IE 361 Module 11
Shewhart Control Charts for Measurements ("Variables" Data)
Reading: Section 3.2, Statistical Quality Assurance Methods for
Engineers
Prof. Steve Vardeman and Prof. Max Morris
Vardeman and Morris (Iowa State University)
Iowa State University
IE 361 Module 11
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Shewhart Control Charts for Measurements
In this module we consider Shewhart control charts for measurements (or
so called "variables data" in old time SQC jargon). As our featured
example, we will use the data from an IE 361 Deming drama. These are
recorded in the …gure on panel 3. (The "red bag" was an earlier version
of the current "brown bag," i.e. had process parameters µ = 5 and
σ = 1.715 and was approximately normal.)
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Deming Drama Data
Figure: Data from an IE 361 Deming Drama
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IE 361 Module 11
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Charts for Means
We introduced the topic of Shewhart control charts in Module 10 using
the most famous of all such charts, the x̄ charts. To review, we saw that
the (approximately)
normal distribution of x̄ (with mean µx̄ = µ and
p
σx̄ = σ/ n) leads to standards given control limits for x̄
σ
UCLx = µ + 3 p
n
and
LCLx = µ
σ
3p
n
Further, we saw that in a retrospective situation like that illustrated on
panel 3 where sample means x̄ and sample ranges R are computed,
estimates
µ̂ = x and σ̂ = R̄/d2
can be substituted to produce retrospective control limits for x̄
UCLx = x + 3
Vardeman and Morris (Iowa State University)
R̄
p
d2 n
and
IE 361 Module 11
LCLx = x
3
R̄
p
d2 n
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Charts for Means
(Example 11-1)
In fact, it is traditional to set
A2 =
3
p
d2 n
and rewrite these retrospective control limits as
UCLx = x + A2 R̄
and
LCLx = x
A2 R̄
Example 11-1 We saw in Module 10 that (since for the brown bag
µ = 5 and σ = 1.715) standards given control limits for x̄ are
1.715
UCLx = 5 + 3 p = 7.3
5
and
LCLx = 5
1.715
3 p = 2.7
5
These limits are marked on the x̄ control chart on panel 3 and we can see
that if they had been applied to x̄’s in real time, process change would
have been detected at sample 16.
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IE 361 Module 11
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Charts for Means
Example 11-1 continued
The 18 sample means and ranges from panel 3 average to
x = 5.744 and R̄ = 4.278
So retrospective limits for x̄ are (since the sample size is n = 5)
UCLx
= 5.744 + .577 (4.278)
= 8.21
LCLx
= 5.744
= 3.28
and
.577 (4.278)
When these limits are applied retrospectively to the 18 sample means, we
see that the last 3 values are outside of these, and there is thus evidence
of process instability in the data on panel 3.
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IE 361 Module 11
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Charts for Ranges
It is traditional (not as it turns out best practice, but traditional) to use an
R chart as a companion to an x̄ chart. (An s chart to be discussed next is
actually a better choice than an R chart, but historical precedent makes R
charts continue to be common.) The x̄ chart is primarily useful for
monitoring process aim, while an R (or s) chart is primarily a tool for
monitoring process spread or short term variation.
In order to identify appropriate control limits for R one needs to know
some probability facts about R based on a sample of size n from a normal
distribution. As it turns out, R has a (non-standard) probability
distribution (not one you met in Stat 231) with mean proportional to the
standard deviation of the sampled process. The constant of
proportionality is the d2 that we have used to turn ranges into estimates of
standard deviations, that is
µR = d2 σ
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IE 361 Module 11
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Charts for Ranges (Standards Given Limits)
Further, the standard deviation of the probability distribution for R is
proportional to the standard deviation of the sampled process. The
constant of proportionality is called d3 . That is,
σR = d3 σ
Taken together, these probability facts about R produce standards given
control limits for R
UCLR = (d2 + 3d3 )σ
and
LCLR = (d2
3d3 )σ
or, if one de…nes
D2 = (d2 + 3d3 ) and D1 = (d2
3d3 )
these standards given limits are
UCLR = D2 σ
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and
IE 361 Module 11
LCLR = D1 σ
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Charts for Ranges (Retrospective Limits)
Further, in a retrospective situation like that illustrated in Figure 1 where
R’s are computed, the estimate
σ̂ = R̄/d2
can be substituted to produce retrospective control limits for R
UCLR = D2 R̄/d2
and
LCLR = D1 R̄/d2
and
D3 =
It is traditional to set
D4 =
D2
d2
D1
d2
and rewrite these retrospective control limits as
UCLR = D4 R̄
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and
IE 361 Module 11
LCLR = D3 R̄
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Charts for Ranges
Example 11-2
Since σ = 1.715 for the brown bag, a standards given upper control limit
for R based on n = 5 is
UCLR = 4.918 (1.715) = 8.43
(No standards given lower control limit is typically used, because for a
sample size of only n = 5, the di¤erence (d2 3d3 ) turns out to be
negative.) This limit is marked on the R control chart in panel 3 and we
can see that if it had been applied to R’s in real time, process change
would have been detected at sample 16.
Recalling that the 18 sample means and ranges from panel 3 average to
R̄ = 4.278, a retrospective upper control limit for R is
UCLR = 2.115 (4.278) = 9.05
When this limit is applied retrospectively to the 18 sample ranges, we see
that the 16th sample range plots "out of control" and there is evidence of
process instability in data in panel 3.
Vardeman and Morris (Iowa State University)
IE 361 Module 11
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Charts for Standard Deviations (Basis)
s charts represent a superior alternative to R charts. At the price of
requiring more than "by hand" calculation (sample standard deviations
being more di¢ cult to compute than sample ranges), they provide
typically quicker detection of process changes. In order to identify
appropriate control limits for s one needs to know some probability facts
about s based on a sample of size n from a normal distribution.
It is a fact mentioned in Stat 231 (that actually stands behind the
standard con…dence limits for σ) that (n 1) s 2 /σ2 has a χ2 probability
distribution. It turns out to follow from this fact that s has mean
proportional to the standard deviation of the sampled process. The
constant of proportionality is something called c4 . That is,
µ s = c4 σ
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Charts for Standard Deviations (Basis and Standards Given
Limits)
Further, the standard deviation of the random variable s is proportional to
the standard deviation of the sampled process. The constant of
proportionality is called c5 . That is,
σ s = c5 σ
Taken together, these probability facts about s produce standards given
control limits
UCLs = (c4 + 3c5 )σ
and
LCLs = (c4
3c5 )σ
or, if one de…nes
B6 = (c4 + 3c5 ) and B5 = (c4
3c5 )
these standards given limits are
UCLs = B6 σ
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and
IE 361 Module 11
LCLs = B5 σ
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Charts for Standard Deviations (Retrospective Limits)
Further, in a retrospective situation like that illustrated on panel 3, where
s values (instead of R values) are computed, the estimate
σ̂ = s̄/c4
can be substituted to produce retrospective control limits for s
UCLs = B6 s̄/c4
and
LCLs = B5 s̄/c4
and
B3 =
It is traditional to set
B4 =
B6
c4
B5
c4
and rewrite these retrospective control limits as
UCLs = B4 s̄
Vardeman and Morris (Iowa State University)
and
IE 361 Module 11
LCLs = B3 s̄
13 / 20
Standard Deviations and Retrospective Charts for Means
A …nal bit of development concerning retrospective s-based calculations is
this. Using σ̂ = s̄/c4 , possible retrospective x̄ chart limits are
UCLx = x + 3
s̄
p
c4 n
LCLx = x
and
and it is traditional to set
A3 =
3
s̄
p
c4 n
3
p
c4 n
and rewrite these retrospective control limits as
UCLx = x + A3 s̄
Vardeman and Morris (Iowa State University)
and
IE 361 Module 11
LCLx = x
A3 s̄
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Charts for Standard Deviations
Example 11-3 (Standards Given)
Since σ = 1.715 for the brown bag, a standards given upper control limit
for s is
UCLs = 1.964 (1.715) = 3.37
(No standards given lower control limit is typically used, because for a
sample size of only n = 5, the di¤erence (c4 3c5 ) is negative.)
Below are the 18 sample standard deviations corresponding to the data on
panel 3.
Sample
1
2
3
4
5
6
7
8
9
10
s
1.82 1.92 1.82 1.14 1.14 1.48 .89 .45 1.82 0
11
12
13
14
15
16
17
18
.84 1.22 1.10 .89 2.35 5.63 2.88 3.58
It is evident that if the standards given control limit had been applied to
these s’s in real time, process change would have been detected at sample
16.
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IE 361 Module 11
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Charts for Standard Deviations
Example 11-3 (Retrospective)
The 18 values s average to s̄ = 30.97/18 = 1.72. So a retrospective
upper control limit for s for the data of panel 3 is
UCLs = 2.089 (1.72) = 3.59
When this limit is applied retrospectively, the 16th s plots "out of control"
and there is thus evidence of process instability. Further, retrospective
control limits for x̄ based on sample standard deviations are
UCLx = 5.774 + 1.427 (1.72) = 8.23
and
LCLx = 5.774 + 1.427 (1.72) = 3.32
When these limits are applied retrospectively to the data of panel 3, the
last 3 values x are outside of them, and there is again evidence of process
instability.
Vardeman and Morris (Iowa State University)
IE 361 Module 11
16 / 20
Charts for Medians
A computationally simpler alternative to the x̄ chart is the Shewhart
median (x̃) chart. Finding a median requires only putting a data set in
order smallest to largest and then …nding the middle value, x̃. This is a
measure of process aim like the mean. But is it generally not as reliable
as the mean. Di¤erently put, it generally takes longer to detect process
change using medians than using means. But in some rare contexts,
considerations of computational simplicity may outweigh this lack of
sensitivity.
In order to identify appropriate control limits for x̃ one needs to know
some probability facts about x̃ based on a sample of size n from a normal
distribution. x̃ has a (non-standard) probability distribution (not one met
in Stat 231) with mean equal to the process mean and standard deviation
larger than that of x̄ by a multiplicative factor that we will call κ. That is
σ
µx̃ = µ and σx̃ = κ p
n
where a small table of values for κ is given on page 72 of SQAME.
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IE 361 Module 11
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Charts for Medians
These facts suggest standards given control limits for x̃
σ
UCLx̃ = µ + 3κ p
n
and
LCLx̃ = µ
σ
3κ p
n
(Any sensible estimates of µ and σ could further be used to make
retrospective limits for x̃.)
Example 11-4 Since for the brown bag process µ = 5 and σ = 1.715,
standards given control limits for x̃ based on n = 5 are
1.715
UCLx̃ = 5 + 3 (1.197) p = 7.75
5
and
LCLx̃ = 5
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1.715
3 (1.197) p = 2.25
5
IE 361 Module 11
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Charts for Medians
Example 11-4
The 18 sample medians for the data of panel 3 are as below
Sample
x̃
1 2 3 4 5 6
4 5 4 5 5 4
15 16 17 18
5 12 9
8
7
5
8
5
9
5
10
4
11
4
12
5
13
6
14
5
So if the standards given control limits had been applied to x̃’s in real
time, process change would have been detected at sample 16.
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IE 361 Module 11
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Two Reminders
It is worth saying again that control limits are NOT engineering
speci…cations nor vice versa. In Module 10 we said and now say again
that a process can be stable without being acceptable and vice versa. The
table below again compares these two fundamentally di¤erent concepts.
Control Limits
have to do with process stability
apply to Q
usually derive from process data
Speci…cations
have to do with product acceptability
apply to individuals, x
derive from performance requirements
It is also worth saying that Table 3.11, page 107 of SQAME, summarizes
the control limit formulas for both the "measurements data" of this
module and for the "attributes data" of Module 13.
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