Applied Statistics and Probability for Engineers

608 CHAPTER 16 STATISTICAL QUALITY CONTROL

The standard deviation of Wis called d 3. Because RW

(16-4)

Let Ribe the range of the ith sample, and let

Rd 2  Rd 3 

The center line and upper and lower control limits for an control chart are

(16-9)

where the constant A 2 is tabulated for various sample sizes in Appendix Table X.

UCLxA 2 r CLx LCLxA 2 r

X

be the average range. Then is an estimator of Rand from Equation 16-4 an unbiased

estimator of is

R

The parameters of the Rchart may also be easily determined. The center line will obvi-

ously be. To determine the control limits, we need an estimate of R, the standard deviation

of R. Once again, assuming the process is in control, the distribution of the relative range, W,

will be useful. We may estimate Rfrom Equation 16-4 as

ˆRd 3 ˆd 3 (16-10)

R

d 2

R

Control Chart

(from )R

X

R (16-5)

1

m^ a

m

i 1

Ri

ˆ (16-6)

R

d 2

Therefore, we may use as our upper and lower control limits for the chart

(16-7)

Define the constant

(16-8)

Now, once we have computed the sample values and , the control chart may be defined

as follows:

x r X

A 2 

3

d 21 n

UCLX 

3

d 21 n

R LCL X 

3

d 21 n

R

X

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