Applied Statistics and Probability for Engineers

610 CHAPTER 16 STATISTICAL QUALITY CONTROL

Assume that there are mpreliminary samples available, each of size n, and let Sidenote the

standard deviation of the ith sample. Define

S (16-14)

1

ma

m

i 1

Si

ˆSc 4 (16-15)

UCLs 3 (16-16)

s

c 421 c

2

4 CLs^ LCLs^3

s

c 421 c

2

4

S Chart

UCLx 3 (16-17)

s

c 41 n

CLx LCLs 3

s

c 41 n

Control Chart

(from )S

X

Because E 1 S 2 c 4 , an unbiased estimator of is That is,Sc 4

A control chart for standard deviations follows.

The LCLfor an Schart can be a negative number, in that case, it is customary to set LCLto zero.

When an Schart is used, the estimate for in Equation 16-15 is commonly used to calculate

the control limits for an Xchart. This produces the following control limits for an Xchart.

EXAMPLE 16-1 A component part for a jet aircraft engine is manufactured by an investment casting

process. The vane opening on this casting is an important functional parameter of the part.

We will illustrate the use of and Rcontrol charts to assess the statistical stability of this

process. Table 16-1 presents 20 samples of five parts each. The values given in the table

have been coded by using the last three digits of the dimension; that is, 31.6 should be

0.50316 inch.

The quantities and are shown at the foot of Table 16-1. The value of A 2

for samples of size 5 is A 2 0.577. Then the trial control limits for the chart are

or

UCL36.67 LCL29.97

xA 2 r33.32 1 0.577 21 5.8 2 33.323.35

X

x33.3 r5.8

X

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