6 Control Charts for Variables

Transcription

1 6 Control Charts for Variables 6.1 Distribution of the To generate R-charts and s-charts it is necessary to work with the sampling distributions of the sample range R and the sample standard deviation s. A brief summary of these distributions when sampling from a normal distribution will be given. The range R of a random sample X 1, X 2,..., X n is R = X (n) X (1) where X (n) and X (1) are, respectively, the largest and smallest order statistics in a sample of size n. When taking a sample of size n from a N(0, 1) distribution, the pdf of R is: g(r; n) = n(n 1) and the CDF of R is: G(r; n) = n = n 0 [Φ(x + r) Φ(x)] n 2 φ(x)φ(x + r)dx r > 0 [Φ(x + r) Φ(x)] n 1 φ(x)dx { [Φ(x + r) Φ(x)] n 1 + [Φ(x r) + Φ(x) 1] n 1} φ(x)dx r > 0 If the sample was taken from a N(0, σ 2 ) distribution, then the relative range W = R/σ has pdf g(r; n). The moments of the range R can be derived from either the pdf above or from the moments of minimum and maximum order statistics X (1) and X (n). The following tables contain the first two moments of X (1), X (n), and R for n = 2,, 4, 5 from a N(0, 1) distribution. Exact values of E(X (1) ), E(X (n) ) and E(R) n E(X (1) ) 1 π 2 ( π a ) 5 ( π π a ) π π E(X (n) ) E(R) 1 π 2 π where a = arcsin(1/) ( a ) π π ( 2 π π 1 + 2a ) π π ( a ) π π ( a ) π π Exact values of E(X(1) 2 ), E(X2 (n) ) and E(R2 ) n E(X(1) 2 ) π π 4π + 5b 2π 2 E(X 2 (n) ) π E(R 2 ) π 1 + π π π + 5b 2π π + 5b π c π 2 where b = arcsin(1.4) and c = arcsin(1/ 6)

2 6.2 Distribution of the Standard Deviation The sample standard deviation S of a random sample X 1, X 2,..., X n is S = 1 n ( Xi X ) 2. n 1 When taking a sample of size n from a N(µ, σ 2 ) distribution, the pdf of S is: i=1 g(s; n) = sν 1 ν ν/2 exp( νs 2 /2σ 2 ) 2 (ν 2)/2 σ ν Γ(ν/2) s > 0 where ν = n 1 1. The first four moments of S are: 2 E(S) = σ n 1 E(S ) = nσ2 E(S) n 1 Γ( n) 2 Γ( n 1) E(S 2 ) = σ 2 2 E(S 4 ) = ( ) n + 1 σ 4 n 1 From Jensen s inequality: E(S) = E[(S 2 ) 1/2 ] < [E(S 2 )] 1/2 = σ. So E(S) < σ. If we define S = n 1 2 Γ( n 1 2 ) Γ( n 2 ) S then E(S ) = σ. Therefore, we can use the sample standard deviation to get an unbiased estimate of σ is we just multiply by the reciprocal of the biasing factor. The same is true if we consider the range R. That is, if multiply by the reciprocal of the appropriate biasing factor then we can get another unbiased estimate of σ. Multipliers for constructing variables control charts The following table will be used throughout this section. It contains multipliers for constructing variables control charts including x, R, s, and individual (IMR) charts. We begin with x and R charts. The x-chart is used to check if the mean of a process characteristic is on aim. Because the variability of the process may cause the process mean to appear off aim, it is also necessary to check that the process variability is not too large. Therefore, the x-chart will be accompanied by either an R-chart or an s-chart, both of which assess the stability of the variability of a process. 46

3 6 Control Charts for Variables 5 47

4 6. x and R-charts Suppose the goal is to control the mean of some quality characteristic. Let random variable X correspond to the quality characteristic from a unit sampled from an in-control process. Suppose it is known that X N(µ, σ 2 ) when the process is running in control. ) If a sample of n independent units is taken from this population, then X N (µ, σ2. n Suppose m samples of size n are collected. For each sample, we can calculate the: s x 1, x 2,..., x m and x = the mean of the m sample means s R 1, R 2,..., R m and R = the mean of the m sample ranges 6..1 For Known µ and σ The µ x + σ x control limits for the x-chart when µ and σ are known are: UCL = µ x + σ x = A = n Centerline = µ x = µ () LCL = µ x σ x = To construct an R-chart, information about the relationship between the sample range R and the standard deviation σ from a normal distribution is needed. Suppose X i N(µ, σ 2 ) for i = 1, 2,..., n. Let x 1, x 2,..., x n be a random sample (realization) of size n. The range R = x max x min. The relative range W = R σ is a random variable with µ W = d 2 and σ W = d. Values of d 2 and d for various sample sizes are given in the table. Motivation: Note that we can rewrite R as R = W σ. Substitution yields: µ R = E(W σ) = σe(w ) = σd 2 where the value of d 2 = E(W ) depends on n. σ 2 R = Var(W σ) = σ2 Var(W ) = σ 2 σ 2 W. Thus, σ R = σ σ W = σd where the value of d depends on n. Using these values, the µ R ± σ R control limits for the R-chart are: UCL = µ R + σ R = D 2 = d 2 + d Centerline = µ R = d 2 σ (4) LCL = µ R σ R = D 1 = d 2 d where D 1 and D 2 are constants that depend on sample size n and can be found in the table. 48

5 EXAMPLE 1: The following data represents m = 100 samples of size n = 4. µ = 100. Assume σ = 2. The data can be found in the file xchart.dat. The target is s 1 to 50 s 51 to *9.55* (91) *92.69* (49)

6 XBAR AND RANGE CHARTS (KNOWN MU AND SIGMA) s and s Chart Summary for response Sigma s with n=4 for Sigma s with n=4 for sample XBAR AND RANGE CHARTS (KNOWN MU AND SIGMA) s and s Chart Summary for response Sigma s with n=4 for Sigma s with n=4 for sample

7 XBAR AND RANGE CHARTS (KNOWN MU AND SIGMA) s and s Chart Summary for response Sigma s with n=4 for Sigma s with n=4 for sample XBAR AND RANGE CHARTS (KNOWN MU AND SIGMA) s and s Chart Summary for response Sigma s with n=4 for Sigma s with n=4 for sample

8 XBAR AND RANGE CHARTS (KNOWN MU AND SIGMA) 52

9 SAS Code for x and R charts for Example 1 assuming µ = 100 and σ = 2: DM LOG; CLEAR; OUT; CLEAR; ; * ODS LISTING; * This is used if you want tables as text output; * ODS PRINTER PDF file= c:\courses\st528\sas\xrchart.pdf ; OPTIONS NODATE NONUMBER LS=120 PS=120; DATA in; INFILE c:\courses\st528\sas\xchart.dat ; DO sample =1 TO 100; DO unit = 1 TO 4; INPUT OUTPUT; END; END; TITLE XBAR AND RANGE CHARTS (KNOWN MU AND SIGMA) ; SYMBOL1 V=DOT WIDTH=.5; PROC SHEWHART DATA=in ; XRCHART response*sample= 1 / NPANELPOS=100 ZONES ZONELABELS MU0=100 XSYMBOL=MU0 SIGMA0=2 RSYMBOL=R0 TESTS = 1 TO 8 LTESTS = 2 TESTS2 = 1 TABLETESTS ALLN SPLIT = / ; LABEL RESPONSE = AVERAGE RESPONSE/RANGE ; RUN; SAS Code for x and s charts for Example 1 assuming µ = 100 and σ = 2: DM LOG; CLEAR; OUT; CLEAR; ; * ODS LISTING; * This is used if you want tables as text output; * ODS PRINTER PDF file= C:\COURSES\ST528\SAS\xschart.pdf ; OPTIONS NODATE NONUMBER; DATA IN; INFILE c:\courses\st528\sas\xchart.dat ; DO sample =1 TO 100; DO unit = 1 TO 4; INPUT OUTPUT; END; END; TITLE XBAR AND S CHARTS (KNOWN MU AND SIGMA) ; SYMBOL1 V=DOT WIDTH=1; PROC SHEWHART DATA=IN ; XSCHART response*sample= 1 / NPANELPOS=100 ZONES ZONELABELS MU0=100 XSYMBOL=MU0 SIGMA0=2 SSYMBOL=S0 TESTS = 1 TO 8 LTESTS = 2 TESTS2 = 1 TABLETESTS ALLN SPLIT = / ; LABEL response = AVERAGE RESPONSE/STANDARD DEVIATION ; RUN; 5

10 6..2 For Unknown µ and σ For new processes, µ and σ are typically not known at startup. Thus, a set of m preliminary samples must be collected in order to compute estimates of µ and σ. x i and R i should be computed for each of the preliminary samples. m i=1 The estimator of the unknown mean µ is µ = x = x i m. m The estimator of σ is σ = R i=1 d 2, where R = R i m. Motivation for estimating σ based on sample ranges: Earlier we showed that µ R = σd 2. This implies σ = µ R /d 2. Replacing µ R with µ R = R, we get σ = R/d 2. Then σ x = σ n = R d 2 n Substitution of the estimators into equations () and (4) for the unknown parameters yields the following trial control limits for the x-chart: UCL = µ + σ n = A 2 = d 2 n Centerline = µ = x (5) LCL = µ σ n = Motivation for estimating σ R based on sample ranges: Earlier we showed that σ R = σd. Replace σ with σ = R/d 2. Then σ R = σd = Rd d 2. We then substitute µ R and σ R into µ R ± σ R. The trial control limits for the R-chart are: UCL = R + σ R = D 4 = 1 + d d 2 Centerline = R = R (6) LCL = R σ R = D = 1 d d 2 where D and D 4 are constants dependent on sample size with values given in the table. Because the x chart is dependent upon the variability of the process being in control, it is good practice to first check if the preliminary values of R i indicate in-control process variability. The trial limits for R must be used to test whether or not the process was in control when the preliminary samples were taken. When testing with the R-chart, it is common to use Rule 1 only to determine if the process variability is out-of-control. If this test on the range indicates no out of control signals, adopt the trial control limits as valid control limits for future process control testing. 54

11 If any range points indicate an out-of-control process, an investigation for assignable causes These can should be be revised carried as more out. in-control samples are collected. If anyassignable range points cause indicate can bean found, out-of-control delete theprocess, point and recompute investigation thefor trial assignable control limits. causes should be carried out. If no an assignable cause can is found, be found, delete one theofpoint two things and recompute can be done. the trial control limits. If (i) nothe assignable point can cause be can deleted be found, and new onelimits of twocomputed. things cancontinue be done. with the preceding (i) test Theuntil pointacceptable can be deleted limitsand are found. new limits computed. Continue with the preceding (ii) Retain test until theacceptable point alonglimits withare thefound. trial control limits. Future points can be plotted to (ii) see Retain if they theplot point inalong control. withif the so, accept trial control the limits limits. as Future valid. points can be plotted to If the R-chart see iftrial theylimits plot inare control. accepted If so, as adopt valid, the thenlimits perform as valid. the same test on the x-chart, using If theany R-chart subset trial of the limits rules are proposed adoptedearlier. as valid, If both thenthe perform x andthe R control same test limits onare theaccepted x-chart, as using valid, anyproceed subset with of theprocess rules proposed control analysis. earlier. If both the x and R control limits are adopted Once as valid, valid proceed controlwith limits process have been control computed, analysis. process control testing can proceed. Once s valid control shouldlimits be collected have been fromcomputed, the sametesting process. for process control can proceed. Compute Collect samples the values fromofthe x i same and Rprocess. i for each sample as the data becomes available. Plot Compute the most x i and current R i forvalues each sample of x andasr the ondata the control becomes charts. available. Use Plotathese subset current of thevalues rules of discussed x i and Rearlier i on the in control this paper charts. to determine if the process is running Use a subset in control. of the rules for the x-chart discussed earlier to determine if the process is If running both charts in control. show an out-of-control signal for the same sample, it is suggested to search for If both an assignable charts showcause an out-of-control for a changesignal in variability for the same firstsample, because it is bringing suggested thetoprocess search variability for an assignable under control cause may for areturn change theinprocess variability to the first in-control becausestate bringing on the the x-chart. process variability under control may return the process to the in-control state on the x-chart. EXAMPLE 2: The melt index of an extrusion grad polyethylene compound is to be studied to determine EXAMPLE variation 2: The inmelt this quality index of property an extrusion and relate grad it polyethylene to raw material, compound shift, is and toother be studied changes to determine the process. variation Ability in of this the quality process property to produce and relate trial control it to raw limits material, is alsoshift, to beand studied. others changes of in n the = process. 4 are selected Abilityand of the melt process index to values produce are collected. trial control In limits an initial is also study, to bedata studied. was collected s over of size 7 days n = yielding 4 are selected m = 20 and samples. the melt The index dataiswith recorded. the sample Data means was collected and ranges overare 7 days givenyielding in the following m = 20 samples. table. The data with the sample means and ranges are given in the following table

12 XBAR and RANGE Charts (Unknown MU and SIGMA) SAMPLE XBAR and RANGE Charts (Unknown MU and SIGMA) s and s Chart Summary for INDEX Sigma s with n=4 for Sigma s with n=4 for

13 XBAR and R Charts ( Removed) XBAR and R Charts (s,4,6,8 Removed) 57

14 XBAR and R Charts ( Removed) s and s Chart Summary for index Sigma s with n=4 for Sigma s with n=4 for sample XBAR and R Charts (s,4,6,8 Removed) s and s Chart Summary for index Sigma s with n=4 for Sigma s with n=4 for sample

15 SAS Code for x and R charts for Example 2 assuming µ and σ are unknown DM LOG; CLEAR; OUT; CLEAR; ; * ODS LISTING; * This is used if you want tables as text output; * ODS PRINTER PDF file= C:\COURSES\ST528\SAS\xrchrt2.pdf ; OPTIONS NODATE NONUMBER LS=120 PS=120; DATA in; INPUT sample day DO item = 1 TO 4; INPUT OUTPUT; END; LINES; ; TITLE XBAR and RANGE Charts (Unknown MU and SIGMA) ; SYMBOL1 V=DOT WIDTH=1; PROC SHEWHART DATA=in; XRCHART index*sample= 1 / NPANELPOS=20 ZONES ZONELABELS TESTS = 1 TO 8 LTESTS = 2 TESTS2 = 1 TABLETEST ALLN SPLIT = / ; LABEL RESPONSE = MEAN/RANGE ; RUN; SAS Code for x and s charts for Example 2 assuming µ and σ are unknown DM LOG; CLEAR; OUT; CLEAR; ; * ODS LISTING; * This is used if you want tables as text output; * ODS PRINTER PDF file= C:\COURSES\ST528\SAS\xschrt2.pdf ; OPTIONS NODATE NONUMBER; DATA in; INPUT sample day DO item = 1 TO 4; INPUT OUTPUT; END; LINES; (same data set as above) ; TITLE XBAR and S Charts (Unknown MU and SIGMA) ; SYMBOL1 V=DOT WIDTH=1; PROC SHEWHART DATA=in; XSCHART index*sample= 1 / NPANELPOS=20 ZONES ZONELABELS TESTS = 1 TO 8 LTESTS = 2 TESTS2 = 1 TABLETEST ALLN SPLIT = / ; LABEL RESPONSE = MEAN/STANDARD DEVIATION ; RUN; 59

16 6.4 x and s-charts The x and R-charts work well when the sample sizes are constant and relatively small. For larger sample sizes, say n > 10, the sample range fails to account for much of the information provided by the sample when the n 2 middle observations are ignored. Therefore, it is suggested that the x and s-charts be used when the sample size is greater than 10. Note: some references say that if n > 5 or 6 then x- and s-charts should be used. It is important to note that E(s 2 ) = σ 2 but E(s) σ. Therefore, there exists ( ) a value c 4 for each sample size n such that µ s = E(s) = c 4 σ where ( ) n 1 Γ ( ) s c 4 = ( ). This implies E = σ. n 1 c 4 Γ n 1 2 It can also be shown that σ s = σ 1 c 2 4. Values of c 4 can be found in the table For Known µ and σ The control limits for the x-chart when both µ and σ are known can be computed using the formulas in (). Motivation for the UCL and LCL: Recall that S is not an unbiased estimator of σ. But, for each n, there exists a constant c 4 such that µ S = E(s) = c 4 σ. Therefore, when plotting sample standard deviations, the centerline should be at c 4 σ. Because σs 2 = σ 2 (1 c 2 4), we get σ s = σ 1 c 2 4. This is substituted to find the UCL and LCL for the s chart. Given a known value of σ and sample size n, the control limits for the s-chart are: UCL = µ s + σ s = Centerline = µ s = c 4 σ (7) LCL = µ s σ s = Values of B 5 and B 6 are given in the table for various values of n. For each sample (i = 1,..., m), compute x i = n j=1 x ij n and s i = n j=1 (x ij x i ) 2. n 1 The value of s i is then plotted against i on the s-chart. Use Rule 1 and the above control limits to determine if the variability of the process characteristic in control. 60

17 XBAR AND S CHARTS (KNOWN MU AND SIGMA) s and Standard Deviations Chart Summary for response Sigma s with n=4 for Sigma s with n=4 for Std Dev sample Std Dev XBAR AND S CHARTS (KNOWN MU AND SIGMA) s and Standard Deviations Chart Summary for response Sigma s with n=4 for Sigma s with n=4 for Std Dev sample Std Dev

18 XBAR AND S CHARTS (KNOWN MU AND SIGMA) s and Standard Deviations Chart Summary for response Sigma s with n=4 for Sigma s with n=4 for Std Dev sample Std Dev XBAR AND S CHARTS (KNOWN MU AND SIGMA) The SHEWHART Procedure s and Standard Deviations Chart Summary for response Sigma s with n=4 for Sigma s with n=4 for Std Dev sample Std Dev XBAR AND S CHARTS (KNOWN MU AND SIGMA) 62

19 6.4.2 For Unknown µ and σ When both µ and σ are unknown, estimates of these parameters must be computed based on m preliminary samples. Let: x = m i=1 x i m be the mean of the sample means and s = m i=1 s i m be the mean of the sample standard deviations. Therefore, the estimator of µ is x. ( ) s Because E(s) = E(s i ) for each i, we have E(s) = c 4 σ. It follows that E = σ. Thus, an unbiased estimator of σ is σ = s c 4 and, σ s The trial control limits for the x-chart are: UCL = µ + σ n = c 4 = σ 1 c 24 = s 1 c 2 4. c4 Centerline = µ = x (8) LCL = µ σ n = The trial control limits for the s-chart are: UCL = µ s + σ s = Centerline = µ s = s (9) LCL = µ s + σ s = where B and B 4 can be found in the table. These trial control limits must be tested in the same fashion as the trial control limits for the x- and R-charts were tested. That is, plot the s i values on the s-chart analogously to the way the R i values are plotted on the R-chart. Once acceptable control limits have been found for both charts, proceed with process control analysis. 6

20 XBAR and S Charts (Unknown MU and SIGMA) sample XBAR and S Charts (Unknown MU and SIGMA) s and Standard Deviations Chart Summary for index Sigma s with n=4 for Sigma s with n=4 for Std Dev Std Dev