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When the parameters and are unknown, we usually estimate them on the basis of preliminary samples, taken when the process is thought to be in control. We recommend the use of at least 20 to 25 preliminary samples. Suppose m preliminary samples are available, each of size n. Typically, n will be 4, 5, or 6; these relatively small sample sizes are widely used and often arise from the construction of rational subgroups. Let the sample mean for the ith sample be Xi. Then we estimate the mean of the population, , by the grand mean


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Cavity Quantum Electrodynamics: The Strange Theory of Light in a Box Sergio M. Dutra Copyright 2005 John Wiley & Sons, Inc.

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Thus, we may take X as the center line on the X control chart. We may estimate from either the standard deviation or the range of the observations within each sample. The sample size is relatively small, so there is little loss in ef ciency in estimating from the sample ranges. The relationship between the range R of a sample from a normal population with known parameters and the standard deviation of that population is needed. Since R is a random variable, the quantity W R , called the relative range, is also a random variable. The parameters of the distribution of W have been determined for any sample size n. The mean of the distribution of W is called d2, and a table of d2 for various n is given in Appendix Table X.

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W d3 (16-4)


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In Appendix A we used the boundary conditions at a perfect conductor, (A.5) and (A.6), to find the modes of a closed cavity made of perfect mirrors. These boundary conditions can be found in any good book on electrodynamics. However, to make our derivation of cavity modes self-contained and save those readers who might be unfamiliar with it the trouble of looking it up somewhere else, we summarize briefly here how these boundary conditions arise. The standard procedure to derive these boundary conditions is to consider an infinitesimally small cylinder positioned so that one of its facets lies on the surface of the cavity wall and the other inside the wall (see Figure B.l). Integrating (2.44)


Therefore, we may use as our upper and lower control limits for the X chart UCL De ne the constant A2 X LCL 3 d2 1n X (16-7)

Figure B.1 Infinitesimal cylinder with one of its facets on the surface of a cavity wall and the other inside the wall. The height of the cylinder is 8h and 88 is the area of each of its facets.

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f dS /\ F,

Now, once we have computed the sample values x and r, the X control chart may be de ned as follows: X Control Chart (from R )

The center line and upper and lower control limits for an X control chart are UCL x A2 r CL x LCL x A2 r (16-9)


where the constant A2 is tabulated for various sample sizes in Appendix Table X. The parameters of the R chart may also be easily determined. The center line will obviously be R. 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 R from Equation 16-4 as



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