Applied Statistics and Probability for Engineers

16-7 PROCESS CAPABILITY 623

estimate of the process capability ratio PCRkis

Note that if PCRPCRk, the process is centered at the nominal dimension. Since

for the vane-opening process and the process is obviously run-

ning off-center, as was first noted in Figs. 16-14 and 16-17. This off-center operation was ul-

timately traced to an oversized wax tool. Changing the tooling resulted in a substantial im-

provement in the process (Montgomery, 2001).

The fractions of nonconforming output (or fallout) below the lower specification limit and

above the upper specification limit are often of interest. Suppose that the output from a normally

distributed process in statistical control is denoted as X. The fractions are determined from

EXAMPLE 16-3 For an electronic manufacturing process a current has specifications of mil-

liamperes. The process mean and standard deviation are 107.0 and 1.5, respectively. The

process mean is nearer to the USL. Consequently,

The small PCRkindicates that the process is likely to produce currents outside of the specifi-

cation limits. From the normal distribution in Appendix Table II

P 1 X USL 2 P 1 Z 1110  1072 1.5 2 P 1 Z 22 0.023

P 1 X LSL 2 P 1 Z 190  1072 1.5 2 P 1 Z11.33 2  0

PCR 1110  902 16 1.5 2 2.22 and PCRk 1110  1072 13 1.5 2 0.67

100

10

P 1 X LSL 2 P 1 Z 1 LSL 2  2 P 1 X USL 2 P 1 Z 1 USL 2  2

^PCRk1.06 ^PCR1.55,

min c

40 33.19

31 2.15 2

1.06,

33.19 20

31 2.15 2

2.04d1.06

^PCRkmin c

USLx

3 ˆ^

,

xLSL

3 ˆ

d

For this example, the relatively large probability of exceeding the USLis a warning of po-

tential problems with this criterion even if none of the measured observations in a preliminary

sample exceed this limit. We emphasize that the fraction-nonconforming calculation assumes

that the observations are normally distributed and the process is in control. Departures from

normality can seriously affect the results. The calculation should be interpreted as an approx-

imate guideline for process performance. To make matters worse, and need to be esti-

mated from the data available and a small sample size can result in poor estimates that further

degrade the calculation.

Montgomery (2001) provides guidelines on appropriate values of the PCRand a table re-

lating fallout for a normally distributed process in statistical control to the value of PCR.

Many U.S. companies use PCR1.33 as a minimum acceptable target and PCR1.66 as a

minimum target for strength, safety, or critical characteristics. Some companies require that

internal processes and those at suppliers achieve a PCRk2.0. Figure 16-15 illustrates a

process with PCRPCRk2.0. Assuming a normal distribution, the calculated fallout for

this process is 0.0018 parts per million. A process with PCRk2.0 is referred to as asix-

sigma process because the distance from the process mean to the nearest specification is six

standard deviations. The reason that such a large process capability is often required is that it

c 16 .qxd 5/8/02 9:58 PM Page 623 RK UL 6 RK UL 6:Desktop Folder: