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# PROB&INFERNC, ENGRS STAT 231

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This 21 page Class Notes was uploaded by Giovani Ullrich PhD on Saturday September 26, 2015. The Class Notes belongs to STAT 231 at Iowa State University taught by Staff in Fall. Since its upload, it has received 3 views. For similar materials see /class/214402/stat-231-iowa-state-university in Statistics at Iowa State University.

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Date Created: 09/26/15

IE 361 Module 11 Shewhart Control Charts for Measurements Profs Stephen B Vardeman and Max D Morris Reading Section 32 Statistical Quality Assurance Methods for Engineers In this module we consider Shewhart control charts for measurements or so called quotvariables data in old time SQC jargon As our featured example we will use the data from an IE 361 Deming drama These are recorded in Figure 1 The quotred bag was an earlier version of the current quotbrown bag ie had process parameters u 5 and a 1715 and was approximately normal Figure 1 Data from an IE 361 Deming Drama 70 Charts We introduced the topic of Shewhart control charts in Module 10 using the most famous of all such charts the 50 charts To review we saw that the approximately normal distribution off with mean M5 u and 053 a leads to standards given control limits for 50 039 039 Further we saw that in a retrospective situation like that illustrated in Figure 1 where sample means 9390 and sample ranges R are computed estimates A p and a Rd2 all can be substituted to produce retrospective control limits for 50 R R Cl LCL 3 d2x an m x d2x In fact it is traditional to set UCLE 3 3 A 2 dg and rewrite these retrospective control limits as UCLE A2R and LCszi AQR Example 111 We saw in Module 10 that since for the brown bag u 5 and a 1715 standards given control limits for 9390 are 1715 1715 UCL 53 73 and LCL 5 3 27 These limits are marked on the 9390 control chart in Figure 1 and we can see that if they had been applied to 50 s in real time process change would have been detected at sample 16 The 18 sample means and ranges from Figure 1 average to E 5744 and R 4278 So retrospective limits for 50 are since the sample size is n 5 UCLE 57445774278 821 and LCLE 5744 577 4278 328 When these limits are applied retrospectively to the 18 sample means we see that the last 3 values are outside of these and there is thus evidence of process instability in the data of Figure 1 R Charts It is traditional not as it turns out best practice but traditional to use an R chart as a companion to an 50 chart An 8 chart to be discussed next is actually a better choice than an R chart but historical precedent makes R charts continue to be common The 50 chart is primarily useful for monitoring process aim while an R or s chart is primarily a tool for monitoring process spread or short term variation In order to identify appropriate control limits for R one needs to know some probability facts about R based on a sample of size n from a normal distribution As it turns out R has a non standard probability distribution not one you met in Stat 231 with mean proportional to the standard deviation of the sampled process The constant of proportionality is the dg that we have used to turn ranges into estimates of standard deviations that is HR d20 Further the standard deviation of the probability distribution for R is propor tional to the standard deviation of the sampled process The constant of proportionality is called d3 That is OR d3a Ta ken together these probability facts about R produce standards given control limits for R UCLR 2 d2 3d3a and LCLR 2 d2 3d3a or if one defines D2 2 d2 l 3d3 and D1 2 d2 3d3 these standards given limits are UCLR 2 D20 and LCLR 2 D10 Further in a retrospective situation like that illustrated in Figure 1 where R s are computed the estimate a Rd2 can be substituted to produce retrospective control limits for R UCLR D2Rd2 and LCLR DlRd2 It is traditional to set D2 d2 D1 D 4 d2 and D3 and rewrite these retrospective control limits as UCLR D4R and LCLR D3R Example 112 Since a 1715 for the brown bag a standards given upper control limit for R based on n 5 is UCLR 4918 1715 843 No standards given lower control limit is typically used because for a sample size of only n 5 the difference d2 3d3 turns out to be negative This limit is marked on the R control chart in Figure 1 and we can see that if it had been applied to R s in real time process change would have been detected at sample 16 10 Recalling that the 18 sample means and ranges from Figure 1 average to R 4278 a retrospective upper control limit for R is UCLR 2115 4278 905 When this limit is applied retrospectively to the 18 sample ranges we see that the 16th sample range plots quotout of control and there is evidence of process instability in data in Figure 1 8 Charts 8 charts represent a superior alternative to R charts At the price of requiring more than quotby hand calculation sample standard deviations being more diffi cult to compute than sample ranges they provide typically quicker detection 11 of process changes In order to identify appropriate control limits for 8 one needs to know some probability facts about 8 based on a sample of size n from a normal distribution It is a fact mentioned in Stat 231 that actually stands behind the standard confidence limits for a that n 1 8202 has a X2 probability distribution It turns out to follow from this fact that s has mean proportional to the standard deviation of the sampled process The constant of proportionality is something called C4 That is Ms C40 Further the standard deviation of the random variable 8 is proportional to the standard deviation of the sampled process The constant of proportionality is called C5 that is as 2 C50 Taken together these probability facts about 8 produce standards given control limits UCLS C4 3C50 and LCLS C4 3C5039 12 or if one defines B6 C4 l 305 and B5 C4 305 these standards given limits are UCLS 2 B60 and LCLS 2 B50 Further in a retrospective situation like that illustrated in Figure 1 where 8 values instead of R values are computed the estimate 6 C4 can be substituted to produce retrospective control limits for s UCLS B6 C4 and LCLS B5 C4 It is traditional to set B B 6 and B3 5 C4 C4 B4 13 and rewrite these retrospective control limits as UCLS B4 and LCLS B3 A final bit of development concerning these retrospective s based calculations is this Using 6 C4 possible retrospective 9390 chart limits are d LCL 3 04 an a x 04 and it is traditional to set UCLE 3 8 04 and rewrite these retrospective control limits as A3 UCLEZ A3 and LCL Z E A3 CIJI 14 Example 113 Since a 1715 for the brown bag a standards given upper control limit for s is UCLS 1964 1715 337 No standards given lower control limit is typically used because for a sample size of only n 5 the difference C4 3C5 turns out to be negative The following table shows the 18 sample standard deviations corresponding to the data in Figure 1 Sample 1 2 3 4 5 6 7 8 9 10 s 182 192 182 114 114 148 89 45 182 0 11 12 13 14 15 16 17 18 84 122 110 89 235 563 288 358 15 It is evident from these 8 values that if the standards given control limit had been applied to 8 s in real time process change would have been detected at sample 16 The 18 values 8 average to E 309718 2 172 So a retrospective upper control limit for s for the data of Figure 1 is UCLS 2089 172 359 and when this limit is applied retrospectively to the 18 sample standard devi ations we see that the 16th plots quotout of control and there is evidence of process instability in data of Figure 1 Further consider making retrospective control limits for 50 based on sample standard deviations These are UCLE 57741427172 823 16 and LCLE 5774 1427 172 332 When these limits are applied retrospectively to the 18 sample means from Figure 1 we see that the last 3 values are outside of these and there is evidence of process instability in the data Median Charts A computationally simpler but not often used alternative to the Shewhart 9390 chart is Shewhart median chart Finding a median requires only putting a 17 data set in order smallest to largest and then finding the middle value 55 This is a measure of process aim like the mean But is it generally not as reliable as the mean Differently put it generally takes longer to detect process change using medians than using means But in some rare contexts computational simplicity may outweigh this lack of sensitivity In order to identify appropriate control limits for 53 one needs to know some probability facts about 53 based on a sample of size n from a normal distribution As it turns out 53 has a non standard probability distribution not one you met in Stat 231 with mean equal to the process mean and standard deviation larger than that of 9390 by a multiplicative factor that we will call Kc That is d a Mm M an 0m quotOW where a small table of values for Fe is given on page 72 of SQAME These facts suggest standards given control limits for 53 UCLi M 3n and LCLi M 3 039 18 Any sensible estimates of u and a could further be used to make retrospective limits for Example 114 Since for the brown bag process u 5 and a 1715 standards given control limits for 53 based on n 5 are 1715 UCL 2 53 1197 m lt 775 and 1715 LCLi 5 31197 xE 225 The 18 sample medians for the data of Figure 1 are as in the table below 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 5 4 5 5 4 5 5 5 4 4 5 6 5 5 12 9 8 19 So if the standards given control limits had been applied to 58 s in real time process change would have been detected at sample 16 An Important Reminder It is worth saying again that control limits are NOT engineering specifications nor vice versa In Module 10 we said and now say again that a process can be stable without being acceptable and vice versa The table below again compares these two fundamentally different concepts 20 Control Limits Specifications have to do with process stability apply to Q usually derived from process data have to do with product acceptability apply to individuals 90 derived from performance requirements 21

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