Page 124 - Six Sigma Advanced Tools for Black Belts and Master Black Belts

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JWBK119-09
Surrogate PCIs for Nonnormal Data 109
9.2.1 Probability plot
A widely accepted approach to PCI computation is to use a normal probability plot 13
so that the normality assumption can be verified simultaneously. Analogous to the
normal probability plot, where the natural process width is between the 0.135 and
99.865 percentiles, surrogate PCI values may be obtained via suitable probability
plots:

USL − LSL
C p =
upper 0.135% point − lower 0.135% point
USL − LSL
= ,
U p − L p

where U p and L p are respectively the 99.865 and 0.135 percentiles of the observations.
These percentile points can easily be obtained from the simple computer code that
performs probability ploting. Since the median is the preferred central value for a
skewed distribution, the equivalent C pu and C pl are defined as

USL − median
C pu = ,
x 0.998 65 − median
median − LSL
C pl = .
median − x 0.998 65

C pk is then taken as the minimum of (C pu ,C pl ).

9.2.2 Distribution-free tolerance intervals
6
Chan et al. adopted a distribution-free tolerance interval approach to compute C p
and C pk for a nonnormal process using

USL − LSL USL − LSL
C p = = 3
6σ (4σ)
2
USL − LSL
= .
3 (2σ)

This results in
USL − LSL USL − LSL
C p = = 3
w
2 w 2
USL − LSL
= ,
3w 3
min [(USL − μ), (μ − LSL)]
C pk = ,
w/2