Eight consecutive points plot on one side of the center line.
4. Eight consecutive points plot on one side of the center line.
Figure 15-6 (a) Variability with the
cyclic pattern. (b) Variability with the cyclic pattern
LSL
μ
USL
eliminated.
(b)
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15-3 X AND R OR S CONTROL CHARTS
Zone A
Zone B
1 σ X
74.0000 Zone C
Zone B
Zone A
Figure 15-7 The Western Electric zone
rules.
Sample number
We have found these rules very effective in practice for enhancing the sensitivity of control charts. Rules 2 and 3 apply to one side of the center line at a time. That is, a point above the upper 2-sigma limit followed immediately by a point below the lower 2-sigma limit would not signal an out-of-control alarm.
Figure 15-7 shows an
control chart for the piston ring process with the 1-sigma,
2-sigma, and 3-sigma limits used in the Western Electric procedure. Notice that these inner limits (sometimes called warning limits ) partition the control chart into three zones A, B, and
C on each side of the center line. Consequently, the Western Electric rules are sometimes called the run rules for control charts. Notice that the last four points fall in zone B or beyond. Thus, since four of five consecutive points exceed the 1-sigma limit, the Western Electric procedure will conclude that the pattern is nonrandom and the process is out of control.
15-3
AND R OR S CONTROL CHARTS
When dealing with a quality characteristic that can be expressed as a measurement, it is cus- tomary to monitor both the mean value of the quality characteristic and its variability. Control over the average quality is exercised by the control chart for averages, usually called the chart. Process variability can be controlled by either a range chart (R chart) or a standard de- viation chart (S chart), depending on how the population standard deviation is estimated.
Suppose that the process mean and standard deviation and are known and that we can
assume that the quality characteristic has a normal distribution. Consider the X chart. As dis-
cussed previously, we can use as the center line for the control chart, and we can place the upper and lower 3-sigma limits at
UCL
3 1n
LCL
3 1n
CL
(15-2)
When the parameters and are unknown, we usually estimate them on the basis of preliminary samples, taken when the process is thought to be in control. We recommend the
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CHAPTER 15 STATISTICAL QUALITY CONTROL
use of at least 20 to 25 preliminary samples. Suppose m preliminary samples are available, each of size n. Typically, n will be 4, 5, or 6; these relatively small sample sizes are widely used and often arise from the construction of rational subgroups. Let the sample mean for the ith sample be . Then we estimate the mean of the population, i , by the grand mean
1 m
ˆ X
ma X i
(15-3)
i 1
Thus, we may take as the center line on the X control chart.
We may estimate from either the standard deviation or the range of the observations within each sample. The sample size is relatively small, so there is little loss in efficiency in estimating from the sample ranges.
The relationship between the range R of a sample from a normal population with known parameters and the standard deviation of that population is needed. Since R is a random variable, the quantity W R兾 , called the relative range, is also a random variable. The parameters of the distribution of W have been determined for any sample size n. The mean and
standard deviation of the distribution of W are called d 2 and d 3 respectively. Because R W, R d 2 R d 3 (15-4)
Let R i
be the range of the ith sample, and let
1 m
i a 1 i
m R
(15-5)
be the average range. Then R is an estimator of R and from Equation 15-4 an unbiased
estimator of is
Estimator of
from R Chart
R
ˆ (15-6)
d 2
where the constant d 2 is tabulated for various sample sizes in Appendix Table XI.
Therefore, we may use as our upper and lower control limits for the X chart
UCL X R LCL
X R (15-7)
d 2 1n
d 2 1n