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3 The control charts discussed in the previous section were designed to control the location (equivalently, central tendency) of a process, with particular attention to the mean as a measure of location. It is equally important to ensure that a process is under control with respect to variation. In fact, most practitioners recommend that control be established on variation prior to constructing an chart or any other chart for controlling location.

4 Control Charts for Process Variation In this section, we consider charts for variation based on the sample standard deviation S and also charts based on the sample range R. The former are generally preferred because the standard deviation gives a more efficient assessment of variation than does the range, but R charts were used first and tradition dies hard.

5 The S Chart

6 We again suppose that k independently selected samples are available, each one consisting of n observations on a normally distributed variable.. Denote the sample standard deviations by s 1, s 2, … s k, with The values s 1, s 2, s 3 are plotted in sequence on an S chart. The center line of the chart will be at height, and the 3-sigma limits necessitate determining 3  S (just as 3-sigma limits of an chart required with  then estimated from the data).

7 The S Chart We know that for any rv Y, V(Y) = E(Y 2 ) – [E(Y)] 2, and that a sample variance S 2 is an unbiased estimator of  2, that is, E(S 2 ) =  2. Thus V(S) = E(S 2 ) – [E(S)] 2 =  2 – (a n  ) 2 =  2 (1 – ) where values of an for n = 3,...,8 are tabulated in the previous section. The standard deviation of S is then

8 The S Chart It is natural to estimate  using s 1,…, s k, as was done in the previous section namely,. Substituting for s in the expression for  S gives the quantity used to calculate 3-sigma limits. The 3-sigma control limits for an S control chart are The expression for LCL will be negative if n  5, in which case it is customary to use LCL = 0.

9 Example 4 Table 16.2 displays observations on stress resistance of plastic sheets (the force, in psi, necessary to crack a sheet). Table 16.2 Stress-Resistance Data for Example 4

10 Example 4 There are k = 22 samples, obtained at equally spaced time points, and n = 4 observations in each sample. It is easily verified that  s i = 51.10 and s = 2.32 so the center of the S chart will be at 2.32 (though because n = 4, LCL = 0 and the center line will not be midway between the control limits). From the previous section, a 4 =.921, from which the UCL is cont’d

11 Example 4 The resulting control chart is shown in Figure 16.3. All plotted points are well within the control limits, suggesting stable process behavior with respect to variation. Figure 16.3 S chart for stress-resistance data for Example 4 cont’d

12 The R Chart

13 The R Chart Let r 1, r 2,..., r k denote the k sample ranges and r =  r i /k. The center line of an R chart will be at height r. Determination of the control limits requires  R, where R denotes the range (prior to making observations—as a random variable) of a random sample of size n from a normal distribution with mean value  and standard deviation . Because R = max(X 1,..., X n ) – min(X 1,..., X n ) =  {max(Z 1,..., Z n ) – min(Z 1,..., Z n )}

14 The R Chart Where Z i = (X i –  )/ , and the Z i ’s are standard normal rv’s, it follows that =   c n The values of c n for n = 3,..., 8 appear in the accompanying table.

15 The R Chart It is customary to estimate  by as discussed in the previous section. This gives as the estimated standard deviation of R. The 3-sigma limits for an R chart are The expression for LCL will be negative if n  6, in which case LCL = 0 should be used.

16 Example 5 In tissue engineering, cells are seeded onto a scaffold that then guides the growth of new cells. The article “On the Process Capability of the Solid Free-Form Fabrication: A Case Study of Scaffold Moulds for Tissue Engineering” (J. of Engr. in Med., 2008: 377–392) used various quality control methods to study a method of producing such scaffolds. An unusual feature is that instead of subgroups being observed over time, each subgroup resulted from a different design dimension (  m).

17 Example 5 Table 16.3 contains data from Table 2 of the cited article on the deviation from target in the perpendicular orientation (these deviations are indeed all positive—the printed beams exhibit larger dimensions than those designed). cont’d Table 16.3 Deviation-from-Target Data for Example 5 (continued)

18 Example 5 Table 16.3 yields  r I = 124, from which = 7.29. Since n = 3, LCL = 0. With b 3 = 1.693 and c 3 =.888, UCL = 7.29 + 3  (.888)(7.29)/1.693 Table 16.3 Deviation-from-Target Data for Example 5 (continued) cont’d = 18.76

19 Example 5 Figure 16.4 shows both an R chart and an chart from the Minitab software package (the cited article also included these charts). cont’d Figure 16.4 Control charts for the deviation-from-target data of Example 5

20 Example 5 All points are within the appropriate control limits, indicating an in-control process for both location and variation. cont’d

21 Charts Based on Probability Limits

22 Charts Based on Probability Limits Consider an chart based on the in-control (target) value  0 and known . When the variable of interest is normally distributed and the process is in control, P( i >  0 + 3  / ) =.0013 = P( i <  0 – 3  / ) That is, the probability that a point on the chart falls above the UCL is.0013, as is the probability that the point falls below the LCL (using 3.09 in place of 3 gives.001 for each probability).

23 Charts Based on Probability Limits When control limits are based on estimates of  and , these 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 P(S i > UCL) = P(S i UCL) = P(R i < LCL) =.0013. 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.

24 Charts Based on Probability Limits 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.