Transcript DevStat8e_16_03

16
Quality Control
Methods
Copyright © Cengage Learning. All rights reserved.
16.3
Control Charts for Process
Variation
Copyright © Cengage Learning. All rights reserved.
Control Charts for Process Variation
The control charts discussed in the previous section were
designed to control the location (equivalently, central
tendency) of a process, with particular attention to the
mean as a measure of location.
It is equally important to ensure that a process is under
control with respect to variation.
In fact, most practitioners recommend that control be
established on variation prior to constructing an chart or
any other chart for controlling location.
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Control Charts for Process Variation
In this section, we consider charts for variation based on
the sample standard deviation S and also charts based on
the sample range R.
The former are generally preferred because the standard
deviation gives a more efficient assessment of variation
than does the range, but R charts were used first and
tradition dies hard.
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The S Chart
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The S Chart
We again suppose that k independently selected samples
are available, each one consisting of n observations on a
normally distributed variable. .
Denote the sample standard deviations by s1, s2, … sk, with
The values s1, s2, s3 are plotted in sequence on
an S chart.
The center line of the chart will be at height , and the
3-sigma limits necessitate determining 3 S (just as 3-sigma
limits of an chart required
with  then
estimated from the data).
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The S Chart
We know that for any rv Y, V(Y) = E(Y2) – [E(Y)]2, and that
a sample variance S2 is an unbiased estimator of  2, that
is, E(S2) =  2. Thus
V(S) = E(S2) – [E(S)]2 =  2 – (an)2 =  2(1 –
)
where values of an for n = 3, . . . ,8 are tabulated in the
previous section. The standard deviation of S is then
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The S Chart
It is natural to estimate  using s1,…, sk, as was done in the
previous section namely,
. Substituting for s in the
expression for S gives the quantity used to calculate
3-sigma limits.
The 3-sigma control limits for an S control chart are
The expression for LCL will be negative if n  5, in which
case it is customary to use LCL = 0.
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Example 4
Table 16.2 displays observations on stress resistance of
plastic sheets (the force, in psi, necessary to crack a
sheet).
Stress-Resistance Data for Example 4
Table 16.2
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Example 4
cont’d
There are k = 22 samples, obtained at equally spaced time
points, and n = 4 observations in each sample.
It is easily verified that si = 51.10 and s = 2.32 so the center
of the S chart will be at 2.32 (though because n = 4, LCL = 0
and the center line will not be midway between the control
limits).
From the previous section, a4 = .921 , from which the UCL is
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Example 4
cont’d
The resulting control chart is shown in Figure 16.3. All
plotted points are well within the control limits, suggesting
stable process behavior with respect to variation.
S chart for stress-resistance data for Example 4
Figure 16.3
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The R Chart
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The R Chart
Let r1, r2, . . . , rk denote the k sample ranges and r = ri /k.
The center line of an R chart will be at height r.
Determination of the control limits requires R, where R
denotes the range (prior to making observations—as a
random variable) of a random sample of size n from a
normal distribution with mean value  and standard
deviation . Because
R = max(X1,. . . , Xn) – min(X1,. . . , Xn)
=  {max(Z1,. . . , Zn) – min(Z1, . . . , Zn)}
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The R Chart
Where Zi = (Xi –  )/, and the Zi’s are standard normal rv’s,
it follows that
=   cn
The values of cn for n = 3, . . . , 8 appear in the
accompanying table.
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The R Chart
It is customary to estimate  by
previous section.
This gives
R.
as discussed in the
as the estimated standard deviation of
The 3-sigma limits for an R chart are
The expression for LCL will be negative if n  6, in which
case LCL = 0 should be used.
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Example 5
In tissue engineering, cells are seeded onto a scaffold that
then guides the growth of new cells. The article “On the
Process Capability of the Solid Free-Form Fabrication:
A Case Study of Scaffold Moulds for Tissue Engineering”
(J. of Engr. in Med., 2008: 377–392) used various quality
control methods to study a method of producing such
scaffolds.
An unusual feature is that instead of subgroups being
observed over time, each subgroup resulted from a
different design dimension (m).
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Example 5
cont’d
Table 16.3 contains data from Table 2 of the cited article on
the deviation from target in the perpendicular orientation
(these deviations are indeed all positive—the printed
beams exhibit larger dimensions than those designed).
Deviation-from-Target Data for Example 5 (continued)
Table 16.3
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Example 5
cont’d
Deviation-from-Target Data for Example 5 (continued)
Table 16.3
Table 16.3 yields rI = 124, from which = 7.29.
Since n = 3, LCL = 0.
With b3 = 1.693 and c3 = .888,
UCL = 7.29 + 3  (.888)(7.29)/1.693 = 18.76
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Example 5
cont’d
Figure 16.4 shows both an R chart and an chart from the
Minitab software package (the cited article also included
these charts).
Control charts for the deviation-from-target data of Example 5
Figure 16.4
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Example 5
cont’d
All points are within the appropriate control limits, indicating
an in-control process for both location and variation.
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Charts Based on Probability
Limits
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Charts Based on Probability Limits
Consider an chart based on the in-control (target) value
0 and known . When the variable of interest is normally
distributed and the process is in control,
P(
i
> 0 + 3/
) = .0013 = P(
i
< 0 – 3/
)
That is, the probability that a point on the chart falls above
the UCL is .0013, as is the probability that the point falls
below the LCL (using 3.09 in place of 3 gives .001 for each
probability).
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Charts Based on Probability Limits
When control limits are based on estimates of  and ,
these probabilities will be approximately correct provided
that n is not too small and k is at least 20.
By contrast, it is not the case for a 3-sigma S chart that
P(Si > UCL) = P(Si < LCL) = .0013, nor is it true for a
3-sigma R chart that P(Ri > UCL) = P(Ri < LCL) = .0013.
This is because neither S nor R has a normal distribution
even when the population distribution is normal.
Instead, both S and R have skewed distributions.
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Charts Based on Probability Limits
The best that can be said for 3-sigma S and R charts is that
an in-control process is quite unlikely to yield a point at any
particular time that is outside the control limits.
Some authors have advocated the use of control limits for
which the “exceedance probability” for each limit is
approximately .001.
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