distribution (process variability is increasing) UCL x-Chart LCL Does not reveal increase UCL R-chart Reveals increase LCL How Non-Random Patterns Show Up (process mean is shifting upward) Sampling Distribution UCL Detects shift x-Chart LCL UCL R-chart LCL Does not detect shift Is a Stable Process a Good Process? п‚· вЂњIn controlвЂќ indicates that the process mean is stable and hence predictable. п‚· A stable process, however, is not necessary a вЂњgoodвЂќ (defect free) process. The process mean, although stable, may be far off target, resulting in the production of defective product. п‚· In this case, we have, as Deming puts it, вЂњA stable process for the production of defective product.вЂќ п‚· Control Limits vs. Spec. Limits п‚· Control limits apply to sample means, not individual values. п‚· п‚· Mean diameter of sample of 5 parts, X-bar Spec limits apply to individual values п‚· Diameter of an individual part, X Control Limits vs. Spec. Limits Sampling distribution, X-bar Process distribution, X Mean= Target LSL Lower control limit Upper control limit USL Underlying Distributions Sampling distribution X of X Distribution of X LSL USL X LCL UCL Control limits are put on distribution of X-bar Spec limits apply to the distribution of X Responsibility for Corrective Action Special Causes (Process out of control) Operators (workers) Random Variation (Process in Control) Management Benefits of Control Charts п‚· Control charts prevent unnecessary adjustments. п‚· If process is in control, do not adjust it. п‚· Adjustments will increase the variance. п‚· Management action is required to improve process. п‚· Adjustments should be made only when special causes occur. Benefits of Control Charts Control charts assign responsibility for corrective action. п‚· Control charts are the only statistical valid way to estimate the mean and variance of a process or product. п‚· Control charts make it possible to predict future performance of a process and thereby take early corrective action. п‚·

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