6.5
x and sp -charts
• When both µ and σ are unknown, estimates of these parameters should be computed based
on m preliminary samples that are assumed to come from an in-control process.
• For larger sample sizes, another option for tracking process variability is the sp -chart which
is based on a pooled estimate of the variance.
Pm
Pm
2
(ni − 1)s2i
2
i=1
i=1 (ni − 1)si
= P
• The pooled variance estimate is sp = Pm
m
i=1 (ni − 1)
i=1 ni − m
where m is the number of samples and ni is the ith sample size.
• s2p is the MSE in a oneway ANOVA with m levels and ni observations for treatment i.
• For equal sample sizes (ni = n for all i), we form control limits in a fashion analogous to the
x and s-charts but use sp instead of s and σ
b = sp /c4 .
• The trial control limits for the x-chart are:
UCL = x + 3
Centerline = x
LCL = x − 3
• The trial control limits for the sp -chart are:
UCL = B4 sp
Centerline = sp
LCL = B3 sp
where B3 and B4 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 and for the x- and s-charts.
SAS Code for x and sp Charts
DATA in;
INPUT sample day shift @@;
DO ITEM = 1 TO 4;
INPUT index @@; OUTPUT;
END;
LINES;
1 1 3 218 224 220 231
2
3 1 4 280 228 228 221
4
5 2 1 243 240 230 230
6
7 3 2 240 238 240 243
8
9 3 4 238 233 252 243
10
11 4 4 218 232 230 226
12
13 5 1 224 221 230 222
14
15 5 3 224 228 226 240
16
17 6 4 243 250 248 250
18
19 7 1 224 228 228 246
20
;
TITLE ’XBAR AND S CHARTS (USING
SYMBOL1 V=DOT WIDTH=3;
1
2
2
3
4
4
5
6
6
7
1
3
4
1
2
3
4
1
3
4
228
210
225
244
228
226
230
232
247
236
236
249
250
248
238
231
220
240
238
230
247
241
258
265
220
236
227
241
244
230
234
246
244
234
230
242
226
232
230
232
POOLED S_P)’;
PROC SHEWHART DATA=in;
XSCHART index*sample=’1’ / SMETHOD = RMSDF TESTS = 1 TO 8
LTESTS = 2 TESTS2 = 1 TABLE ALLN
SPLIT = ’/’;
LABEL index = ’AVERAGE RESPONSE/POOLED STD DEV’;
RUN;
65
XBAR AND S CHARTS (USING POOLED S_P)
The SHEWHART Procedure
XBAR AND S CHARTS (USING POOLED S_P)
The SHEWHART Procedure
Means and Standard Deviations Chart Summary for index
3 Sigma Limits with n=4 for
Mean
Subgroup
Sample
sample
Size
Lower Subgroup
Limit
Mean
3 Sigma Limits with n=4 for
Std Dev
Special
Upper
Tests Lower Subgroup
Limit Signaled Limit Std Dev
Special
Upper
Tests
Limit Signaled
1
4 219.71509
223.25000
250.35991
0
5.737305
21.326240
2
4 219.71509
236.25000
250.35991
0
7.932003
21.326240
3
4 219.71509
239.25000
250.35991
0
27.366342
21.326240
4
4 219.71509
236.50000
250.35991
0
17.972201
21.326240
5
4 219.71509
235.75000
250.35991
0
6.751543
21.326240
6
4 219.71509
244.25000
250.35991
0
14.056434
21.326240
7
4 219.71509
240.25000
250.35991
0
2.061553
21.326240
8
4 219.71509
247.75000
250.35991
0
12.919623
21.326240
9
4 219.71509
241.50000
250.35991
0
8.103497
21.326240
10
4 219.71509
229.00000
250.35991
0
7.393691
21.326240
11
4 219.71509
226.50000
250.35991
0
6.191392
21.326240
12
4 219.71509
233.75000
250.35991
0
6.849574
21.326240
13
4 219.71509
224.25000
250.35991
0
4.031129
21.326240
14
4 219.71509
225.75000
250.35991
0
4.193249
21.326240
15
4 219.71509
229.50000
250.35991
0
7.187953
21.326240
16
4 219.71509
236.25000
250.35991
0
4.924429
21.326240
17
4 219.71509
247.75000
250.35991
0
3.304038
21.326240
18
4 219.71509
239.75000
250.35991
0
7.500000
21.326240
19
4 219.71509
231.50000
250.35991
0
9.848858
21.326240
20
4 219.71509
232.00000
250.35991
0
2.828427
21.326240
6
6
66
1
6.6
6.6.1
Control Charts for Unequal Sample Sizes
x and R Charts
• So far we have considered samples of constant size (n). Often, however, this cannot be
achieved. We will, therefore, consider the case of control charts with variable sample size.
That is, all samples do not consist of the same number of observations.
• Quality control charting software will create variable sample size x and R-charts.
• SAS calculates the variable R-chart centerline and control limits based on the following:
– Let d2 (ni ) = the d2 value when the ith sample size is ni .
– Recall: Under the normality assumptions, µRi = d2 (ni )σ.
Ri
(i = 1, . . . , m) is an unbiased estimate of σ.
Then each
d2 (ni )
– σ
b is the mean of these m estimates of σ: σ
b=
th
Hence, for the i
R1
d2 (n1 )
+
R2
d2 (n2 )
+ ... +
Rm
d2 (nm )
m
sample, Ri = d2 (ni )b
σ
• The control limits for the ith sample are:
UCL =
Centerline = Ri
LCL =
where D3 (ni ) and D4 (ni ) are constants in the table corresponding to sample size ni .
6.6.2
x and s Charts
• The x and s-charts also lead to a changing centerline and control limits on the s-chart when
the sample size is not constant.
• SAS calculates the variable s-chart centerline and control limits based on the following:
– Let c4 (ni ) = the c4 value when the ith sample size is ni .
– Recall: Under the normality assumptions, µsi = c4 (ni )σ.
si
(i = 1, . . . , m) is an unbiased estimate of σ.
Then each
c4 (ni )
s1
+ c4 s(n2 2 ) + . . . +
c4 (n1 )
– σ
b is the mean of these m estimates of σ: σ
b=
m
sm
c4 (nm )
• If we define si = c4 (ni )b
σ , then the control limits for the ith sample are:
UCL =
Centerline = si
LCL =
where c4 (ni ), B3 (ni ) and B4 (ni ) are constants in the table corresponding to sample size ni .
67
SAS Code for x and R Charts with Unequal ni
DM ’LOG; CLEAR; OUT; CLEAR;’;
* ODS LISTING;
* ODS PRINTER PDF file=’C:\COURSES\ST528\xrunequa.pdf’;
OPTIONS NODATE NONUMBER LS=120 PS=120;
*******************************************************;
*** Mean and Range Charts with Varying Sample Sizes ***;
*******************************************************;
DATA wire (DROP=i size);
INPUT day size @;
INFORMAT day DATE7.; FORMAT day DATE7.;
DO i=1 TO size;
INPUT brstr @@; OUTPUT;
END;
LINES;
20JUN94 5 60.6 62.3 62.0 60.4 59.9
21JUN94 5 61.9 62.1 60.6 58.9 65.3
22JUN94 4 57.8 60.5 60.1 57.7
23JUN94 5 56.8 62.5 60.1 62.9 58.9
24JUN94 5 63.0 60.7 57.2 61.0 53.5
25JUN94 7 58.7 60.1 59.7 60.1 59.1 57.3 60.9
26JUN94 5 59.3 61.7 59.1 58.1 60.3
27JUN94 5 61.3 58.5 57.8 61.0 58.6
28JUN94 6 59.5 58.3 57.5 59.4 61.5 59.6
29JUN94 5 61.7 60.7 57.2 56.5 61.5
30JUN94 3 63.9 61.6 60.9
01JUL94 5 58.7 61.4 62.4 57.3 60.5
02JUL94 5 56.8 58.5 55.7 63.0 62.7
03JUL94 5 62.1 60.6 62.1 58.7 58.3
04JUL94 5 59.1 60.4 60.4 59.0 64.1
05JUL94 5 59.9 58.8 59.2 63.0 64.9
06JUL94 6 58.8 62.4 59.4 57.1 61.2 58.6
07JUL94 5 60.3 58.7 60.5 58.6 56.2
08JUL94 5 59.2 59.8 59.7 59.3 60.0
09JUL94 5 62.3 56.0 57.0 61.8 58.8
10JUL94 4 60.5 62.0 61.4 57.7
11JUL94 4 59.3 62.4 60.4 60.0
12JUL94 5 62.4 61.3 60.5 57.7 60.2
13JUL94 5 61.2 55.5 60.2 60.4 62.4
14JUL94 5 59.0 66.1 57.7 58.5 58.9
;
SYMBOL V=dot WIDTH=3;
TITLE ’XBAR/R CHARTS -- UNEQUAL SAMPLE SIZES’;
PROC SHEWHART data=wire;
XRCHART brstr*day / NPANELPOS=25 TABLETEST ALLN
HMINOR=1 VMINOR=3 SPLIT = ’/’;
LABEL brstr = ’AVERAGE BREAKING STRENGTH/RANGE’;
RUN;
SAS Code for x and s Charts with Unequal ni
(all code before the TITLE statement is the same as above)
TITLE ’XBAR/S CHARTS -- UNEQUAL SAMPLE SIZES’;
PROC SHEWHART data=wire;
XSCHART brstr*day / NPANELPOS=25 TABLETEST ALLN
HMINOR=1 VMINOR=3 SPLIT = ’/’;
LABEL brstr = ’AVERAGE BREAKING STRENGTH/STD DEV’;
RUN;
68
XBAR/R CHARTS -- UNEQUAL SAMPLE SIZES
The SHEWHART Procedure
XBAR/S CHARTS -- UNEQUAL SAMPLE SIZES
The SHEWHART Procedure
69
70
5 57.086004
5 57.086004
4 56.744814
5 57.086004
5 57.086004
7 57.533602
5 57.086004
5 57.086004
6 57.337860
5 57.086004
3 56.244853
5 57.086004
5 57.086004
5 57.086004
5 57.086004
5 57.086004
6 57.337860
5 57.086004
5 57.086004
5 57.086004
4 56.744814
4 56.744814
5 57.086004
5 57.086004
5 57.086004
21JUN94
22JUN94
23JUN94
24JUN94
25JUN94
26JUN94
27JUN94
28JUN94
29JUN94
30JUN94
01JUL94
02JUL94
03JUL94
04JUL94
05JUL94
06JUL94
07JUL94
08JUL94
09JUL94
10JUL94
11JUL94
12JUL94
13JUL94
14JUL94
60.040000
59.940000
60.420000
60.525000
60.400000
59.180000
59.600000
58.860000
59.583333
61.160000
60.600000
60.360000
59.340000
60.060000
62.133333
59.520000
59.300000
59.440000
59.700000
59.414286
59.080000
60.240000
59.025000
61.760000
61.040000
Lower Subgroup
Limit
Mean
20JUN94
Subgroup
Sample
day
Size
62.867222
62.867222
62.867222
63.208412
63.208412
62.867222
62.867222
62.867222
62.615366
62.867222
62.867222
62.867222
62.867222
62.867222
63.708373
62.867222
62.615366
62.867222
62.867222
62.419623
62.867222
62.867222
63.208412
62.867222
62.867222
Upper
Limit
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.06224788
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.06224788
0.00000000
0.00000000
0.24325396
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
3.4260765
2.6283074
1.7426991
1.3301002
1.9026298
2.8110496
0.3391165
1.7271364
1.9145931
2.6632687
2.0700242
1.8105248
3.3575289
2.0574256
1.5695010
2.4783059
1.3579396
1.5946787
1.3638182
1.1767591
3.7532652
2.5412595
1.4818344
2.3554193
1.0502381
Lower Subgroup
Limit Std Dev
4.2307006
4.2307006
4.2307006
4.4981243
4.4981243
4.2307006
4.2307006
4.2307006
4.0379695
4.2307006
4.2307006
4.2307006
4.2307006
4.2307006
4.9036757
4.2307006
4.0379695
4.2307006
4.2307006
3.8907290
4.2307006
4.2307006
4.4981243
4.2307006
4.2307006
Upper
Limit
14JUL94
13JUL94
12JUL94
11JUL94
10JUL94
09JUL94
08JUL94
07JUL94
06JUL94
05JUL94
04JUL94
03JUL94
02JUL94
01JUL94
30JUN94
29JUN94
28JUN94
27JUN94
26JUN94
25JUN94
24JUN94
23JUN94
22JUN94
21JUN94
20JUN94
5 57.143795
5 57.143795
5 57.143795
4 56.809426
4 56.809426
5 57.143795
5 57.143795
5 57.143795
6 57.390615
5 57.143795
5 57.143795
5 57.143795
5 57.143795
5 57.143795
3 56.319460
5 57.143795
6 57.390615
5 57.143795
5 57.143795
7 57.582444
5 57.143795
5 57.143795
4 56.809426
5 57.143795
60.040000
59.940000
60.420000
60.525000
60.400000
59.180000
59.600000
58.860000
59.583333
61.160000
60.600000
60.360000
59.340000
60.060000
62.133333
59.520000
59.300000
59.440000
59.700000
59.414286
59.080000
60.240000
59.025000
61.760000
61.040000
Lower Subgroup
Limit
Mean
5 57.143795
Subgroup
Sample
day
Size
62.809431
62.809431
62.809431
63.143800
63.143800
62.809431
62.809431
62.809431
62.562610
62.809431
62.809431
62.809431
62.809431
62.809431
63.633766
62.809431
62.562610
62.809431
62.809431
62.370781
62.809431
62.809431
63.143800
62.809431
62.809431
Upper
Limit
3 Sigma Limits for Mean
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.43230151
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
8.4000000
6.9000000
4.7000000
3.1000000
4.3000000
6.3000000
0.8000000
4.3000000
5.3000000
6.1000000
5.1000000
3.8000000
7.3000000
5.1000000
3.0000000
5.2000000
4.0000000
3.5000000
3.6000000
3.6000000
9.5000000
6.1000000
2.8000000
6.4000000
2.4000000
Lower Subgroup
Limit
Range
10.384520
10.384520
10.384520
9.920001
9.920001
10.384520
10.384520
10.384520
10.723107
10.384520
10.384520
10.384520
10.384520
10.384520
9.201044
10.384520
10.723107
10.384520
10.384520
10.987971
10.384520
10.384520
9.920001
10.384520
10.384520
Upper
Limit
3 Sigma Limits for Range
Means and Ranges Chart Summary for brstr
Means and Standard Deviations Chart Summary for brstr
3 Sigma Limits for Std Dev
The SHEWHART Procedure
The SHEWHART Procedure
3 Sigma Limits for Mean
XBAR/R CHARTS -- UNEQUAL SAMPLE SIZES
XBAR/S CHARTS -- UNEQUAL SAMPLE SIZES
6.6.3
The I-chart and MR-chart
• In some circumstances, it may be necessary to restrict the sample size to n=1. In such a case,
the methods discussed so far for estimating process variability are not applicable.
• Instead, we use the moving range MR as an estimate of the process variability.
• A moving range is computed by taking the absolute value of the difference between two
consecutive observations.
• The ith moving range is M Ri = |xi − xi−1 |.
• To set up trial control limits for the MR-chart and I-chart (where I represents ‘individual
measurement’) with unknown µ and σ, a set of m preliminary samples must be taken.
Pm−1
M Ri
MR
• The estimator of σ is σ
b = d2 , where M R = i=1
is the average of the m−1 consecutive
m−1
moving ranges in m observations.
• In essence, the estimate of the process variability is based on samples of size n=2, the two consecutive observations (ignoring the fact that the samples are no longer independent). Therefore, we use the d2 table value for n = 2. SAS allows for moving ranges for n > 2.
• The trial control limits for the MR-chart are:
UCL = µ
bM R + 3b
σM R =
Centerline = µ
bR = M R
(10)
LCL = µ
bM R + 3b
σM R =
where D3 and D4 for n = 2 can be found in the table. These control limits are the same as
the UCL and LCL for an R-chart but with M R replacing R.
• These trial limits must be tested to see if they are acceptable for monitoring the process
variability. To do so, plot the M Ri values and use the same procedure as discussed for testing
the trial control limits for the R-chart.
• Once acceptable control limits have been computed for the MR-chart, construct the I-chart.
• The trial control limits for the I-chart are:
MR
UCL = x + 3
d2
Centerline = x
MR
LCL = x − 3
d2
Pm
where x =
i=1
xi
m
(11)
• These control limits are the same as the UCL and LCL for an x-chart but with M R replacing
R and with n = 1.
71
• As before, check to see if these limits are acceptable for monitoring the process characteristic.
• To do so, plot the xi values on the I-chart and use the same procedure as discussed for testing
the trial control limits for the x-chart.
• If the trial control limits are satisfactory, proceed with process control analysis.
SAS Code for IMR Charts
DM ’LOG; CLEAR; OUT; CLEAR;’;
ODS PRINTER PDF file=’C:\COURSES\ST528\CRSNOTES\imr1.pdf’;
OPTIONS NODATE NONUMBER;
**************************************;
*** Individual Measurements Chart ***;
*** (1) Mu And Sigma are unknown
***;
*** (2) Mu And Sigma are specified ***;
**************************************;
DATA engines;
INPUT id weight @@;
LABEL weight = ’Engine Weight in lbs’
id
= ’Engine ID Number’;
LINES;
1711 1270
1712 1258
1713 1248
1714
1715 1263
1716 1260
1717 1259
1718
1719 1260
1720 1246
1721 1238
1722
1723 1249
1724 1245
1725 1251
1726
1727 1249
1728 1274
1729 1258
1730
1731 1248
1732 1295
1733 1243
1734
1735 1258
;
SYMBOL1 VALUE=dot WIDTH=2;
1260
1240
1253
1252
1268
1253
/* IMR CHART FOR UNKNOWN MU AND SIGMA */
PROC SHEWHART DATA=engines;
IRCHART weight*id=’1’ /
TESTS = 1 to 8 TEST2RUN = 7
LTESTS = 2 TABLETESTS
ZONES ZONES2 ZONELABELS ZONE2LABELS
VMINOR=1 HMINOR=1
SPLIT = ’/’;
LABEL weight = ’ENGINE WEIGHT/MOVING RANGE’;
TITLE ’IMR CHARTS (MU and SIGMA unknown)’;
RUN;
/* IMR CHART WITH SPECIFIED VALUES OF MU AND SIGMA */
PROC SHEWHART DATA=engines;
IRCHART weight*id=’1’ /
MU0 = 1250 SIGMA0 = 8
XSYMBOL = MU0 RSYMBOL = R0
TESTS = 1 to 8 TEST2RUN = 7
LTESTS = 2 TABLETESTS
ZONES ZONES2 ZONELABELS ZONE2LABELS
VMINOR=1 HMINOR=1
SPLIT = ’/’;
LABEL weight = ’ENGINE WEIGHT/MOVING RANGE’;
TITLE ’IMR CHARTS (MU and SIGMA specified)’;
RUN;
72
73
0
0
0
0 19.000000
0 20.000000
0 14.000000
0
0 15.000000
0
0
0
0
1715 1220.4709 1263.0000 1291.3691
1716 1220.4709 1260.0000 1291.3691
1717 1220.4709 1259.0000 1291.3691
1718 1220.4709 1240.0000 1291.3691
1719 1220.4709 1260.0000 1291.3691
1720 1220.4709 1246.0000 1291.3691
1721 1220.4709 1238.0000 1291.3691
1722 1220.4709 1253.0000 1291.3691
1723 1220.4709 1249.0000 1291.3691
1724 1220.4709 1245.0000 1291.3691
1725 1220.4709 1251.0000 1291.3691
1726 1220.4709 1252.0000 1291.3691
5.000000
0
1735 1220.4709 1258.0000 1291.3691
0 47.000000
1732 1220.4709 1295.0000 1291.3691
0 10.000000
0 20.000000
1731 1220.4709 1248.0000 1291.3691
0 52.000000
0 10.000000
1730 1220.4709 1268.0000 1291.3691
1734 1220.4709 1253.0000 1291.3691
0 16.000000
1729 1220.4709 1258.0000 1291.3691
1733 1220.4709 1243.0000 1291.3691
0 25.000000
1728 1220.4709 1274.0000 1291.3691
3.000000
0
1.000000
6.000000
4.000000
4.000000
8.000000
1.000000
3.000000
1727 1220.4709 1249.0000 1291.3691
1
0 12.000000
1714 1220.4709 1260.0000 1291.3691
2
0 10.000000
1713 1220.4709 1248.0000 1291.3691
3.000000
0 12.000000
1712 1220.4709 1258.0000 1291.3691
Upper
Limit
43.553759
43.553759
43.553759
43.553759
43.553759
43.553759
43.553759
43.553759
43.553759
43.553759
43.553759
43.553759
43.553759
43.553759
43.553759
43.553759
43.553759
43.553759
43.553759
43.553759
43.553759
43.553759
43.553759
43.553759
. 43.553759
Moving
Range
0
weight
Special
Upper
Tests Lower
Limit Signaled Limit
1711 1220.4709 1270.0000 1291.3691
id
Lower
Limit
weight
1735 1226.0000 1258.0000 1274.0000
1734 1226.0000 1253.0000 1274.0000
1733 1226.0000 1243.0000 1274.0000
1732 1226.0000 1295.0000 1274.0000
1731 1226.0000 1248.0000 1274.0000
1730 1226.0000 1268.0000 1274.0000
1729 1226.0000 1258.0000 1274.0000
1728 1226.0000 1274.0000 1274.0000
1727 1226.0000 1249.0000 1274.0000
1726 1226.0000 1252.0000 1274.0000
1725 1226.0000 1251.0000 1274.0000
1724 1226.0000 1245.0000 1274.0000
1723 1226.0000 1249.0000 1274.0000
1722 1226.0000 1253.0000 1274.0000
1721 1226.0000 1238.0000 1274.0000
1720 1226.0000 1246.0000 1274.0000
1719 1226.0000 1260.0000 1274.0000
1718 1226.0000 1240.0000 1274.0000
1717 1226.0000 1259.0000 1274.0000
1716 1226.0000 1260.0000 1274.0000
1715 1226.0000 1263.0000 1274.0000
1714 1226.0000 1260.0000 1274.0000
1713 1226.0000 1248.0000 1274.0000
1712 1226.0000 1258.0000 1274.0000
1
5
6
1.000000
3.000000
8.000000
3.000000
1.000000
6.000000
4.000000
4.000000
0
5.000000
0 10.000000
0 52.000000
0 47.000000
0 20.000000
0 10.000000
0 16.000000
0 25.000000
0
0
0
0
0
0 15.000000
0
0 14.000000
0 20.000000
0 19.000000
0
0
3.000000
0 12.000000
0 10.000000
0
Upper
Limit
29.487093
29.487093
29.487093
29.487093
29.487093
29.487093
29.487093
29.487093
29.487093
29.487093
29.487093
29.487093
29.487093
29.487093
29.487093
29.487093
29.487093
29.487093
29.487093
29.487093
29.487093
29.487093
29.487093
29.487093
. 29.487093
0 12.000000
0
Moving
Range
3 Sigma Limits with n=2 for
Moving Range
Special
Upper
Tests Lower
Limit Signaled Limit
1711 1226.0000 1270.0000 1274.0000
id
Lower
Limit
3 Sigma Limits with n=2 for
weight
Individual Measurements Chart Summary for weight
Individual Measurements Chart Summary for weight
3 Sigma Limits with n=2 for
Moving Range
The SHEWHART Procedure
The SHEWHART Procedure
3 Sigma Limits with n=2 for
weight
IMR CHARTS (MU and SIGMA specified)
IMR CHARTS (MU and SIGMA unknown)
IMR CHARTS (MU and SIGMA unknown)
The SHEWHART Procedure
IMR CHARTS (MU and SIGMA specified)
The SHEWHART Procedure
74
SAS Code for IMR Charts with Transformed Data
DM ’LOG; CLEAR; OUT; CLEAR;’;
* ODS PRINTER PDF file=’C:\COURSES\ST528\CRSNOTES\imrtrns.pdf’;
OPTIONS NODATE NONUMBER;
*****************************************;
*** Individual / Moving Range Charts ***;
*** With and Without Transformation
***;
*****************************************;
DATA liquid;
INPUT sample viscos @@;
logvis = log(viscos);
LINES;
1 13.68
2 17.49
3 18.87
6 6.84
7 8.46
8 48.60
11 37.20
12 28.78
13 21.17
16 31.85
17 20.02
18 39.24
21 10.95
22 20.46
23 33.22
26 13.72
27 40.28
28 18.05
31 23.49
32 15.72
33 9.01
36 37.30
37 18.25
38 48.16
;
SYMBOL1 VALUE=dot WIDTH=2 ;
4
9
14
19
24
29
34
39
13.79
19.38
20.79
23.99
13.76
16.54
17.14
24.96
5
10
15
20
25
30
35
40
15.36
20.57
28.27
17.05
12.71
18.06
19.92
26.54
PROC SHEWHART DATA=liquid;
IRCHART viscos*sample=’1’ / NPANELPOS=40
LTMPLOT = SCHEMATICID
LTMARGIN = 8 TESTS = 1 to 8
TABLETESTs LTESTS = 2
ZONELABELS ZONE2LABELS
VMINOR=3 HMINOR=4 SPLIT = ’/’;
LABEL viscos = ’VISCOSITY/MOVING RANGE’;
TITLE ’IMR CHARTS (no transformation)’;
PROC SHEWHART DATA=liquid;
IRCHART logvis*sample / NPANELPOS=40
LTMPLOT = SCHEMATICID
LTMARGIN = 8 TESTS = 1 to 8
TABLETESTs LTESTS = 2
ZONELABELS ZONE2LABELS
VMINOR=3 HMINOR=4 SPLIT = ’/’;
LABEL logvis = ’LOG(VISCOSITY)/MOVING RANGE’;
TITLE ’IMR CHARTS (with log transformation)’;
RUN;
75
39 -5.9491968 24.960000 50.431197
0 23.200000
34.635289
40 -5.9491968 26.540000 50.431197
0
34.635289
1.580000
IMR CHARTS (with log transformation)
The SHEWHART Procedure
Individual Measurements Chart Summary for logvis
3 Sigma Limits with n=2 for
logvis
sample
Lower
Limit
logvis
3 Sigma Limits with n=2 for
Moving Range
Special
Upper
Tests Lower
Limit Signaled Limit
Moving
Range
Upper
Limit
30 1.7914316 2.8936995 4.2120603
0 0.0879179
1.4870271
31 1.7914316 3.1565748 4.2120603
0 0.2628753
1.4870271
32 1.7914316 2.7549338 4.2120603
0 0.4016410
1.4870271
33 1.7914316 2.1983351 4.2120603
0 0.5565987
1.4870271
34 1.7914316 2.8414149 4.2120603
0 0.6430798
1.4870271
35 1.7914316 2.9917243 4.2120603
0 0.1503093
1.4870271
36 1.7914316 3.6189933 4.2120603
0 0.6272691
1.4870271
37 1.7914316 2.9041651 4.2120603
0 0.7148282
1.4870271
38 1.7914316 3.8745288 4.2120603
0 0.9703637
1.4870271
39 1.7914316 3.2172745 4.2120603
0 0.6572543
1.4870271
40 1.7914316 3.2786530 4.2120603
0 0.0613785
1.4870271
76
IMR CHARTS (no transformation)
IMR CHARTS (with log transformation)
The SHEWHART Procedure
The SHEWHART Procedure
Individual Measurements Chart Summary for viscos
3 Sigma Limits with n=2 for
viscos
sample
Lower
Limit
viscos
Individual Measurements Chart Summary for logvis
3 Sigma Limits with n=2 for
Moving Range
Special
Upper
Tests Lower
Limit Signaled Limit
Moving
Range
3 Sigma Limits with n=2 for
logvis
Upper
Limit
sample
Lower
Limit
logvis
3 Sigma Limits with n=2 for
Moving Range
Special
Upper
Tests Lower
Limit Signaled Limit
Moving
Range
Upper
Limit
1 -5.9491968 13.680000 50.431197
0
1 1.7914316 2.6159349 4.2120603
0
2 -5.9491968 17.490000 50.431197
0
3.810000
34.635289
2 1.7914316 2.8616293 4.2120603
0 0.2456944
1.4870271
3 -5.9491968 18.870000 50.431197
0
1.380000
34.635289
3 1.7914316 2.9375734 4.2120603
0 0.0759441
1.4870271
4 -5.9491968 13.790000 50.431197
0
5.080000
34.635289
4 1.7914316 2.6239437 4.2120603
0 0.3136297
1.4870271
5 -5.9491968 15.360000 50.431197
0
1.570000
34.635289
5 1.7914316 2.7317667 4.2120603
0 0.1078230
1.4870271
6 -5.9491968
6.840000
50.431197
0
8.520000
34.635289
6 1.7914316 1.9227877 4.2120603
0 0.8089790
1.4870271
7 -5.9491968
8.460000
50.431197
0
1.620000
34.635289
7 1.7914316 2.1353492 4.2120603
0 0.2125614
1.4870271
8 -5.9491968 48.600000 50.431197
0 40.140000
34.635289
8 1.7914316 3.8836235 4.2120603
0 1.7482744
1.4870271
9 -5.9491968 19.380000 50.431197
0 29.220000
34.635289
9 1.7914316 2.9642416 4.2120603
0 0.9193819
1.4870271
10 -5.9491968 20.570000 50.431197
0
1.190000
34.635289
10 1.7914316 3.0238337 4.2120603
0 0.0595921
1.4870271
11 -5.9491968 37.200000 50.431197
0 16.630000
34.635289
11 1.7914316 3.6163088 4.2120603
0 0.5924751
1.4870271
12 -5.9491968 28.780000 50.431197
0
8.420000
34.635289
12 1.7914316 3.3596807 4.2120603
0 0.2566281
1.4870271
13 -5.9491968 21.170000 50.431197
0
7.610000
34.635289
13 1.7914316 3.0525851 4.2120603
0 0.3070956
1.4870271
14 -5.9491968 20.790000 50.431197
0
0.380000
34.635289
14 1.7914316 3.0344721 4.2120603
0 0.0181130
1.4870271
15 -5.9491968 28.270000 50.431197
0
7.480000
34.635289
15 1.7914316 3.3418012 4.2120603
0 0.3073291
1.4870271
16 -5.9491968 31.850000 50.431197
0
3.580000
34.635289
16 1.7914316 3.4610374 4.2120603
0 0.1192362
1.4870271
17 -5.9491968 20.020000 50.431197
0 11.830000
34.635289
17 1.7914316 2.9967318 4.2120603
0 0.4643056
1.4870271
18 -5.9491968 39.240000 50.431197
0 19.220000
34.635289
18 1.7914316 3.6696966 4.2120603
0 0.6729649
1.4870271
19 -5.9491968 23.990000 50.431197
0 15.250000
34.635289
19 1.7914316 3.1776371 4.2120603
0 0.4920596
1.4870271
20 -5.9491968 17.050000 50.431197
0
6.940000
34.635289
20 1.7914316 2.8361502 4.2120603
0 0.3414869
1.4870271
21 -5.9491968 10.950000 50.431197
0
6.100000
34.635289
21 1.7914316 2.3933395 4.2120603
0 0.4428107
1.4870271
22 -5.9491968 20.460000 50.431197
0
9.510000
34.635289
0 0.6251323
1.4870271
IMR CHARTS (no transformation)
0 12.760000
22 1.7914316 3.0184718 4.2120603
34.635289
23 1.7914316 3.5031521 4.2120603
IMR CHARTS (with log transformation)
0 0.4846803
1.4870271
34.635289
24 1.7914316 2.6217658
4.2120603
The SHEWHART
Procedure 0 0.8813863 1.4870271
. 34.635289
23 -5.9491968 33.220000 50.431197
24 -5.9491968 13.760000
50.431197
The SHEWHART
Procedure
0 19.460000
25 -5.9491968 12.710000 50.431197
0 1.050000 34.635289
Individual Measurements Chart Summary for viscos
26 -5.9491968 13.720000 50.431197
0 1.010000 34.635289
3 Sigma Limits with n=2 for
3 Sigma Limits with n=2 for
27 -5.9491968 40.280000
50.431197
0 Moving
26.560000
34.635289
viscos
Range
28 -5.9491968 18.050000 50.431197 Special
0 22.230000 34.635289
Lower 16.540000 50.431197
Upper
Tests Lower0 1.510000
Moving 34.635289
Upper
29 -5.9491968
sample
Limit
viscos
Limit Signaled Limit
Range
Limit
5
. 1.4870271
25 1.7914316 2.5423891 4.2120603
0 0.0793767 1.4870271
Individual Measurements Chart Summary for logvis
26 1.7914316 2.6188546 4.2120603
0 0.0764655 1.4870271
3 Sigma Limits with n=2 for
3 Sigma Limits with n=2 for
27 1.7914316 3.6958551
4.2120603
0 Moving
1.0770004
1.4870271
logvis
Range
28 1.7914316 2.8931457 4.2120603 Special
0 0.8027094 1.4870271
Lower 2.8057817 4.2120603
Upper
Tests Lower0 0.0873640
Moving 1.4870271
Upper
29 1.7914316
sample
Limit
logvis
Limit Signaled Limit
Range
Limit
30 -5.9491968 18.060000 50.431197
0
1.520000
34.635289
30 1.7914316 2.8936995 4.2120603
0 0.0879179
1.4870271
31 -5.9491968 23.490000 50.431197
0
5.430000
34.635289
31 1.7914316 3.1565748 4.2120603
0 0.2628753
1.4870271
32 -5.9491968 15.720000 50.431197
0
7.770000
34.635289
32 1.7914316 2.7549338 4.2120603
0 0.4016410
1.4870271
33 -5.9491968
50.431197
0
6.710000
34.635289
33 1.7914316 2.1983351 4.2120603
0 0.5565987
1.4870271
34 -5.9491968 17.140000 50.431197
0
8.130000
34.635289
34 1.7914316 2.8414149 4.2120603
0 0.6430798
1.4870271
35 -5.9491968 19.920000 50.431197
0
2.780000
34.635289
35 1.7914316 2.9917243 4.2120603
0 0.1503093
1.4870271
36 -5.9491968 37.300000 50.431197
0 17.380000
34.635289
36 1.7914316 3.6189933 4.2120603
0 0.6272691
1.4870271
37 -5.9491968 18.250000 50.431197
0 19.050000
34.635289
37 1.7914316 2.9041651 4.2120603
0 0.7148282
1.4870271
38 -5.9491968 48.160000 50.431197
0 29.910000
34.635289
38 1.7914316 3.8745288 4.2120603
0 0.9703637
1.4870271
39 -5.9491968 24.960000 50.431197
0 23.200000
34.635289
39 1.7914316 3.2172745 4.2120603
0 0.6572543
1.4870271
40 -5.9491968 26.540000 50.431197
0
34.635289
40 1.7914316 3.2786530 4.2120603
0 0.0613785
1.4870271
9.010000
1.580000
77
6.7
Specification, Control, and Natural Tolerance Limits
• For a manufactured product, the specifications are the desired measurements for the quality
characteristic of interest.
• The upper specification limit (USL) and the lower specification limit (LSL) are,
respectively, the largest and smallest allowable values for a quality characteristic of interest.
• The specification limits for individual measurements, USL and LSL, are determined externally.
They are often set by management, the manufacturing engineers, the customer, or by product
developers and designers.
• For means, the control limits, UCL and LCL, are driven by the natural variability of the
process (measured by the process standard deviation σ).
• Thus, there is no mathematical or statistical relationship between data-derived process control
limits (LCL and UCL) and specification limits (USL and LSL).
• Natural tolerance limits UNTL and LNTL for individual measurements are also driven by
the process standard deviation σ. They are typically set at ±3σ around the process mean.
• When control charting individual measurements, it is helpful to include the specification limits
with the tolerance limits. Do not include specification limits on control charts when n ≥ 2.
6.8
Introduction to Process Capability
We will look at two ways of assessing the capability of a process.
1. Find the probability of meeting process specifications given the distribution of the individual
process measurements.
2. Calculate the process capability ratio Cp =
• If σ is not known, we use an estimate σ
b = R/d2
cp = USL − LSL
C
6R/d2
or
or
σ
b = s/c4 :
cp = USL − LSL
C
6s/c4
The larger the Cp , the better the process is for potentially meeting specifications.
• Warning: This still does not mean that in practice we are actually meeting specifications
because the Cp does not involve the mean of the process.
• Interpretation of Cp values: Assuming the process is running on aim (i.e., the process
mean µ = (LSL+USL)/2), then Cp is a measure of the capability of the process for
meeting specifications.
– Cp ≥ 1 indicates a process that is capable of meeting specifications because the natural tolerance limits can be contained within the specification limits. If measurements
are normally distributed, we would expect at least 99.7% of the measurements to
fall within specifications.
– Cp < 1 indicates a process not capable of meeting specifications because the natural
tolerance limits cannot be contained within the specification limits. If measurements
are normally distributed, we would expect less than 99.7% of the measurements to
fall within specifications (with the percentage decreasing as Cp → 0).
78
• The Cp can be interpreted in another way. The ratio P =
1
100% is the percentage
Cp
of the specification band that the process covers.
1
• When σ is unknown we use Pb =
100% as an estimate of P .
cp
C
6.9
Process Performance Measures
• The operating characteristic (OC) curve for X associated with samples of size n is a plot
of
β = P {LCL ≤ x ≤ UCL |µ = µ1 = µ0 + kσ}
against various process shifts from the on-aim process mean µ0 . The shifts are represented in
standard deviation units.
• For a process whose measurements are N (µ0 + kσ, σ) with k 6= 0, we get
β =
where typically L = 3. When k = 0, the process is in-control yielding
1−α =
• Thus, 1 − β is the probability of getting an out-of-control signal from a single sample of size
n. It is a measure of the ability of the X-chart to detect shifts in the process average.
• For any Shewhart control chart, the average run length (ARL) = the average number of
samples before an out-of-control signal is detected is
1
when the process is in-control.
α
1
ARL1 =
when the process is out-of-control having mean µ1 .
1−β
ARL0 =
• If samples are taken are regularly spaced time intervals h, another measure of process performance is the average time to signal (ATS) = ARL × h. The ATS is the average total
time until an out-of-control signal occurs.
• The SAS code to generate the following OC and ARL curves is posted on the course webpage.
79
OC Curves for Xbar Charts -- mu, sigma known
beta
1.00
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Shift in Population Mean (Unit=Std Dev)
Sample Size n:
1
2
3
4
80
5
10
15
20
ARL Curves for Xbar Charts -- mu, sigma known
arl
380
360
340
320
300
280
260
240
220
200
180
160
140
120
100
80
60
40
20
0
0.00
0.25
0.50
0.75
1.00
1.25
1.50
Shift in Population Mean (Unit=Std Dev)
Sample Size n:
1
2
3
4
81
5
10
15
20