When S2 is the sample variance of a normal random sample, has a chi-squared distribution with df, so from which This suggests that an alternative chart for controlling process variation involves plotting the sample variances and using the control limits Construct the corresponding chart for the data of Exercise 11.

[Hint: The lower- and upper-tailed chi-squared critical values for 5 df are .210 and 20.515, respectively.]

UCL 5 s2x.001,n21 2 /(n 2 1)

LCL 5 s2x.999,n21 2 /(n 2 1)

Pa s2 x.999,n21 2

n 2 1

, S2 , s2 x.001,n21 2

n 2 1 b 5 .998 Pax.999,n21 2 , (n 2 1)S2 s2 , x.001,n21 2 b 5 .998 n 2 1

(n 2 1)S2

/s2 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

, nor is it true for a 3-sigma R chart that

. 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. 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. The book Statistical Methods for Quality Improvement (see the chapter bibliography) contains more information on this topic.

P(Ri , LCL) 5 .0013 P(Si , LCL) 5 .0013 P(Ri . UCL) 5 P(Si . UCL) 5