624 CHAPTER 16 STATISTICAL QUALITY CONTROLis difficult to maintain a process mean at the center of the specifications for long periods of
time. A common model that is used to justify the importance of a six-sigma process is illus-
trated by referring to Fig. 16-15. If the process mean shifts off-center by 1.5 standard devia-
tions, the PCRkdecreases to 4.5 3 1.5. Assuming a normally distributed process, the
fallout of the shifted process is 3.4 parts per million.Consequently, the mean of a 6-sigma
process can shift 1.5 standard deviations from the center of the specifications and still main-
tain a fallout of 3.4 parts per million.
In addition, some U.S. companies, particularly the automobile industry, have adopted the
terminology CpPCRand CpkPCRk.Because Cphas another meaning in statistics (in
multiple regression) we prefer the traditional notation PCRand PCRk.
We repeat that process capability calculations are meaningful only for stable
processes; that is, processes that are in control. A process capability ratio indicates
whether or not the natural or chance variability in a process is acceptable relative to the
specifications.LSL μ USL(^3) σ (^3) σ
1.5σ
PCRk = 2 PCRk = 1.5
Figure 16-15 Mean
of a six-sigma process
shifts by 1.5 standard
deviations.
16-13. A normally distributed process uses 66.7% of the
specification band. It is centered at the nominal dimension, lo-
cated halfway between the upper and lower specification limits.
(a) Estimate PCRand PCRk. Interpret these ratios.
(b) What fallout level (fraction defective) is produced?
16-14. Reconsider Exercise 16-1. Use the revised control
limits and process estimates.
(a) Estimate PCRand PCRk. Interpret these ratios.
(b) What percentage of defectives is being produced by this
process?
16-15. Reconsider Exercise 16-2, where the specification
limits are 14.50 0.50.
(a) What conclusions can you draw about the ability of the
process to operate within these limits? Estimate the per-
centage of defective items that will be produced.
(b) Estimage PCRand PCRk. Interpret these ratios.
16-16. Reconsider Exercise 16-3. Using the process esti-
mates, what is the fallout level if the coded specifications are
10 5 mm? Estimate PCRand interpret this ratio.
16-17. A normally distributed process uses 85% of the spec-
ification band. It is centered at the nominal dimension, located
halfway between the upper and lower specification limits.
EXERCISES FOR SECTION 16-7
(a) Estimate PCRand PCRk. Interpret these ratios.
(b) What fallout level (fraction defective) is produced?
16-18. Reconsider Exercise 16-5. Suppose that the quality
characteristic is normally distributed with specification at 220
- What is the fallout level? Estimate PCRand PCRkand in-
terpret these ratios.
16-19. Reconsider Exercise 16-6. Suppose that the variable
is normally distributed with specifications at 220 50. What
is the proportion out of specifications? Estimate and interpret
PCRand PCRk.
16-20. Reconsider Exercise 16-4(a). Assuming that both
charts exhibit statistical control and that the process specifica-
tions are at 20 5, estimate PCRand PCRkand interpret these
ratios.
16-21. Reconsider Exercise 16-8. Given that the specifica-
tions are at 6.01.0, estimate PCRand PCRkand interpret
these ratios.
16-22. Reconsider 16-7(b). What are the natural tolerance
limits of this process?
16-23. Reconsider 16-12. The viscosity specifications are at
500 25. Calculate estimates of the process capability ratios
PCRand PCRkfor this process and provide an interpretation.
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