IE 361 Module 11Shewhart Control Charts for MeasurementsProf.s Stephen B. Vardeman and Max D . MorrisReading: Section 3.2, Statistical Quality Assurance Methods for Engineers1In this module we consider Shewhart control charts for measurements (or socalled " variables data" in old time SQC jargon). As our featured example, wewill use the data from an IE 361 Deming drama. These are recorded in Figure1. (The "red bag" was an earlier version of the current "brown bag," i.e. hadprocess parameters μ =5and σ =1.715 and was approximately normal.)2Figure 1: Data from an IE 361 Deming Drama3¯x ChartsWe introduced the topic of Shewhart control charts in Module 10 using themost famous of all such charts, the ¯x charts. To review, we saw that the(approximately) normal distribution of ¯x (with mean μ¯x= μ and σ¯x= σ/√n)leads to standards given control limits for ¯xUCLx= μ +3σ√nand LCLx= μ − 3σ√nFurther, we saw that in a retrospective situation like that illustrated in Figure1 where sample means ¯x and sample ranges R are computed, estimatesˆμ =x and ˆσ =¯R/d24can be substituted to produce retrospective control limits for ¯xUCLx= x +3¯Rd2√nand LCLx= x − 3¯Rd2√nInfact,itistraditionaltosetA2=3d2√nand rewrite these retrospective control limits asUCLx= x + A2¯R and LCLx= x − A2¯RExample 11-1 We saw in Module 10 that (since for the brown bag μ =5and σ =1.715) standards given control limits for ¯x areUCLx=5+31.715√5=7.3andLCLx=5− 31.715√5=2.75These limits are marked on the ¯x control chart in Figure 1 and we can see thatif they had been applied to ¯x’s in real time, process change would have beendetected at sample 16.The 18 sample means and ranges from Figure 1 average tox =5.744 and¯R =4.278So retrospective limits for ¯x are (since the sample size is n =5)UCLx=5.744 + .577 (4.278)=8.21andLCLx=5.744 − .577 (4.278)=3.286When these limits are applied retrospectively to the 18 sample means, we seethat the last 3 values are outside of these, and there is thus evidence of processinstability in the data of Figure 1.R ChartsIt is traditional (not as it turns out best practice, but traditional) to use anR chart as a companion to an ¯x chart. (An s chart to be discussed next isactually a better choice than an R chart, but historical precedent makes Rcharts continue to be common.) The ¯x chart is primarily useful for monitoringprocess aim, while an R (or s) chart is primarily a tool for monitoring processspread or short term variation.7In order to identify appropriate control limits for R one needs to know someprobability 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 metin Stat 231) with mean proportional to the standard deviation of the sampledprocess. The constant of proportionality is the d2that we have used to turnranges into estimates of standard deviations, that isμR= d2σFurther, the standard deviation of the probability distribution for R is propor-tional to the standard deviation of the sampled process. The constant ofproportionality is called d3.Thatis,σR= d3σTaken together, these probability facts about R produce standards given controllimits for RUCLR=(d2+3d3)σ and LCLR=(d2− 3d3)σ8or, if one definesD2=(d2+3d3) and D1=(d2− 3d3)these standards given limits areUCLR= D2σ and LCLR= D1σFurther, in a retrospective situation like that illustrated in Figure 1 where R’sare computed, the estimateˆσ =¯R/d2can be substituted to produce retrospective control limits for RUCLR= D2¯R/d2and LCLR= D1¯R/d2It is traditional to setD4=D2d2and D3=D1d29and rewrite these retrospective control limits asUCLR= D4¯R and LCLR= D3¯RExample 11-2 Since σ =1.715 for the brown bag, a standards given uppercontrol limit for R based on n =5isUCLR=4.918 (1.715) = 8.43(No standards given lower control limit is typically used, because for a samplesize of only n =5,thedifference (d2− 3d3) turns out to be negative.) ThislimitismarkedontheR control chart in Figure 1 and we can see that if it hadbeen applied to R’s in real time, process change would have been detected atsample 16.10Recalling that the 18 sample means and ranges from Figure 1 average to¯R =4.278, a retrospective upper control limit for R isUCLR=2.115 (4.278) = 9.05When this limit is applied retrospectively to the 18 sample ranges, we see thatthe 16th sample range plots "out of control" and there is evidence of processinstability in data in Figure 1.s Chartss charts represent a superior alternative to R charts. At the price of requiringmore than "by hand" calculation (sample standard deviations being more diffi-cult to compute than sample ranges), they provide typically quicker detection11of process changes. In order to identify appropriate control limits for s oneneeds to know some probability facts about s basedonasampleofsizenfrom a normal distribution. It is a fact mentioned in Stat 231 (that actuallystands behind the standard confidence limits for σ)that(n − 1) s2/σ2has aχ2probability distribution. It turns out to follow from this fact that s has meanproportional to the standard deviation of the sampled process. The constantof proportionality is something called c4.Thatis,μs= c4σFurther, the standard deviation of the random variable s is proportional to thestandard deviation of the sampled process. The constant of proportionality iscalled c5that isσs= c5σTaken together, these probability facts about s produce standards given controllimitsUCLs=(c4+3c5)σ and LCLs=(c4− 3c5)σ12or, if one definesB6=(c4+3c5) and B5=(c4− 3c5)these standards given limits areUCLs= B6σ and LCLs= B5σFurther, in a retrospective situation like that illustrated in Figure 1 where svalues (instead of R values) are computed, the estimateˆσ =¯s/c4can be substituted to produce retrospective control limits for sUCLs= B6¯s/c4and LCLs= B5¯s/c4It is traditional to setB4=B6c4and B3=B5c413and rewrite these retrospective control limits asUCLs= B4¯s and LCLs= B3¯sA final bit of development concerning these retrospective s-based calculationsis this. Using ˆσ =¯s/c4, possible retrospective ¯x cha rt limits areUCLx= x +3¯sc4√nand LCLx= x − 3¯sc4√nand it is traditional to setA3=¯sc4√nand rewrite these retrospective control limits asUCLx= x + A3¯s and LCLx= x − A3¯s14Example 11-3 Since σ =1.715 for the brown bag, a standards given uppercontrol limit for s isUCLs=1.964 (1.715) = 3.37(No