Suppose that with the samples of Exercise 15.6, it is desired to establish control also over the variability of the process. Using the method of Exercise 15.5 and the values of \(\overline{\bar{x}}\) and \(\bar{s}\) given in Exercise 15.6, calculate the central line and the central limits for a \(\sigma\) chart with \(n=5\).

Data From Exercise 15.5

Data From Exercise 15.6

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15.5 If the sample standard deviations instead of the sam- ple ranges are used to estimate , the control limits for the resulting x chart are given by A 5, where 5 is the mean of the sample standard deviations obtained from given data, and A can be found in Table 8W. Note that in connection with problems of quality con- trol the sample standard deviation is defined using the divisor n instead of n-1. The corresponding R chart is replaced by a chart, having the central line c2 5 and the lower and upper control limits B3 5 and B4 5, where c2, B3 and B4 can be obtained from Table 8W. (a) Construct an chart and a chart for these sam- ples of size 3 which had x equal to 25, 28, 26, 29, 30, 32, 22, 24, 27, 26, 29, 20, 25, 24, 27, 28, 24, 23, 27, 28 and s equal to 1, 2, 3, 5, 4, 2, 1, 3, 1, 4, 4, 3, 2, 1, 5, 4, 1, 3, 2, 2. (b) Would it be reasonable to use these control limits for subsequent data? Why?