control charts for variables distribution of the sample range · 6 control charts for variables 6.1...
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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 thesample range R and the sample standard deviation s. A brief summary of these distributionswhen 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:
g(r;n) = n(n− 1)
∫ ∞−∞
[Φ(x+ r)− Φ(x)]n−2 φ(x)φ(x+ r)dx r > 0
and the CDF of R is:
G(r;n) = n
∫ ∞−∞
[Φ(x+ r)− Φ(x)]n−1 φ(x)dx
= n
∫ ∞0
{[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, σ2) distribution, then the relative range W = R/σ haspdf g(r;n).
• The moments of the range R can be derived from either the pdf above or from the momentsof minimum and maximum order statistics X(1) and X(n). The following tables contain thefirst two moments of X(1), X(n), and R for n = 2, 3, 4, 5 from a N(0, 1) distribution.
Exact values of E(X(1)), E(X(n)) and E(R)
n 2 3 4 5
E(X(1)) − 1√π
− 3
2√π
− 3
2√π
(1 +
2a
π
)− 5
4√π
(1 +
6a
π
)
E(X(n))1√π
3
2√π
3
2√π
(1 +
2a
π
)5
4√π
(1 +
6a
π
)
E(R)2√π
3√π
3√π
(1 +
2a
π
)5
2√π
(1 +
6a
π
)where a = arcsin(1/3) ≈ 0.3398369094.
Exact values of E(X2(1)), E(X2
(n)) and E(R2)
n 2 3 4 5
E(X2(1)) 1 1 +
√3
2π1 +
√3
π1 +
5√3
4π+
5b√3
2π2
E(X2(n)) 1 1 +
√3
2π1 +
√3
π1 +
5√3
4π+
5b√3
2π2
E(R2) 2 2 +3√3
π2 +
6 + 3√3
π2 +
5√3
2π+
5b√3
π2+
60c
π2
where b = arcsin(1.4) ≈ .2526802552 and c = arcsin(1/√6) ≈ 0.42053434.
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6.2 Distribution of the Sample Standard Deviation
• The sample standard deviation S of a random sample X1, X2, . . . , Xn is
S =
√√√√ 1
n− 1
n∑i=1
(Xi −X
)2.
• When taking a sample of size n from a N(µ, σ2) distribution, the pdf of S is:
g(s;n) =sν−1 νν/2 exp(−νs2/2σ2)
2(ν−2)/2 σν Γ(ν/2)s > 0
where ν = n− 1 ≥ 1.
• The first four moments of S are:
E(S) = σ
√2
n− 1
Γ(n2)
Γ(n−12
)E(S2) = σ2
E(S3) =nσ2E(S)
n− 1E(S4) =
(n+ 1
n− 1
)σ4
• From Jensen’s inequality: E(S) = E[(S2)1/2] < [E(S2)]1/2 = σ. So E(S) < σ.
• If we define
S∗ =
√n− 1
2
Γ(n−12
)
Γ(n2)S
then E(S∗) = σ.
• Therefore, we can use the sample standard deviation to get an unbiased estimate of σ is wejust 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 theappropriate 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 construct-ing 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 alsonecessary 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 whichassess the stability of the variability of a process.
46
6 Control Charts for Variables
5347
6.3 x and R-charts
• Suppose the goal is to control the mean of some quality characteristic. Let random variableX correspond to the quality characteristic from a unit sampled from an in-control process.
• Suppose it is known that X ∼ N(µ, σ2) 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:
Means x1, x2, . . . , xm and x = the mean of the m sample meansRanges R1, R2, . . . , Rm and R = the mean of the m sample ranges
6.3.1 For Known µ and σ
• The µx + 3σx control limits for the x-chart when µ and σ are known are:
UCL = µx + 3σx = A =3√n
Centerline = µx = µ (3)
LCL = µx − 3σx =
• To construct an R-chart, information about the relationship between the sample range R andthe standard deviation σ from a normal distribution is needed.
• Suppose Xi ∼ N(µ, σ2) 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 d2and d3 for various sample sizes are given in the table.
• Motivation: Note that we can rewrite R as R = Wσ. Substitution yields:
µR = E(Wσ) = σE(W ) = σd2 where the value of d2 = E(W ) depends on n.
σ2R = Var(Wσ) = σ2Var(W ) = σ2σ2
W .
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 = D2 = d2 + 3d3
Centerline = µR = d2σ (4)
LCL = µR − 3σR = D1 = d2 − 3d3
where D1 and D2 are constants that depend on sample size n and can be found in the table.
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EXAMPLE 1: The following data represents m = 100 samples of size n = 4. The target isµ = 100. Assume σ = 2. The data can be found in the file xchart.dat.
Samples 1 to 50 Samples 51 to 100
99.98 102.05 102.85 99.81 98.73 103.48 99.77 98.2297.37 100.78 101.42 102.44 99.03 99.50 98.56 99.43
103.93 97.15 99.67 100.89 99.80 97.13 98.90 101.22100.58 101.07 99.76 99.68 101.33 100.14 99.75 97.69100.80 99.65 102.63 97.96 100.86 101.43 100.62 100.77100.21 101.27 97.39 100.52 98.53 98.77 100.17 103.22100.56 101.65 96.94 97.94 102.10 99.96 99.27 98.9898.24 100.44 100.74 102.11 99.72 100.48 99.80 99.91
100.62 99.12 100.07 100.49 97.17 97.61 98.86 101.73100.86 97.49 100.54 98.49 99.63 100.99 98.96 100.71100.23 99.42 103.33 100.98 102.49 98.10 100.65 101.0798.63 98.48 99.97 100.50 99.33 101.59 100.29 99.6595.74 102.26 102.33 101.09 103.09 100.38 105.95 98.8097.56 98.71 94.98 94.72 101.78 102.26 103.68 102.2197.99 98.60 98.74 95.99 99.22 98.28 100.44 98.20
101.14 101.37 98.23 97.53 98.16 100.34 98.10 102.7498.23 98.98 96.46 96.65 101.02 103.31 97.08 97.73
101.10 97.78 104.07 103.32 100.23 96.63 98.63 98.9599.41 100.52 102.26 99.70 103.35 100.01 99.73 98.3597.91 99.94 97.67 98.03 101.35 97.71 101.09 97.53
100.43 98.67 98.27 101.03 100.34 99.65 98.44 100.43102.45 98.51 102.47 98.46 98.72 97.35 99.86 101.2299.59 98.72 103.04 97.34 100.52 97.75 97.62 100.49
101.00 98.95 99.71 98.39 99.08 98.41 99.29 102.3797.41 98.71 102.95 98.80 100.03 99.31 100.71 99.41
102.27 103.36 96.41 95.91 99.43 100.13 96.95 103.00105.07 100.53 104.04 103.47 99.04 98.30 99.94 98.7099.78 101.18 100.92 100.32 99.03 99.86 100.43 101.2999.65 99.22 96.73 99.38 100.25 101.64 101.58 100.3298.89 102.56 99.34 97.45 100.14 102.45 102.76 100.3999.35 100.12 98.96 100.76 103.67 101.29 100.17 99.36
105.51 102.54 98.15 100.27 100.67 100.65 101.58 102.50103.96 99.51 97.26 101.02 99.24 100.83 99.76 100.42101.06 99.01 100.27 99.54 101.35 98.09 102.01 100.5296.77 98.26 103.82 99.93 103.45 101.28 103.74 100.1098.71 101.92 104.04 99.17 98.20 101.88 102.30 102.09
100.08 100.94 103.39 97.88 99.01 99.22 98.73 100.1596.82 101.23 99.18 98.39 100.83 103.30 102.47 102.8498.85 96.96 103.40 98.53 100.47 99.48 98.38 101.31
100.78 95.06 95.35 100.35 99.24 102.65 99.67 98.3898.97 102.39 102.22 100.36 98.76 *93.55*103.26 99.12 (91)96.82 98.59 97.85 102.41 100.15 96.61 100.49 102.5699.17 98.03 99.72 100.00 99.72 101.10 100.29 97.37
103.47 99.01 103.65 100.67 101.58 101.71 99.79 96.8497.11 98.83 98.87 99.29 99.00 100.39 100.55 98.5998.29 99.22 98.71 98.66 98.96 101.35 105.69 100.54
101.70 100.23 100.56 98.67 99.01 99.71 101.34 97.7699.36 98.50 100.23 102.87 99.67 98.49 99.88 100.8597.42 103.90 *92.69*101.20 (49) 95.65 101.33 95.25 101.46
102.09 103.48 98.27 99.51 99.22 98.84 100.29 98.72
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art
Su
mm
ary
for
resp
onse
3 S
igm
a L
imit
s w
ith
n=
4 fo
rM
ean
3 S
igm
a L
imit
s w
ith
n=
4 fo
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ange
sam
ple
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oup
Sam
ple
Siz
eL
ower
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ub
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0.21
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0000
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2.26
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51
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=MU0SIGMA0=2 RSYMBOL=R0TESTS = 1 TO 8 LTESTS = 2TESTS2 = 1TABLETESTS 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=MU0SIGMA0=2 SSYMBOL=S0TESTS = 1 TO 8 LTESTS = 2 TESTS2 = 1TABLETESTS 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 preliminarysamples must be collected in order to compute estimates of µ and σ.
– xi and Ri should be computed for each of the preliminary samples.
– The estimator of the unknown mean µ is µ̂ = x =
∑mi=1 xim
.
– The estimator of σ is σ̂ = Rd2
, where R =
∑mi=1Ri
m.
• Motivation for estimating σ based on sample ranges:
– Earlier we showed that µR = σd2. This implies σ = µR/d2.
– Replacing µR with µ̂R = R, we get σ̂ = R/d2. Then σ̂x =σ̂√n
=R
d2√n
• Substitution of the estimators into equations (3) and (4) for the unknown parameters yieldsthe following trial control limits for the x-chart:
UCL = µ̂+ 3σ̂√n
= A2 =3
d2√n
Centerline = µ̂ = x (5)
LCL = µ̂− 3σ̂√n
=
• Motivation for estimating σR based on sample ranges:
– Earlier we showed that σR = σd3. Replace σ with σ̂ = R/d2.
– Then σ̂R = σ̂d3 =Rd3d2
. We then substitute µ̂R and σ̂R into µ̂R ± 3σ̂R.
• The trial control limits for the R-chart are:
UCL = R̂ + 3σ̂R = D4 = 1 + 3d3d2
Centerline = R̂ = R (6)
LCL = R̂− 3σ̂R = D3 = 1− 3d3d2
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 goodpractice 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 thepreliminary samples were taken. When testing with the R-chart, it is common to use Rule 1only 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 limitsas valid control limits for future process control testing.
54
– These can be revised as more in-control samples are collected.
– If any range points indicate an out-of-control process, an investigation for assignablecauses should be carried out.
– If an assignable cause is found, delete the point and recompute the trial control limits.
– If no assignable cause can be found, one of two things can be done.
(i) The point can be deleted and new limits computed. Continue with the precedingtest until acceptable limits are found.
(ii) Retain the point along with the trial control limits. Future points can be plotted tosee if they plot in control. If so, adopt the limits as valid.
• If the R-chart trial limits are adopted as valid, then perform the same test on the x-chart,using any subset of the rules proposed earlier. If both the x and R control limits are adoptedas valid, proceed with process control analysis.
• Once valid control limits have been computed, testing for process control can proceed.
– Collect samples from the same process.
– Compute xi and Ri for each sample as the data becomes available.
– Plot these current values of xi and Ri on the control charts.
– Use a subset of the rules for the x-chart discussed earlier to determine if the process isrunning in control.
– If both charts show an out-of-control signal for the same sample, it is suggested to searchfor an assignable cause for a change in variability first because bringing the processvariability 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 todetermine variation in this quality property and relate it to raw material, shift, and other changesin the process. Ability of the process to produce trial control limits is also to be studied. Samplesof size n = 4 are selected and the melt index is recorded. Data was collected over 7 days yieldingm = 20 samples. The data with the sample means and ranges are given in the following table.
– If any range points indicate an out-of-control process, an investigation for assignablecauses should be carried out.
– If an assignable cause can be found, delete the point and recompute the trial controllimits.
– If no assignable cause can be found, one of two things can be done.
(i) The point can be deleted and new limits computed. Continue with the precedingtest until acceptable limits are found.
(ii) Retain the point along with the trial control limits. Future points can be plotted tosee if they plot in control. If so, accept the limits as valid.
• If the R-chart trial limits are accepted as valid, then perform the same test on the x-chart,using any subset of the rules proposed earlier. If both the x and R control limits are acceptedas valid, proceed with process control analysis.
• Once valid control limits have been computed, process control testing can proceed.
– Samples should be collected from the same process.
– Compute the values of xi and Ri for each sample as the data becomes available.
– Plot the most current values of x and R on the control charts.
– Use a subset of the rules discussed earlier in this paper to determine if the process isrunning in control.
– If both charts show an out-of-control signal for the same sample, it is suggested to searchfor an assignable cause for a change in variability first because bringing the processvariability 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 todetermine variation in this quality property and relate it to raw material, shift, and other changesin the process. Ability of the process to produce trial control limits is also to be studied. Samplesof n = 4 are selected and melt index values are collected. In an initial study, data was collectedover 7 days yielding m = 20 samples. The data with the sample means and ranges are given in thefollowing table.
6055
XBAR and RANGE Charts (Unknown MU and SIGMA)
The SHEWHART Procedure
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 forMean
3 Sigma Limits with n=4 forRange
SAMPLE
SubgroupSample
SizeLower
LimitSubgroup
MeanUpper
Limit
SpecialTests
SignaledLower
LimitSubgroup
RangeUpper
Limit
SpecialTests
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 1
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 56 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
XBAR and R Charts (Sample 3 Removed)
The SHEWHART Procedure
XBAR and R Charts (Samples 3,4,6,8 Removed)
The SHEWHART Procedure
57
XB
AR
an
d R
Ch
arts
(S
ampl
es 3
,4,6
,8 R
emov
ed)
Th
e S
HE
WH
AR
T P
roce
dure
XB
AR
an
d R
Ch
arts
(S
ampl
es 3
,4,6
,8 R
emov
ed)
Th
e S
HE
WH
AR
T P
roce
dure
Mea
ns
and
Ran
ges
Ch
art
Su
mm
ary
for
ind
ex
3 S
igm
a L
imit
s w
ith
n=
4 fo
rM
ean
3 S
igm
a L
imit
s w
ith
n=
4 fo
rR
ange
sam
ple
Su
bgr
oup
Sam
ple
Siz
eL
ower
Lim
itS
ub
grou
pM
ean
Up
per
Lim
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ests
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nal
edL
ower
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ub
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ests
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nal
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14
223.
6130
522
3.25
000
243.
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51
013
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7981
1
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50
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0000
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811
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50
13.0
0000
030
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811
74
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6130
524
0.25
000
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0119
50
5.00
0000
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7981
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6130
524
1.50
000
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0119
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019
.000
000
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7981
1
104
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6130
522
9.00
000
243.
0119
50
18.0
0000
030
.379
811
114
223.
6130
522
6.50
000
243.
0119
50
14.0
0000
030
.379
811
124
223.
6130
523
3.75
000
243.
0119
50
16.0
0000
030
.379
811
134
223.
6130
522
4.25
000
243.
0119
55
09.
0000
0030
.379
811
144
223.
6130
522
5.75
000
243.
0119
56
010
.000
000
30.3
7981
1
154
223.
6130
522
9.50
000
243.
0119
50
16.0
0000
030
.379
811
164
223.
6130
523
6.25
000
243.
0119
50
9.00
0000
30.3
7981
1
174
223.
6130
524
7.75
000
243.
0119
51
07.
0000
0030
.379
811
184
223.
6130
523
9.75
000
243.
0119
50
17.0
0000
030
.379
811
194
223.
6130
523
1.50
000
243.
0119
50
22.0
0000
030
.379
811
204
223.
6130
523
2.00
000
243.
0119
50
6.00
0000
30.3
7981
1
XB
AR
an
d R
Ch
arts
(S
ampl
e 3
Rem
oved
)
Th
e S
HE
WH
AR
T P
roce
dure
XB
AR
an
d R
Ch
arts
(S
ampl
e 3
Rem
oved
)
Th
e S
HE
WH
AR
T P
roce
dure
Mea
ns
and
Ran
ges
Ch
art
Su
mm
ary
for
ind
ex
3 S
igm
a L
imit
s w
ith
n=
4 fo
rM
ean
3 S
igm
a L
imit
s w
ith
n=
4 fo
rR
ange
sam
ple
Su
bgr
oup
Sam
ple
Siz
eL
ower
Lim
itS
ub
grou
pM
ean
Up
per
Lim
it
Sp
ecia
lT
ests
Sig
nal
edL
ower
Lim
itS
ub
grou
pR
ange
Up
per
Lim
it
Sp
ecia
lT
ests
Sig
nal
ed
14
222.
6980
722
3.25
000
246.
9335
10
13.0
0000
037
.954
121
24
222.
6980
723
6.25
000
246.
9335
10
19.0
0000
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.954
121
44
222.
6980
723
6.50
000
246.
9335
10
39.0
0000
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121
1
54
222.
6980
723
5.75
000
246.
9335
10
13.0
0000
037
.954
121
64
222.
6980
724
4.25
000
246.
9335
10
33.0
0000
037
.954
121
74
222.
6980
724
0.25
000
246.
9335
10
5.00
0000
37.9
5412
1
84
222.
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724
7.75
000
246.
9335
11
50
31.0
0000
037
.954
121
94
222.
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724
1.50
000
246.
9335
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019
.000
000
37.9
5412
1
104
222.
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9.00
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246.
9335
10
18.0
0000
037
.954
121
114
222.
6980
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6.50
000
246.
9335
10
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121
124
222.
6980
723
3.75
000
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9335
10
16.0
0000
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121
134
222.
6980
722
4.25
000
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9335
15
09.
0000
0037
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121
144
222.
6980
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5.75
000
246.
9335
16
010
.000
000
37.9
5412
1
154
222.
6980
722
9.50
000
246.
9335
10
16.0
0000
037
.954
121
164
222.
6980
723
6.25
000
246.
9335
10
9.00
0000
37.9
5412
1
174
222.
6980
724
7.75
000
246.
9335
11
07.
0000
0037
.954
121
184
222.
6980
723
9.75
000
246.
9335
10
17.0
0000
037
.954
121
194
222.
6980
723
1.50
000
246.
9335
10
22.0
0000
037
.954
121
204
222.
6980
723
2.00
000
246.
9335
10
6.00
0000
37.9
5412
1
58
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 1 228 236 247 2343 1 4 280 228 228 221 4 2 3 210 249 241 2465 2 1 243 240 230 230 6 2 4 225 250 258 2447 3 2 240 238 240 243 8 3 1 244 248 265 2349 3 4 238 233 252 243 10 4 2 228 238 220 230
11 4 4 218 232 230 226 12 4 3 226 231 236 24213 5 1 224 221 230 222 14 5 4 230 220 227 22615 5 3 224 228 226 240 16 6 1 232 240 241 23217 6 4 243 250 248 250 18 6 3 247 238 244 23019 7 1 224 228 228 246 20 7 4 236 230 230 232;
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 = 1TABLETEST 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 = 1TABLETEST 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 infor-mation 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 than10. 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) = σ2 but E(s) 6= σ.
• Therefore, there exists a value c4 for each sample size n such that µs = E(s) = c4σ where
c4 =
(2
n− 1
) 12
Γ
(n2
)Γ
(n−12
) . This implies E
(s
c4
)= σ.
• It can also be shown that σs = σ√
1− c24. Values of c4 can be found in the table.
6.4.1 For Known µ and σ
• The control limits for the x-chart when both µ and σ are known can be computed using theformulas 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 constantc4 such that µS = E(s) = c4σ. Therefore, when plotting sample standard deviations, thecenterline should be at c4σ.
– Because σ2s = σ2(1 − c24), 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 = c4σ (7)
LCL = µs − 3σs =
Values of B5 and B6 are given in the table for various values of n.
• For each sample (i = 1, . . . ,m), compute xi =
∑nj=1 xij
nand si =
√∑nj=1(xij − xi)2n− 1
.
The value of si is then plotted against i on the s-chart. Use Rule 1 and the above controllimits to determine if the variability of the process characteristic in control.
60
XB
AR
AN
D S
CH
AR
TS
(K
NO
WN
MU
AN
D S
IGM
A)
Th
e S
HE
WH
AR
T P
roce
dure
Mea
ns
and
Sta
nd
ard
Dev
iati
ons
Ch
art
Su
mm
ary
for
resp
onse
3 S
igm
a L
imit
s w
ith
n=
4 fo
rM
ean
3 S
igm
a L
imit
s w
ith
n=
4 fo
rS
td D
ev
sam
ple
Su
bgr
oup
Sam
ple
Siz
eL
ower
Lim
itS
ub
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Up
per
Lim
it
Sp
ecia
lT
ests
Sig
nal
edL
ower
Lim
itS
ub
grou
pS
tdD
evU
pp
erL
imit
Sp
ecia
lT
ests
Sig
nal
ed
304
97.0
0000
099
.560
0010
3.00
000
02.
1563
395
4.17
5498
7
314
97.0
0000
099
.797
5010
3.00
000
00.
8025
117
4.17
5498
7
324
97.0
0000
010
1.61
750
103.
0000
00
3.15
3932
74.
1754
987
334
97.0
0000
010
0.43
750
103.
0000
00
2.81
0935
54.
1754
987
344
97.0
0000
099
.970
0010
3.00
000
00.
8915
530
4.17
5498
7
354
97.0
0000
099
.695
0010
3.00
000
03.
0378
556
4.17
5498
7
364
97.0
0000
010
0.96
000
103.
0000
00
2.49
4968
34.
1754
987
374
97.0
0000
010
0.57
250
103.
0000
00
2.27
7826
14.
1754
987
384
97.0
0000
098
.905
0010
3.00
000
01.
8342
755
4.17
5498
7
394
97.0
0000
099
.435
0010
3.00
000
02.
7693
621
4.17
5498
7
404
97.0
0000
097
.885
0010
3.00
000
03.
1018
328
4.17
5498
7
414
97.0
0000
010
0.98
500
103.
0000
00
1.62
7892
34.
1754
987
424
97.0
0000
098
.917
5010
3.00
000
02.
4388
436
4.17
5498
7
434
97.0
0000
099
.230
0010
3.00
000
00.
8711
295
4.17
5498
7
444
97.0
0000
010
1.70
000
103.
0000
00
2.25
3323
54.
1754
987
454
97.0
0000
098
.525
0010
3.00
000
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9660
055
4.17
5498
7
464
97.0
0000
098
.720
0010
3.00
000
00.
3823
611
4.17
5498
7
474
97.0
0000
010
0.29
000
103.
0000
00
1.25
0200
04.
1754
987
484
97.0
0000
010
0.24
000
103.
0000
00
1.89
0238
14.
1754
987
494
97.0
0000
098
.802
5010
3.00
000
04.
8650
617
4.17
5498
71
504
97.0
0000
010
0.83
750
103.
0000
00
2.37
3876
94.
1754
987
514
97.0
0000
010
0.05
000
103.
0000
00
2.37
5892
84.
1754
987
524
97.0
0000
099
.130
0010
3.00
000
00.
4327
432
4.17
5498
7
534
97.0
0000
099
.262
5010
3.00
000
01.
7126
661
4.17
5498
7
544
97.0
0000
099
.727
5010
3.00
000
01.
5154
840
4.17
5498
7
554
97.0
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0.92
000
103.
0000
00
0.35
4118
64.
1754
987
564
97.0
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0.17
250
103.
0000
00
2.15
6546
24.
1754
987
574
97.0
0000
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0.07
750
103.
0000
00
1.40
9595
14.
1754
987
584
97.0
0000
099
.977
5010
3.00
000
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3439
356
4.17
5498
7
XB
AR
AN
D S
CH
AR
TS
(K
NO
WN
MU
AN
D S
IGM
A)
Th
e S
HE
WH
AR
T P
roce
dure
Mea
ns
and
Sta
nd
ard
Dev
iati
ons
Ch
art
Su
mm
ary
for
resp
onse
3 S
igm
a L
imit
s w
ith
n=
4 fo
rM
ean
3 S
igm
a L
imit
s w
ith
n=
4 fo
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td D
ev
sam
ple
Su
bgr
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Sam
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Siz
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Lim
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14
97.0
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1.17
250
103.
0000
00
1.51
2445
64.
1754
987
24
97.0
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0.50
250
103.
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00
2.19
7367
74.
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987
34
97.0
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0.41
000
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2.81
6380
74.
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987
44
97.0
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0.27
250
103.
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00
0.66
9396
54.
1754
987
54
97.0
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0.26
000
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00
1.96
3890
74.
1754
987
64
97.0
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099
.847
5010
3.00
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30
1.69
7692
14.
1754
987
74
97.0
0000
099
.272
5010
3.00
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02.
2004
753
4.17
5498
7
84
97.0
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0.38
250
103.
0000
00
1.60
2672
24.
1754
987
94
97.0
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0.07
500
103.
0000
00
0.67
8552
44.
1754
987
104
97.0
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099
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0010
3.00
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6222
721
4.17
5498
7
114
97.0
0000
010
0.99
000
103.
0000
00
1.68
5051
94.
1754
987
124
97.0
0000
099
.395
0010
3.00
000
00.
9956
740
4.17
5498
7
134
97.0
0000
010
0.35
500
103.
0000
00
3.12
8796
44.
1754
987
144
97.0
0000
096
.492
5010
3.00
000
10
1.95
6721
34.
1754
987
154
97.0
0000
097
.830
0010
3.00
000
50
1.26
9146
74.
1754
987
164
97.0
0000
099
.567
5010
3.00
000
01.
9716
385
4.17
5498
7
174
97.0
0000
097
.580
0010
3.00
000
01.
2249
898
4.17
5498
7
184
97.0
0000
010
1.56
750
103.
0000
00
2.82
2379
34.
1754
987
194
97.0
0000
010
0.47
250
103.
0000
00
1.28
1025
04.
1754
987
204
97.0
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098
.387
5010
3.00
000
01.
0457
653
4.17
5498
7
214
97.0
0000
099
.600
0010
3.00
000
01.
3376
098
4.17
5498
7
224
97.0
0000
010
0.47
250
103.
0000
00
2.29
5072
64.
1754
987
234
97.0
0000
099
.672
5010
3.00
000
02.
4286
262
4.17
5498
7
244
97.0
0000
099
.512
5010
3.00
000
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1296
128
4.17
5498
7
254
97.0
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099
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5010
3.00
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02.
4069
673
4.17
5498
7
264
97.0
0000
099
.487
5010
3.00
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8733
308
4.17
5498
7
274
97.0
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010
3.27
750
103.
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01
01.
9476
717
4.17
5498
7
284
97.0
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0.55
000
103.
0000
00
0.62
7056
64.
1754
987
294
97.0
0000
098
.745
0010
3.00
000
01.
3550
031
4.17
5498
7
61
XB
AR
AN
D S
CH
AR
TS
(K
NO
WN
MU
AN
D S
IGM
A)
Th
e S
HE
WH
AR
T P
roce
dure
Mea
ns
and
Sta
nd
ard
Dev
iati
ons
Ch
art
Su
mm
ary
for
resp
onse
3 S
igm
a L
imit
s w
ith
n=
4 fo
rM
ean
3 S
igm
a L
imit
s w
ith
n=
4 fo
rS
td D
ev
sam
ple
Su
bgr
oup
Sam
ple
Siz
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Lim
itS
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Lim
it
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ests
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nal
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Lim
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ub
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pp
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imit
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ests
Sig
nal
ed
594
97.0
0000
098
.842
5010
3.00
000
02.
0537
993
4.17
5498
7
604
97.0
0000
010
0.07
250
103.
0000
00
0.94
5458
44.
1754
987
614
97.0
0000
010
0.57
750
103.
0000
00
1.82
9706
34.
1754
987
624
97.0
0000
010
0.21
500
103.
0000
00
0.99
9783
34.
1754
987
634
97.0
0000
010
2.05
500
103.
0000
00
3.14
3400
54.
1754
987
644
97.0
0000
010
2.48
250
103.
0000
05
00.
8268
968
4.17
5498
7
654
97.0
0000
099
.035
0010
3.00
000
01.
0449
083
4.17
5498
7
664
97.0
0000
099
.835
0010
3.00
000
02.
1992
347
4.17
5498
7
674
97.0
0000
099
.785
0010
3.00
000
02.
9149
557
4.17
5498
7
684
97.0
0000
098
.610
0010
3.00
000
01.
4900
559
4.17
5498
7
694
97.0
0000
010
0.36
000
103.
0000
00
2.12
1288
94.
1754
987
704
97.0
0000
099
.420
0010
3.00
000
02.
0824
665
4.17
5498
7
714
97.0
0000
099
.715
0010
3.00
000
00.
9186
403
4.17
5498
7
724
97.0
0000
099
.287
5010
3.00
000
01.
6470
453
4.17
5498
7
734
97.0
0000
099
.095
0010
3.00
000
01.
6290
386
4.17
5498
7
744
97.0
0000
099
.787
5010
3.00
000
01.
7620
892
4.17
5498
7
754
97.0
0000
099
.865
0010
3.00
000
00.
6471
218
4.17
5498
7
764
97.0
0000
099
.877
5010
3.00
000
02.
4889
271
4.17
5498
7
774
97.0
0000
098
.995
0010
3.00
000
00.
6988
324
4.17
5498
7
784
97.0
0000
010
0.15
250
103.
0000
00
0.95
1573
34.
1754
987
794
97.0
0000
010
0.94
750
103.
0000
00
0.76
5914
54.
1754
987
804
97.0
0000
010
1.43
500
103.
0000
00
1.36
0747
34.
1754
987
814
97.0
0000
010
1.12
250
103.
0000
00
1.87
3630
64.
1754
987
824
97.0
0000
010
1.35
000
103.
0000
00
0.88
0870
84.
1754
987
834
97.0
0000
010
0.06
250
103.
0000
00
0.70
3532
84.
1754
987
844
97.0
0000
010
0.49
250
103.
0000
00
1.71
3755
64.
1754
987
854
97.0
0000
010
2.14
250
103.
0000
00
1.74
9025
94.
1754
987
864
97.0
0000
010
1.11
750
103.
0000
02
01.
9525
432
4.17
5498
7
874
97.0
0000
099
.277
5010
3.00
000
00.
6153
251
4.17
5498
7
XB
AR
AN
D S
CH
AR
TS
(K
NO
WN
MU
AN
D S
IGM
A)
Th
e S
HE
WH
AR
T P
roce
dure
Mea
ns
and
Sta
nd
ard
Dev
iati
ons
Ch
art
Su
mm
ary
for
resp
onse
3 S
igm
a L
imit
s w
ith
n=
4 fo
rM
ean
3 S
igm
a L
imit
s w
ith
n=
4 fo
rS
td D
ev
sam
ple
Su
bgr
oup
Sam
ple
Siz
eL
ower
Lim
itS
ub
grou
pM
ean
Up
per
Lim
it
Sp
ecia
lT
ests
Sig
nal
edL
ower
Lim
itS
ub
grou
pS
tdD
evU
pp
erL
imit
Sp
ecia
lT
ests
Sig
nal
ed
884
97.0
0000
010
2.36
000
103.
0000
00
1.07
5019
44.
1754
987
894
97.0
0000
099
.910
0010
3.00
000
01.
2648
320
4.17
5498
7
904
97.0
0000
099
.985
0010
3.00
000
01.
8558
466
4.17
5498
7
914
97.0
0000
098
.672
5010
3.00
000
03.
9788
221
4.17
5498
7
924
97.0
0000
099
.952
5010
3.00
000
02.
4697
689
4.17
5498
7
934
97.0
0000
099
.620
0010
3.00
000
01.
6033
091
4.17
5498
7
944
97.0
0000
099
.980
0010
3.00
000
02.
2692
583
4.17
5498
7
954
97.0
0000
099
.632
5010
3.00
000
00.
9836
115
4.17
5498
7
964
97.0
0000
010
1.63
500
103.
0000
00
2.87
9751
14.
1754
987
974
97.0
0000
099
.455
0010
3.00
000
01.
4932
403
4.17
5498
7
984
97.0
0000
099
.722
5010
3.00
000
00.
9691
706
4.17
5498
7
994
97.0
0000
098
.422
5010
3.00
000
40
3.43
6639
64.
1754
987
100
497
.000
000
99.2
6750
103.
0000
00
0.71
4207
04.
1754
987
XBAR AND S CHARTS (KNOWN MU AND SIGMA)
The SHEWHART Procedure
62
6.4.2 For Unknown µ and σ
• When both µ and σ are unknown, estimates of these parameters must be computed based onm preliminary samples. Let:
x =
∑mi=1 xim
be the mean of the sample means and
s =
∑mi=1 sim
be the mean of the sample standard deviations.
• Therefore, the estimator of µ is x.
• Because E(s) = E(si) for each i, we have E(s) = c4σ. It follows that E
(s
c4
)= σ.
Thus, an unbiased estimator of σ is σ̂ =s
c4and, σ̂s = σ̂
√1− c24 =
s
c4
√1− c24.
• The trial control limits for the x-chart are:
UCL = µ̂+ 3σ̂√n
=
Centerline = µ̂ = x (8)
LCL = µ̂− 3σ̂√n
=
• The trial control limits for the s-chart are:
UCL = µ̂s + 3σ̂s =
Centerline = µ̂s = s (9)
LCL = µ̂s + 3σ̂s =
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 thex- and R-charts were tested.
• That is, plot the si values on the s-chart analogously to the way the Ri values are plotted onthe R-chart.
• Once acceptable control limits have been found for both charts, proceed with process controlanalysis.
63
XBAR and S Charts (Unknown MU and SIGMA)
The SHEWHART Procedure
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 forMean
3 Sigma Limits with n=4 forStd Dev
sample
SubgroupSample
SizeLower
LimitSubgroup
MeanUpper
Limit
SpecialTests
SignaledLower
LimitSubgroup
Std DevUpper
Limit
SpecialTests
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 1
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 56 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
64