Applied Statistics and Probability for Engineers

16-5XAND ROR SCONTROL CHARTS 607

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 rulesfor 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.

16-5 AND R ORSCONTROL 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 (Rchart) or a standard de-

viation chart (Schart), 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 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

X

X

X

CL (16-2)

LCL 3  1 n

UCL 3  1 n

ˆX  i (16-3)

1

ma

m

i 1

X

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

use of at least 20 to 25 preliminary samples. Suppose mpreliminary samples are available,

each of size n. Typically, nwill 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, Xi , by the grand mean

Thus, we may take as the center line on the 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 Rof a sample from a normal population with known

parameters and the standard deviation of that population is needed. Since Ris a random

variable, the quantity WR, called the relative range, is also a random variable. The

parameters of the distribution of Whave been determined for any sample size n. The mean of

the distribution of Wis called d 2 , and a table of d 2 for various nis given in Appendix Table X.

X X

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