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**Communications in Statistics - Simulation and**

**Computation**

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**The performance of process capability index c**

**s**

**On**

**Skewed Distributions**

W. L. Pearn a & C. S. Chang a a

Department of Industrial Engineering & Management , National Chiao Tung University , Hsinchu , Taiwan ROC

Published online: 27 Jun 2007.

**To cite this article: W. L. Pearn & C. S. Chang (1997) The performance of process capability index c**sOn
Skewed Distributions, Communications in Statistics - Simulation and Computation, 26:4, 1361-1377, DOI:
10.1080/03610919708813444

**To link to this article: **http://dx.doi.org/10.1080/03610919708813444

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**COMMUN. **

**STATIST.-SIMULA.,**

**26(4),**1361-1377 (1997)

**THE PERFORMANCE OF **

**PROCESS CAPABILITY INDEX **

**Cs **

**Cs**

**ON SKEWED DISTRIBUTIONS **

W. L. PEARN and C. S. **CHANG **
Department of Industrial Engineering & Management

National Chiao Tung University
**Hsinchu, Taiwan ROC **

* Keywords and Phrases: * Process capability indices; specification limits;
process mean; process standard deviation, skewed distributions.

**ABSTRACT **

**Wright (1995) considered a new process capability index C,, which extends **
**the most useful index to date for processes with two-sided specification limits, Cpmk **
**proposed by Pearn, Kotz and Johnson (1992). The new index C, not only takes **
into account the process variation as well as the location of the process mean
relative to the specification limits, but also considers the asymmetry of the
distribution by incorporating a penalty for skewness. Wright (1995) investigated
an estimator of C, and studied its bias and variance by simulation. The simulation
study, however, was restricted to normal distributions where skewness is not
present. In this paper, we extend Wright's simulation study to cover some skewed
distributions including chi-square, lognormal, and Weibull distributions for some
parameter values. The results show that the percentage bias of the estimator
increases as the skewness coefficient Il,/o'l increases. Extensive simulation
results, comparisons, and analysis are provided.

1361

**Copyright O 1997 by Marcel Dekker, Inc. **

1362 **PEARN AND CHANG **

**1. INTRODUCTION **

Process capability indices (PCIs) have become widely used in the
manufacturing industry to provide measures for process quality. Several basic
indices including C,, Cpk, and C,,, (Kane (1986), Chan, Cheng and Spiring
(1988)) have been proposed to monitor the process potential and process
performance. These indices are useful management tools, which provide numerical
measures of a process characteristic standardized by the process target and
specifications. Combining the three basic indices, Pearn, Kotz and Johnson (1992)
developed the index Cplnk. which is considered to be the most useful index to date
for processes with two-sided specification limits. This index, designed for normal
and near-normal processes, is constructed as with all process capability indices, that
the larger the index, the more capable the process. The index Cpmk is defined **as **the
following:

USL

### -

**p**

Cpmk = min p

### -

LSL*3Jd *

### +

**(p **

### -

T)' '*3Jd *

### +

(p### -

T)'] 'where USL and LSL are the upper and lower specification limits, **p **and **o **are
process mean and process standard deviation, and T is the target value.

The index **Cpmk **provides warnings of the increase of process variation and
process departure (the deviation of process mean from its target), but provides no
sensitivity to the changes in the shape of the distribution, particularly, the
skewness. To detect the shape changes of the processes due to skewness, Wright
(1995) considered a new process capability index C, to extend CPmk. The new
index C, not only takes into account the process variation **as **well as the departure of
process mean from the target, but also the asymmetry of the distribution by
incorporating a penalty for skewness. Utilizing the third central moment **pg = **E(X

### -

**p)3 as **

### a measure of skewness, the new index C, is defined

**as**the following:

with **p3 **divided by **o **to ensure that the skewness term is expressed in the same units
**as **the other terms in the denominator (Wright (1995)). Utilizing the identity min(x,
**y) = (x **

### +

y)/2### -

Ix### -

y1/2, the index C, can be rewritten as the following, where d = (USL### -

LSL)/2:**PROCESS CAPABILITY INDEX Cs **

**2. ESTIMATION **

**OF **

**Cs**

T o estimate the index C,, Wright (1995) considered a complicated estimator (defined in the following). W e note that for normal samples, E[X (xi

### -

T)I/n] = aZ### +

**(p**

### -

T)2, E [ ( I ~ ~ ) ~ / ~ ] = {(n-l)/n}lR c4 a , and E(m3) = (n-l)(n-2) ~ 3 / n ' , where m, = ( l / n )### 2

(xi### -

**TI)'**is the r-th sample central moment, and c4 = {2/(n-l)}1/2T(n/2) T{(n-1)/2}-I (see Wright (1995)).

Obviously, if the third sample central moment is zero, then the estimator

### C^,

**A **

defined above reduces to the estimator Cpmk considered by Pearn, Kotz and Johnson (1992) for the index Cpnlk The distribution of

### e,

is intractable even under normality assumption. Wright (1995) used a simulation technique to compute the expected value and valiance of### e,.

The simulation study, however, was restricted to normal distributions where skewness is not present.Before investigating the performance of the estimator

### e,

under nonnormal*

(skewed) samples, we repeat the calculation on the moments of C, based on 15,000,000 random samples of size n from the uniform distribution, U(0, 1). which are generated by AS183 generator (Wichmann and Hill (1987)) with multiple seeds using IBM RISCl600 work stations. Note that we have extended the sample size for the simulation to n = 500. Tables l(a) and l(b) display the expected

**h ** **h **

**values, variances, and the performance of C, in terms of percentage bias, {E(Cs) - **
C , ) / C , , in normal samples for various values of d / a , and I(F

### -

T)/csl. Our simulation results are almost identical to those presented in Wright (1995). In the next section, we extend Wright's simulation study on percentage bias of the estimator### ?,

to cover some skewed distributions including the chi-square**samples with n=10. 20. 30. 40. 50 **

### I

( P - T ) / a### I

**0.0**

**0.5**

**1.0**

**1.5**

**2.0**

**-20.9% -6.9%**

**-.6% 5.5%*****%

**-17.3% -7.3% -2.8% -.2% 1.8%**

**-15.4% -7.5% -3.5% -1.4% -.I%**

**-14.4% -7.6% -3.8% -1.9% -.a%**

**-13.6% -7.6% -4.1% -2.1% -1.1%**

**-18.3% -7.6% -3.3%**

**.8%*****%

**-15.7% -8.0% -4.4% -2.0% -.4%**

**-14.4% -8.1% -4.8% -2.6% -1.3%**

**-13.6% -8.2% -4.9% -2.9% -1.7%**

**-13.1% -8.3% -5.1% -3.0% -1.8%**

**-16.4% -7.4% -3.8% -.4%*****%

**-14.2% -7.7% -4.5% -2.4%**

**-.9%**

**-13.1% -7.8% -4.8% -2.7% -1.5%**

**-12.5% -7.9% -4.9% -2.9% -1.7%**

**-12.1% -7.9% -4.9% -3.0% -1.9%**

**-15.0% -7.1% -3.8% -.9%*****%

**-13.1% -7.3% -4.4% -2.4% -1.1%**

**-12.1% -7.4% -4.6% -2.7% -1.6%**

**-11.6% -7.4% -4.6% -2.8% -1.7%**

**-11.2% -7.5% -4.7% -2.9% -1.8%**

**-13.9% -6.7% -3.7% -1.1%**

### ***%

**-12.2% -6.9% -4.2% -2.3% -1.1%**

**-11.4% -7.0% -4.4% -2.6% -1.5%**

**-10.9% -7.0% -4.4% -2.7% -1.7%**

**-10.5% -7.1% -4.5% -2.7% -1.7%**

**Table l(a). Expected value, variance. and percentage bias o f **

### e,

**for normal**

**50 **
**3 **
**4 **
**5 **
**6 **
**.8779 .0134 .6937 .0106 .4516 .0047 .2709 .OD19 .I474 .0008 **
**1.1818 .0234 .9704 .0179 .6763 -0079 .4503 .0032 .2936 .0014 **
**1.4858 .03651.2471 .0272 .go10 .0121 .6298 .0049 .4398 .0021 **
**1.7897 .05241.5238 .03861.1258 -0171 .a093 .0070 .5860 .0030 **

**Table l(b). Expected value. variance. and percentage bias o f **

### e,

**for normal samples w i t h n=100. 200. 300. 400. 500**

1366 **PEARN AND CHANG **
Table 2. Characteristics of the three distributions.

### ...

Mean 3.00 4.00 5.00 6.00 7.00 Variance 6.00 8.00 10.00 12.00 14.00 Skewness 1.63 1.41 1.26 1.15 1.07### ...

### ...

Mean I . 1.13 1.06 1.03 1.02 V d a n c e 4.67 0.36 0.13 0.07 0.04 Skewness 6.18 1.75 1.07 0.78 0.61 ~ ( 1 , s ) 1 1 W(1,2) ~ ( l , ? )### ...

Mean 0.91 0.90 0.89 0.89 0.89 Variance 0.44 0.33 0.26 0.22 0.18 Skewness 1.20 0.96 0.78 0.63 0.51distribution X2(r), the lognormal distribution logN(p, **oz), **and the Weibull
distribution *W ( a , *

**P). **

**3. ESTIMATION OF Cs FOR SKEWED DISTRIBUTIONS **
For skewed distributions, we consider the following three distributions: (a)
chi-square distribution, X?(r), with probability density function f(x) = {T(r/2)}-1
(1 * /2)r12 *(x)'"

**- I**(e)-x/2, for 0 < x < -, and degrees of freedom r =

**3,**4, 5, 6, and

**7;**

(b) lognormal distribution, logN(p, 02), with probability density function f(x) = ( x (2n)ln 0 ) - I exp{- [In(x)

### -

p]2/(202)), for*0*< x <

### -,

### -

## -

*p < -, and parameter values*

**c****p**= 0, and o = 1,

**1/2,**1 / 3 , 1/4, and 1/5; (c) Weibull distribution, W ( a , p), with probability density function f(x) = { p (x)bl a+) exp { -

**(x/a)P), for 0**

### <

x <### -,

a > 0,### p

> 0, and parameters a = 1 , p = 1.4, 1.6, 1.8, 2.0, and**2.2.**

The characteristics, including the means, the variances, and the skewness
coefficients 1 p d d of the three distributions are summarized in Table **2. **W e note
that for chi-square distribution, XZ(r), the skewness coefficient decreases as the

**PROCESS CAPABILITY INDEX Cs ** 1367

value of the degrees of freedom increases. For lognormal distribution, logN(0, 02). the skewness coefficient decreases as the value of o2 decreases. For Weibull distribution, W(1,

**P), **

the skewness coefficient decreases as the value of **P **

increases.
Table 3 displays the results from the simulation for the chi-square
distribution, X2(r), with degrees of freedom r = 3,4, 5, 6, and **7. **Table 4 displays
the results from the simulation for the lognormal distribution, IogN(p, 02), with p =

0, and o = 1, 1/2, 1/3, 1/4, and 1/5. Table 5 displays the results from the simulation for the Weibull distribution, W(a,

**P), **

with a = 1, and **P **

= 1.4, 1.6, 1.8,
2.0, and 2.2. The simulation was carried out for the following values, d / o = 2, 3,
4, 5, and 6, and I(p ### -

T ) / d = 0.0, 0.5, 1.0, 1.5, and 2.0. For simplicity of the presentation, the variance columns are omitted, only values of the percentages bias,**{E(& **

### -

C,}/C,, are presented.In Figures 1 (a)- l(c), we plot the percentage bias versus skewness coefficient,
1k3/031, for the three distributions, with d / b = **3, ** 1(p

### -

T ) / d = 0.5, and n = 20, 30, 50. The figures show that for all three distributions, the percentage bias, ( ~ ( c , )### -

C,} /C,, increases as the skewness coefficient, Ip,/031, increases. From Tables 3, 4, and 5, we observed that this relationship remains intact for all values of d/o, l(p### -

T)/ol, and sample size n. In fact, Chen and Kotz (1996) have pointed out that the**A **

asymptotic behavior of the estimator C, is highly sensitive to the skewness of the
process distribution regardless of whether p = T or **p **# T.

Chen and Kotz (1996) showed that

### c,

is a consistent and asymptotic unbiased estimator of### C,.

But, they did not investigate the direction of bias. For normal distribution, the bias is negative except for some cases with small n regardless of whether**p**= T or p # T (see Wright (1995)). For skewed distributions, the

direction of bias is quite different. Tables 3, 4, and 5 show that for the three
distributions, Xz(r), logN(0, * 02). * and W(1.

**P), **

the bias is positive for all n if the
process is off-target **(p**# T). On the other hand, if the process is on-target (p = T), the bias tends to be positive for small n, and negative for large n.

In order to find the interpretations for such different behaviors of

### e,

between the normal distribution and skewed distributions, we consider the estimator, L / s , of the term p3/o in the denominator of C, defined in Wright (1995) We perform the same simulation for ;3/s and calculate the percentage bias { E G 3 / s )### -

k3/o}/(p3/o). The results, which are displayed in Tables 6(a)-6(c), indicate that**PEARN AND CHANG **
**Table 3. Percentage bias of **

*e, *

**for chi-square distribution.**

**x2(r), with r=3, 4, 5, 6, and 7. **

**PROCESS CAPABILITY INDEX Cs **

**Table 3. (continued) Percentage bias of **

*e, *

**for chi-square distribution,**

**x2(r), with r=3, 4, 5, 6, and 7.**

**Skewness **

Figure l(a). Percentage bias plot for P ( r ) distribution,

r = 3.4.5, **6, 7, d/u **= **3, ** **1(p **

### -

**n/ul**=

**0.5,**n =

**20.30, 50.**

### .a

**100,**.a

**n = 20**

**5 **

**5**

### =

**80**

### -

n = 3 0

**2****20**

_{n = }_{50 }**Skewness**0

Figure l(b). Percentage bias plot for **logN(0,az) **distribution,

**o **= 1, **112, 113, 114, 115, d/a **= **3, 1(p **

### -

**T)/al**=

**0.5,**n =

**20, 30, 50.**

**2 **

**Skewness **

0 **I **

### ,

(c) **0.51 **0.63 **0.78 ** 0.96 **1.2 **

**Figure l(c). Percentage bias plot for W(1,P) distribution, **

**p **

= 1.4, **1.6,**1.8,

**2.0, 2.2, d/o**=

**3, 1(p**

### -

**T)/d**=

**0.5, n**=

**20, 30, 50.**

**PROCESS **

**CAPABILITY INDEX **

**Cs**

**1371**

**Table 4. Percentage bias of **

*e, *

**for lognormal distribution, logN(0,u2),**

**with u = l ,**

*112,*

**113, 114,****and**

**115.****(continued) **

**1372 ** **PEARN AND CHANG **
**Table 4. (continued) Percentage bias of **

### e,

**for lognormal distribution, logN(0,u2),**

**with u=1, 112, 113, 114, and 115. **

**PROCESS CAPABILITY INDEX Cs **

**Table 5. Percentage bias of **

### e,

**for Weibull distribution, W(l,P),**

**with P=1.4, 1.6, 1.8, 2.0, and 2.2.**

**(continued) **

**1374 ** **PEARN AND CHANG **
**Table 5. (continued) Percentage bias of **

### e,

**for Weibull distribution, W(l,P),**

**with P=1.4, 1.6, 1.8, 2.0, and 2.2. **

**P - 2 . 0 ****P . 2 . 2 **

**PROCESS CAPABILITY INDEX Cs **

Table 6(a). Percentages bias ofG3/s for X2(r).

Table 6(b). Percentages bias of p3/s for l o g ~ ( 0 , **0 2 ) . **

Table 6(c). Percentages bias of G3/s for ~ ( 1 , (3).

**1376 ** **PEARN AND CHANG **
for all three skewed distributions, the bias of C3/s is negative. The under-estimate
of f;3/s for the term p 3 / 0 results in a reduction for the value of the denominator of

### ?,.

Consequently,### ?,

over-estimates C,, and the bias becomes positive.**4. CONCLUSIONS **

Wright (1995) considered a new process capability index C,, which takes into account the process variation, the location of the process mean relative to the specification limits, and the asymmetry of the distribution. Wright (1995) investigated an estimator of C, and studied its bias and variance by simulation for normal distributions where skewness is not present. In this paper, we extend Wright's simulation study to cover some skewed distributions including chi-square, lognormal, and Weibull distributions.

The result show that for all three skewed distributions, the percentage bias,

**{E(?,) **

### -

C,)/C,, increases as the skewness coefficient, Ip3/afl, increases. For the normal distribution, the bias is negative except for some cases with small n regardless of whether**p**= T or p # T. For skewed distributions, the bias is positive

for all n if the process is off-target (p # T). On the other hand, if the process is on-

target (p = T), the bias tends to he positive for small n, and negative for large n.
Although the index C, is sensitive to skewed distributions, and has some interesting
properties over **CPlllk. **but the estimator

### ?,

proposed by Wright (1995) is highly unstable in the presence of skewness. In fact,### .-.

we demonstrated that the percentage bias, {E(C,)### -

C,) /C,, increases as the skewness coefficient 1p3/031 increases for the three typical skewed distributions we investigated. Evidently, for the index C, to be acceptable by the practitioners a more stable estimator is needed.**5. ACKNOWLEDGEMENT **

The authors would like to thank the anonymous referees for their careful reading of the paper and several suggestions which improved the paper.

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Chen H. and Kotz, S. (1996). **An asymptotic distribution of Wright's process **
* capability index sensitive to skewness. Jounlnl of Statistical Computation *&

* Simulation. *To appear.

**PROCESS CAPABILITY INDEX Cs ** 1377
Chan, L.K., Cheng, S.W. and Spiring, **F.A. (1988). A new measure of process **

capability: C,,. *Journal of Qrtrrlity Technology, *20(3), 162-175.

*Kane, V.E. (1986). Process capability indices. Journal of Quality Technology, 18(1), *
41-52.

Pearn, W.L., Kotz, S. and Johnson. N.L. (1992). Distributional and inferential
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**Wichmann, B.A. and Hill, I.D. (1982). An efficient and portable pseudo-random **
*number generator. Jorirnol of the Royol Stntistical Society, Series C, *31(2), 188-190.
**Wichmann, B.A. and Hill, I.D. (1984). Correction to AS183: an efficient and **
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*Series C, *33(1), 123.

**Wichmann, B.A. and Hill, I.D. (1987). Programming insight: building a random **
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*Statistical Cornprimtion *& *Simrrltrtion, *52, 195-203.

Received October, 1996; Revised *May, * 1997.

.-L