6
Control Charts for Variables
6.1
Distribution of the Sample Range
• 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 X1 , X2 , . . . , Xn 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:
Z ∞
[Φ(x + r) − Φ(x)]n−2 φ(x)φ(x + r)dx
r>0
g(r; n) = n(n − 1)
−∞
and the CDF of R is:
Z ∞
[Φ(x + r) − Φ(x)]n−1 φ(x)dx
G(r; n) = n
Z−∞
∞
= n
[Φ(x + r) − Φ(x)]n−1 + [Φ(x − r) + Φ(x) − 1]n−1 φ(x)dx r > 0
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, 3, 4, 5 from a N (0, 1) distribution.
n
2
Exact values of E(X(1) ), E(X(n) ) and E(R)
3
4
5
3
3
2a
6a
5
− √
− √
1+
1+
− √
π
π
2 π
2 π
4 π
E(X(1) )
1
−√
π
E(X(n) )
1
√
π
3
√
2 π
3
√
2 π
2a
1+
π
3
√
π
3
√
π
E(R)
2
√
π
2a
1+
π
5
√
6a
1+
π
5
√
6a
1+
π
4 π
2 π
where a = arcsin(1/3) ≈ 0.3398369094.
2 ), E(X 2 ) and E(R2 )
Exact values of E(X(1)
(n)
n
2
3√
4√
√5
√
3
3
5 3 5b 3
2
E(X(1) )
1
1+
1+
1+
+
2π
π
4π
2π 2
√
√
√
√
3
3
5 3 5b 3
2
1+
1+
+
E(X(n) )
1
1+
2π
π
4π
2π 2
√
√
√
√
6+3 3
5 3 5b 3 60c
3 3
2
E(R )
2
2+
2+
2+
+
+ 2
π
π
2π
π2
π
√
where b = arcsin(1.4) ≈ .2526802552 and c = arcsin(1/ 6) ≈ 0.42053434.
45
6.2
Distribution of the Sample Standard Deviation
• The sample standard deviation S of a random sample X1 , X2 , . . . , Xn is
v
u
n
u 1 X
2
t
S =
Xi − X .
n − 1 i=1
• When taking a sample of size n from a N (µ, σ 2 ) distribution, the pdf of S is:
g(s; n) =
sν−1 ν ν/2 exp(−νs2 /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:
r
E(S) = σ
Γ( n2 )
2
n − 1 Γ( n−1
)
2
nσ 2 E(S)
E(S ) =
n−1
4
3
• From Jensen’s inequality:
E(S 2 ) = σ 2
E(S ) =
n+1
n−1
σ4
E(S) = E[(S 2 )1/2 ] < [E(S 2 )]1/2 = σ. So E(S) < σ.
• If we define
r
S
∗
=
)
n − 1 Γ( n−1
2
S
2
Γ( n2 )
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
6
Control Charts for Variables
53
47
6.3
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
If a sample
in 2control.
σ
of n independent units is taken from this population, then X ∼ N µ,
.
n
• Suppose m samples of size n are collected. For each sample, we can calculate the:
Means x1 , x2 , . . . , xm
Ranges R1 , R2 , . . . , Rm
6.3.1
and x = the mean of the m sample means
and R = the mean of the m sample ranges
For Known µ and σ
• The µx + 3σx control limits for the x-chart when µ and σ are known are:
3
A= √
n
UCL = µx + 3σx =
Centerline = µx = µ
(3)
LCL = µx − 3σx =
• 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 Xi ∼ N (µ, σ 2 ) for i = 1, 2, . . . , n. Let x1 , x2 , . . . , xn be a random sample (realization)
of size n.
• The range R = xmax − xmin .
• The relative range W = Rσ is a random variable with µW = d2 and σW = d3 . Values of d2
and d3 for various sample sizes are given in the table.
• Motivation: Note that we can rewrite R as R = W σ. Substitution yields:
µR = E(W σ) = σE(W ) = σd2 where the value of d2 = E(W ) depends on n.
2
σR2 = Var(W σ) = σ 2 Var(W ) = σ 2 σW
.
Thus, σR = σ σW = σd3 where the value of d3 depends on n.
• Using these values, the µR ± 3σR control limits for the R-chart are:
UCL = µR + 3σR =
Centerline = µR = d2 σ
LCL = µR − 3σR =
D2 = d2 + 3d3
(4)
D1 = d2 − 3d3
where D1 and D2 are constants that depend on sample size n and can be found in the table.
48
EXAMPLE 1: The following data represents m = 100 samples of size n = 4. The target is
µ = 100. Assume σ = 2. The data can be found in the file xchart.dat.
Samples 1 to 50
99.98
97.37
103.93
100.58
100.80
100.21
100.56
98.24
100.62
100.86
100.23
98.63
95.74
97.56
97.99
101.14
98.23
101.10
99.41
97.91
100.43
102.45
99.59
101.00
97.41
102.27
105.07
99.78
99.65
98.89
99.35
105.51
103.96
101.06
96.77
98.71
100.08
96.82
98.85
100.78
98.97
96.82
99.17
103.47
97.11
98.29
101.70
99.36
97.42
102.09
102.05
100.78
97.15
101.07
99.65
101.27
101.65
100.44
99.12
97.49
99.42
98.48
102.26
98.71
98.60
101.37
98.98
97.78
100.52
99.94
98.67
98.51
98.72
98.95
98.71
103.36
100.53
101.18
99.22
102.56
100.12
102.54
99.51
99.01
98.26
101.92
100.94
101.23
96.96
95.06
102.39
98.59
98.03
99.01
98.83
99.22
100.23
98.50
103.90
103.48
102.85 99.81
101.42 102.44
99.67 100.89
99.76 99.68
102.63 97.96
97.39 100.52
96.94 97.94
100.74 102.11
100.07 100.49
100.54 98.49
103.33 100.98
99.97 100.50
102.33 101.09
94.98 94.72
98.74 95.99
98.23 97.53
96.46 96.65
104.07 103.32
102.26 99.70
97.67 98.03
98.27 101.03
102.47 98.46
103.04 97.34
99.71 98.39
102.95 98.80
96.41 95.91
104.04 103.47
100.92 100.32
96.73 99.38
99.34 97.45
98.96 100.76
98.15 100.27
97.26 101.02
100.27 99.54
103.82 99.93
104.04 99.17
103.39 97.88
99.18 98.39
103.40 98.53
95.35 100.35
102.22 100.36
97.85 102.41
99.72 100.00
103.65 100.67
98.87 99.29
98.71 98.66
100.56 98.67
100.23 102.87
*92.69*101.20 (49)
98.27 99.51
Samples 51 to 100
98.73
99.03
99.80
101.33
100.86
98.53
102.10
99.72
97.17
99.63
102.49
99.33
103.09
101.78
99.22
98.16
101.02
100.23
103.35
101.35
100.34
98.72
100.52
99.08
100.03
99.43
99.04
99.03
100.25
100.14
103.67
100.67
99.24
101.35
103.45
98.20
99.01
100.83
100.47
99.24
98.76
100.15
99.72
101.58
99.00
98.96
99.01
99.67
95.65
99.22
49
103.48 99.77
99.50 98.56
97.13 98.90
100.14 99.75
101.43 100.62
98.77 100.17
99.96 99.27
100.48 99.80
97.61 98.86
100.99 98.96
98.10 100.65
101.59 100.29
100.38 105.95
102.26 103.68
98.28 100.44
100.34 98.10
103.31 97.08
96.63 98.63
100.01 99.73
97.71 101.09
99.65 98.44
97.35 99.86
97.75 97.62
98.41 99.29
99.31 100.71
100.13 96.95
98.30 99.94
99.86 100.43
101.64 101.58
102.45 102.76
101.29 100.17
100.65 101.58
100.83 99.76
98.09 102.01
101.28 103.74
101.88 102.30
99.22 98.73
103.30 102.47
99.48 98.38
102.65 99.67
*93.55*103.26
96.61 100.49
101.10 100.29
101.71 99.79
100.39 100.55
101.35 105.69
99.71 101.34
98.49 99.88
101.33 95.25
98.84 100.29
98.22
99.43
101.22
97.69
100.77
103.22
98.98
99.91
101.73
100.71
101.07
99.65
98.80
102.21
98.20
102.74
97.73
98.95
98.35
97.53
100.43
101.22
100.49
102.37
99.41
103.00
98.70
101.29
100.32
100.39
99.36
102.50
100.42
100.52
100.10
102.09
100.15
102.84
101.31
98.38
99.12 (91)
102.56
97.37
96.84
98.59
100.54
97.76
100.85
101.46
98.72
50
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
98.74500
100.55000
103.27750
99.48750
99.46750
99.51250
99.67250
100.47250
99.60000
98.38750
100.47250
101.56750
97.58000
99.56750
97.83000
96.49250
100.35500
99.39500
100.99000
99.34500
100.07500
100.38250
99.27250
99.84750
100.26000
100.27250
100.41000
100.50250
101.17250
Lower Subgroup
Limit
Mean
1
Subgroup
Sample
sample
Size
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
1
5
1
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2.920000
1.400000
4.540000
7.450000
5.540000
2.610000
5.700000
4.010000
2.760000
2.270000
2.850000
6.290000
2.520000
3.840000
2.750000
3.990000
6.590000
2.020000
3.910000
3.370000
1.500000
3.870000
4.710000
3.880000
4.670000
1.390000
6.780000
5.070000
3.040000
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
Special
Upper
Tests
Limit Signaled
3 Sigma Limits with n=4 for
Range
Special
Upper
Tests Lower Subgroup
Limit Signaled Limit
Range
3 Sigma Limits with n=4 for
Mean
Means and Ranges Chart Summary for response
The SHEWHART Procedure
XBAR AND RANGE CHARTS (KNOWN MU AND SIGMA)
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
99.97750
100.07750
100.17250
100.92000
99.72750
99.26250
99.13000
100.05000
100.83750
98.80250
100.24000
100.29000
98.72000
98.52500
101.70000
99.23000
98.91750
100.98500
97.88500
99.43500
98.90500
100.57250
100.96000
99.69500
99.97000
100.43750
101.61750
99.79750
99.56000
Lower Subgroup
Limit
Mean
30
Subgroup
Sample
sample
Size
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.760000
3.120000
4.690000
0.810000
3.640000
4.090000
0.940000
5.260000
5.210000
11.210000
4.370000
3.030000
0.930000
2.180000
4.640000
1.970000
5.590000
3.420000
5.720000
6.440000
4.410000
5.510000
5.330000
7.050000
2.050000
6.700000
7.360000
1.800000
5.110000
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
1
Special
Upper
Tests
Limit Signaled
3 Sigma Limits with n=4 for
Range
Special
Upper
Tests Lower Subgroup
Limit Signaled Limit
Range
3 Sigma Limits with n=4 for
Mean
Means and Ranges Chart Summary for response
The SHEWHART Procedure
XBAR AND RANGE CHARTS (KNOWN MU AND SIGMA)
51
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
99.27750
101.11750
102.14250
100.49250
100.06250
101.35000
101.12250
101.43500
100.94750
100.15250
98.99500
99.87750
99.86500
99.78750
99.09500
99.28750
99.71500
99.42000
100.36000
98.61000
99.78500
99.83500
99.03500
102.48250
102.05500
100.21500
100.57750
100.07250
98.84250
Lower Subgroup
Limit
Mean
59
Subgroup
Sample
sample
Size
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
2
5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1.420000
4.100000
3.640000
3.920000
1.590000
1.850000
4.310000
2.620000
1.390000
2.260000
1.640000
6.050000
1.400000
3.960000
2.900000
3.870000
1.990000
3.820000
5.000000
3.600000
6.230000
4.640000
2.240000
1.900000
7.150000
2.260000
4.390000
2.030000
4.560000
Special
Upper
Tests Lower Subgroup
Limit Signaled Limit
Range
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
Special
Upper
Tests
Limit Signaled
100
99
98
97
96
95
94
93
92
91
90
89
88
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
99.26750
98.42250
99.72250
99.45500
101.63500
99.63250
99.98000
99.62000
99.95250
98.67250
99.98500
99.91000
102.36000
Lower Subgroup
Limit
Mean
4 97.000000
Subgroup
Sample
sample
Size
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
4
0
0
0
0
0
0
0
0
0
0
0
0
0
1.570000
6.210000
2.360000
3.580000
6.730000
1.960000
4.870000
3.730000
5.950000
9.710000
4.270000
2.930000
2.470000
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
9.3963507
1
Special
Upper
Tests
Limit Signaled
3 Sigma Limits with n=4 for
Range
Special
Upper
Tests Lower Subgroup
Limit Signaled Limit
Range
3 Sigma Limits with n=4 for
Mean
Means and Ranges Chart Summary for response
Means and Ranges Chart Summary for response
3 Sigma Limits with n=4 for
Range
The SHEWHART Procedure
The SHEWHART Procedure
3 Sigma Limits with n=4 for
Mean
XBAR AND RANGE CHARTS (KNOWN MU AND SIGMA)
XBAR AND RANGE CHARTS (KNOWN MU AND SIGMA)
XBAR AND RANGE CHARTS (KNOWN MU AND SIGMA)
The SHEWHART Procedure
52
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 response @@; 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 response @@; 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;
53
6.3.2
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.
– xi and Ri should be computed for each of the preliminary samples.
Pm
xi
– The estimator of the unknown mean µ is µ
b = x = i=1 .
m
Pm
Ri
– The estimator of σ is σ
b = dR2 , where R = i=1 .
m
• Motivation for estimating σ based on sample ranges:
– Earlier we showed that µR = σd2 . This implies σ = µR /d2 .
σ
b
R
– Replacing µR with µ
bR = R, we get σ
b = R/d2 . Then σ
bx = √ = √
n
d2 n
• Substitution of the estimators into equations (3) and (4) for the unknown parameters yields
the following trial control limits for the x-chart:
σ
b
UCL = µ
b + 3√
=
n
Centerline = µ
b = x
σ
b
=
LCL = µ
b − 3√
n
A2 =
3
√
d2 n
(5)
• Motivation for estimating σR based on sample ranges:
– Earlier we showed that σR = σd3 . Replace σ with σ
b = R/d2 .
– Then σ
bR = σ
bd3 =
Rd3
. We then substitute µ
bR and σ
bR into µ
bR ± 3b
σR .
d2
• The trial control limits for the R-chart are:
b + 3b
UCL = R
σR =
D4 = 1 + 3
d3
d2
b = R
Centerline = R
(6)
b − 3b
LCL = R
σR =
D3 = 1 − 3
d3
d2
where D3 and D4 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 Ri 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
–
–
––
If any range points indicate an out-of-control process, an investigation for assignable
causes
should
carried
out. in-control samples are collected.
These can
be be
revised
as more
If
causeindicate
can be an
found,
delete theprocess,
point and
thefor
trial
control
If an
anyassignable
range points
out-of-control
an recompute
investigation
assignable
limits.
causes should be carried out.
–– If
be found,
one
two things
can be done.
If no
an assignable
assignable cause
cause can
is found,
delete
theofpoint
and recompute
the trial control limits.
point can
be can
deleted
and new
– If(i)noThe
assignable
cause
be found,
onelimits
of twocomputed.
things can Continue
be done. with the preceding
limits and
are found.
(i) test
The until
pointacceptable
can be deleted
new limits computed. Continue with the preceding
(ii) Retain
theacceptable
point alonglimits
withare
thefound.
trial control limits. Future points can be plotted to
test until
if they
inalong
control.
If the
so, accept
the limits
as Future
valid. points can be plotted to
(ii) see
Retain
theplot
point
with
trial control
limits.
see if trial
they limits
plot inare
control.
If so,
as valid.
• If the R-chart
accepted
as adopt
valid, the
thenlimits
perform
the same test on the x-chart,
subsettrial
of the
rules
proposed
If both
x andthe
R control
limits
x-chart,
• using
If theany
R-chart
limits
are
adoptedearlier.
as valid,
then the
perform
same test
on are
theaccepted
as
valid,
proceed
with
process
control
analysis.
using any subset of the rules proposed earlier. If both the x and R control limits are adopted
as valid,
proceed
process
control
analysis. process control testing can proceed.
• Once
valid
controlwith
limits
have been
computed,
• Once
valid control
have been
computed,
for process control can proceed.
– Samples
shouldlimits
be collected
from
the sametesting
process.
–– Compute
the values
xi same
and Rprocess.
Collect samples
fromofthe
i for each sample as the data becomes available.
–– Plot
the most
current
of x andasRthe
on data
the control
charts.
xi and
Ri forvalues
each sample
becomes
available.
Compute
–– Use
subset
of thevalues
rules of
discussed
in control
this paper
to determine if the process is
xi and Rearlier
charts.
Plotathese
current
i on the
in control.
– running
Use a subset
of the rules for the x-chart discussed earlier to determine if the process is
running
in control.
both charts
show an out-of-control signal for the same sample, it is suggested to search
– If
an assignable
for a changesignal
in variability
firstsample,
becauseit is
bringing
thetoprocess
– for
If both
charts showcause
an out-of-control
for the same
suggested
search
variability
under
control
may
return
the
process
to
the
in-control
state
on
the
x-chart.
for an assignable cause for a change in variability first because bringing the 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
variation
in melt
this quality
and relate
to raw material,
shift, is
and
changes
EXAMPLE
2: The
index ofproperty
an extrusion
grad it
polyethylene
compound
to other
be studied
to
in
the process.
Ability
process
to produce
trial control
limits
is also shift,
to be and
studied.
determine
variation
in of
thisthe
quality
property
and relate
it to raw
material,
otherSamples
changes
of
= process.
4 are selected
meltprocess
index to
values
are collected.
In limits
an initial
study,
was collected
in nthe
Abilityand
of the
produce
trial control
is also
to bedata
studied.
Samples
over
7 days
m = 20
samples.
The
dataiswith
the sample
and ranges
givenyielding
in the
of size
n = yielding
4 are selected
and
the melt
index
recorded.
Data means
was collected
overare
7 days
following
table. The data with the sample means and ranges are given in the following table.
m = 20 samples.
55
60
XBAR and RANGE Charts (Unknown MU and SIGMA)
The SHEWHART Procedure
XBAR and RANGE Charts (Unknown MU and SIGMA)
The SHEWHART Procedure
Means and Ranges Chart Summary for INDEX
3 Sigma Limits with n=4 for
Mean
SAMPLE
Subgroup
Sample
Size
Lower Subgroup
Limit
Mean
3 Sigma Limits with n=4 for
Range
Special
Upper
Tests Lower Subgroup
Limit Signaled Limit
Range
Special
Upper
Tests
Limit Signaled
1
4 221.37630
223.25000
248.69870
0
13.000000
42.788467
2
4 221.37630
236.25000
248.69870
0
19.000000
42.788467
3
4 221.37630
239.25000
248.69870
0
59.000000
42.788467
4
4 221.37630
236.50000
248.69870
0
39.000000
42.788467
5
4 221.37630
235.75000
248.69870
0
13.000000
42.788467
6
4 221.37630
244.25000
248.69870
0
33.000000
42.788467
7
4 221.37630
240.25000
248.69870
0
5.000000
42.788467
8
4 221.37630
247.75000
248.69870
5
0
31.000000
42.788467
9
4 221.37630
241.50000
248.69870
6
0
19.000000
42.788467
10
4 221.37630
229.00000
248.69870
0
18.000000
42.788467
11
4 221.37630
226.50000
248.69870
0
14.000000
42.788467
12
4 221.37630
233.75000
248.69870
0
16.000000
42.788467
13
4 221.37630
224.25000
248.69870
0
9.000000
42.788467
14
4 221.37630
225.75000
248.69870
0
10.000000
42.788467
15
4 221.37630
229.50000
248.69870
0
16.000000
42.788467
16
4 221.37630
236.25000
248.69870
0
9.000000
42.788467
17
4 221.37630
247.75000
248.69870
0
7.000000
42.788467
18
4 221.37630
239.75000
248.69870
0
17.000000
42.788467
19
4 221.37630
231.50000
248.69870
0
22.000000
42.788467
20
4 221.37630
232.00000
248.69870
0
6.000000
42.788467
56
56
1
XBAR and R Charts (Sample 3 Removed)
The SHEWHART Procedure
XBAR and R Charts (Samples 3,4,6,8 Removed)
The SHEWHART Procedure
57
58
4 222.69807
4 222.69807
4 222.69807
18
19
20
4 222.69807
12
4 222.69807
4 222.69807
11
17
4 222.69807
10
4 222.69807
4 222.69807
9
16
4 222.69807
8
4 222.69807
4 222.69807
7
4 222.69807
4 222.69807
6
15
4 222.69807
5
14
4 222.69807
4
4 222.69807
4 222.69807
13
4 222.69807
2
232.00000
231.50000
239.75000
247.75000
236.25000
229.50000
225.75000
224.25000
233.75000
226.50000
229.00000
241.50000
247.75000
240.25000
244.25000
235.75000
236.50000
236.25000
223.25000
Lower Subgroup
Limit
Mean
1
Subgroup
Sample
sample
Size
246.93351
246.93351
246.93351
246.93351
246.93351
246.93351
246.93351
246.93351
246.93351
246.93351
246.93351
246.93351
246.93351
246.93351
246.93351
246.93351
246.93351
246.93351
246.93351
1
6
5
6
1 5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
6.000000
22.000000
17.000000
7.000000
9.000000
16.000000
10.000000
9.000000
16.000000
14.000000
18.000000
19.000000
31.000000
5.000000
33.000000
13.000000
39.000000
19.000000
13.000000
Special
Upper
Tests Lower Subgroup
Limit Signaled Limit
Range
37.954121
37.954121
37.954121
37.954121
37.954121
37.954121
37.954121
37.954121
37.954121
37.954121
37.954121
37.954121
37.954121
37.954121
37.954121
37.954121
37.954121
37.954121
37.954121
1
Special
Upper
Tests
Limit Signaled
20
19
18
17
16
15
14
13
12
11
10
9
7
5
2
1
4 223.61305
4 223.61305
4 223.61305
4 223.61305
4 223.61305
4 223.61305
4 223.61305
4 223.61305
4 223.61305
4 223.61305
4 223.61305
4 223.61305
4 223.61305
4 223.61305
4 223.61305
232.00000
231.50000
239.75000
247.75000
236.25000
229.50000
225.75000
224.25000
233.75000
226.50000
229.00000
241.50000
240.25000
235.75000
236.25000
223.25000
Lower Subgroup
Limit
Mean
4 223.61305
Subgroup
Sample
sample
Size
243.01195
243.01195
243.01195
243.01195
243.01195
243.01195
243.01195
243.01195
243.01195
243.01195
243.01195
243.01195
243.01195
243.01195
243.01195
243.01195
1
6
5
5
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
6.000000
22.000000
17.000000
7.000000
9.000000
16.000000
10.000000
9.000000
16.000000
14.000000
18.000000
19.000000
5.000000
13.000000
19.000000
13.000000
Special
Upper
Tests Lower Subgroup
Limit Signaled Limit
Range
30.379811
30.379811
30.379811
30.379811
30.379811
30.379811
30.379811
30.379811
30.379811
30.379811
30.379811
30.379811
30.379811
30.379811
30.379811
30.379811
Special
Upper
Tests
Limit Signaled
3 Sigma Limits with n=4 for
Range
Means and Ranges Chart Summary for index
3 Sigma Limits with n=4 for
Mean
Means and Ranges Chart Summary for index
3 Sigma Limits with n=4 for
Range
The SHEWHART Procedure
The SHEWHART Procedure
3 Sigma Limits with n=4 for
Mean
XBAR and R Charts (Samples 3,4,6,8 Removed)
XBAR and R Charts (Sample 3 Removed)
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 shift @@;
DO item = 1 TO 4;
INPUT index @@; OUTPUT;
END;
LINES;
1 1 3 218 224 220 231
2 1 1 228
3 1 4 280 228 228 221
4 2 3 210
5 2 1 243 240 230 230
6 2 4 225
7 3 2 240 238 240 243
8 3 1 244
9 3 4 238 233 252 243
10 4 2 228
11 4 4 218 232 230 226
12 4 3 226
13 5 1 224 221 230 222
14 5 4 230
15 5 3 224 228 226 240
16 6 1 232
17 6 4 243 250 248 250
18 6 3 247
19 7 1 224 228 228 246
20 7 4 236
;
TITLE ’XBAR and RANGE Charts (Unknown MU
SYMBOL1 V=DOT WIDTH=1;
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
and SIGMA)’;
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 shift @@;
DO item = 1 TO 4;
INPUT index @@; 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
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(s2 ) = σ 2 but E(s) 6= σ.
• Therefore, there exists
a value c4 for each sample size n such that µs = E(s) = c4 σ where
12 Γ n2
s
2
. This implies E
c4 =
= σ.
n−1
c4
n−1
Γ 2
• It can also be shown that σs = σ
6.4.1
p
1 − c24 . Values of c4 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 (3).
• Motivation for the UCL and LCL:
– Recall that S is not an unbiased estimator of σ. But, for each n, there exists a constant
c4 such that µS = E(s) = c4 σ. Therefore, when plotting sample standard deviations, the
centerline should be at c4 σ.
q
2
2
2
– Because σs = σ (1 − c4 ), we get σs = σ 1 − c24 . 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 = µs + 3σs =
Centerline = µs = c4 σ
(7)
LCL = µs − 3σs =
Values of B5 and B6 are given in the table for various values of n.
Pn
• For each sample (i = 1, . . . , m), compute xi =
j=1
n
xij
sP
and si =
n
j=1 (xij
− xi )2
.
n−1
The value of si 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
61
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
98.74500
100.55000
103.27750
99.48750
99.46750
99.51250
99.67250
100.47250
99.60000
98.38750
100.47250
101.56750
97.58000
99.56750
97.83000
96.49250
100.35500
99.39500
100.99000
99.34500
100.07500
100.38250
99.27250
99.84750
100.26000
100.27250
100.41000
100.50250
101.17250
Lower Subgroup
Limit
Mean
1
Subgroup
Sample
sample
Size
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
1
5
1
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1.3550031
0.6270566
1.9476717
3.8733308
2.4069673
1.1296128
2.4286262
2.2950726
1.3376098
1.0457653
1.2810250
2.8223793
1.2249898
1.9716385
1.2691467
1.9567213
3.1287964
0.9956740
1.6850519
1.6222721
0.6785524
1.6026722
2.2004753
1.6976921
1.9638907
0.6693965
2.8163807
2.1973677
1.5124456
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
Special
Upper
Tests
Limit Signaled
3 Sigma Limits with n=4 for
Std Dev
Special
Upper
Tests Lower Subgroup
Limit Signaled Limit Std Dev
3 Sigma Limits with n=4 for
Mean
Means and Standard Deviations Chart Summary for response
The SHEWHART Procedure
XBAR AND S CHARTS (KNOWN MU AND SIGMA)
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
99.97750
100.07750
100.17250
100.92000
99.72750
99.26250
99.13000
100.05000
100.83750
98.80250
100.24000
100.29000
98.72000
98.52500
101.70000
99.23000
98.91750
100.98500
97.88500
99.43500
98.90500
100.57250
100.96000
99.69500
99.97000
100.43750
101.61750
99.79750
99.56000
Lower Subgroup
Limit
Mean
4 97.000000
Subgroup
Sample
sample
Size
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.3439356
1.4095951
2.1565462
0.3541186
1.5154840
1.7126661
0.4327432
2.3758928
2.3738769
4.8650617
1.8902381
1.2502000
0.3823611
0.9660055
2.2533235
0.8711295
2.4388436
1.6278923
3.1018328
2.7693621
1.8342755
2.2778261
2.4949683
3.0378556
0.8915530
2.8109355
3.1539327
0.8025117
2.1563395
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
1
Special
Upper
Tests
Limit Signaled
3 Sigma Limits with n=4 for
Std Dev
Special
Upper
Tests Lower Subgroup
Limit Signaled Limit Std Dev
3 Sigma Limits with n=4 for
Mean
Means and Standard Deviations Chart Summary for response
The SHEWHART Procedure
XBAR AND S CHARTS (KNOWN MU AND SIGMA)
XBAR AND S CHARTS (KNOWN MU AND SIGMA)
The SHEWHART Procedure
62
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
78
79
80
81
82
83
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
5
103.00000
Procedure
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.7035328
0.8808708
1.8736306
1.3607473
0.7659145
0.9515733
0.6988324
2.4889271
0.6471218
1.7620892
1.6290386
1.6470453
0.9186403
2.0824665
2.1212889
1.4900559
2.9149557
2.1992347
1.0449083
0.8268968
3.1434005
0.9997833
1.8297063
0.9454584
2.0537993
100.49250
103.00000
4 97.000000
3 Sigma Limits with n=4 for
102.14250
103.00000
Mean
4 97.000000
1.7137556
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
3 Sigma Limits with n=4 for
0 1.7490259
Std Dev 4.1754987
0
Means and Standard Deviations Chart Summary for response
100.06250
101.35000
103.00000
The SHEWHART
101.12250
101.43500
100.94750
100.15250
98.99500
99.87750
99.86500
99.78750
99.09500
99.28750
99.71500
99.42000
100.36000
98.61000
99.78500
99.83500
99.03500
102.48250
102.05500
100.21500
100.57750
103.00000
103.00000
Special
Upper
Tests
Limit Signaled
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
4 97.000000
88
89
90
91
92
93
94
95
96
97
98
99
100
99.26750
98.42250
99.72250
99.45500
101.63500
99.63250
99.98000
99.62000
99.95250
98.67250
99.98500
99.91000
102.36000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
103.00000
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0.7142070
3.4366396
0.9691706
1.4932403
2.8797511
0.9836115
2.2692583
1.6033091
2.4697689
3.9788221
1.8558466
1.2648320
1.0750194
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
4.1754987
86 Subgroup4 97.000000 101.11750 103.00000 Special
2
0 1.9525432 4.1754987 Special
Sample4 97.000000
Lower Subgroup
Upper
Tests Lower0 Subgroup
Upper
Tests
87
99.27750 103.00000
0.6153251 4.1754987
sample
Size
Limit
Mean
Limit Signaled Limit Std Dev
Limit Signaled
85
84
4 97.000000
77
98.84250
100.07250
Special
Upper
Tests Lower Subgroup
Limit Signaled Limit Std Dev
3 Sigma Limits with n=4 for
Std Dev
XBAR AND S CHARTS (KNOWN MU AND SIGMA)
4 97.000000
4 97.000000
70
76
4 97.000000
69
4 97.000000
4 97.000000
68
75
4 97.000000
67
4 97.000000
4 97.000000
66
74
4 97.000000
65
4 97.000000
4 97.000000
64
73
4 97.000000
63
4 97.000000
4 97.000000
62
72
4 97.000000
61
4 97.000000
4 97.000000
71
4 97.000000
Lower Subgroup
Limit
Mean
60
Subgroup
Sample
Size
59
sample
3 Sigma Limits with n=4 for
Mean
Means and Standard Deviations Chart Summary for response
The SHEWHART Procedure
XBAR AND S CHARTS (KNOWN MU AND SIGMA)
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:
Pm
xi
x = i=1 be the mean of the sample means and
m
Pm
si
s = i=1 be the mean of the sample standard deviations.
m
• Therefore, the estimator of µ is x.
• Because E(s) = E(si ) for each i, we have E(s) = c4 σ. It follows that E
s
Thus, an unbiased estimator of σ is σ
b=
and, σ
bs = σ
b
c4
s
c4
= σ.
q
q
s
2
1 − c4 =
1 − c24 .
c4
• The trial control limits for the x-chart are:
σ
b
=
UCL = µ
b + 3√
n
Centerline = µ
b=x
(8)
σ
b
=
LCL = µ
b − 3√
n
• The trial control limits for the s-chart are:
UCL = µ
bs + 3b
σs =
Centerline = µ
bs = s
(9)
LCL = µ
bs + 3b
σs =
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 were tested.
• That is, plot the si values on the s-chart analogously to the way the Ri values are plotted on
the R-chart.
• Once acceptable control limits have been found for both charts, proceed with process control
analysis.
63
XBAR and S Charts (Unknown MU and SIGMA)
The SHEWHART Procedure
XBAR and S Charts (Unknown MU and SIGMA)
The SHEWHART Procedure
Means and Standard Deviations Chart Summary for index
3 Sigma Limits with n=4 for
Mean
sample
Subgroup
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
4 221.43037
223.25000
248.64463
0
5.737305
18.938856
2
4 221.43037
236.25000
248.64463
0
7.932003
18.938856
3
4 221.43037
239.25000
248.64463
0
27.366342
18.938856
4
4 221.43037
236.50000
248.64463
0
17.972201
18.938856
5
4 221.43037
235.75000
248.64463
0
6.751543
18.938856
6
4 221.43037
244.25000
248.64463
0
14.056434
18.938856
7
4 221.43037
240.25000
248.64463
0
2.061553
18.938856
8
4 221.43037
247.75000
248.64463
5
0
12.919623
18.938856
9
4 221.43037
241.50000
248.64463
6
0
8.103497
18.938856
10
4 221.43037
229.00000
248.64463
0
7.393691
18.938856
11
4 221.43037
226.50000
248.64463
0
6.191392
18.938856
12
4 221.43037
233.75000
248.64463
0
6.849574
18.938856
13
4 221.43037
224.25000
248.64463
0
4.031129
18.938856
14
4 221.43037
225.75000
248.64463
0
4.193249
18.938856
15
4 221.43037
229.50000
248.64463
0
7.187953
18.938856
16
4 221.43037
236.25000
248.64463
0
4.924429
18.938856
17
4 221.43037
247.75000
248.64463
0
3.304038
18.938856
18
4 221.43037
239.75000
248.64463
0
7.500000
18.938856
19
4 221.43037
231.50000
248.64463
0
9.848858
18.938856
20
4 221.43037
232.00000
248.64463
0
2.828427
18.938856
56
64
1