Cplog: A Capability Index for Non-Normal Processes

Article

O'Neill, Edwin E., Jr.; Krishnamoorthi, K. S.   (1997, ASQ)   Caterpillar Inc.; Mossville, IL; Bradley University, Peoria, IL

Annual Quality Congress, Orlando, FL    Vol. 51    No. 0
QICID: 10622    May 1997    pp. 774-781
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Article Abstract

Unilateral-plus tolerance processes are more suited to a new Cplog capability index than to Cp and Cpk indices. The non-normal characteristics of unilateral-plus tolerance processes are found in machine shops when characteristics have only non-negative values and one upper specification limit. Cp and Cpk are inappropriate in this environment because they assume a normal distribution in the processes they analyze. A typical modification is to change the multiplier of s in the denominator of Cp, but this works only for some non-normal processes. Cplog is a more general solution. It is based on the lognormal distribution. A variable, X, is lognormal if lnX has a normal distribution. The parameters for lnX are the m, the log-mean, and s, the log-standard deviation. Then, Cplog = (USL - em)/(em(e3s -1)), where USL is the upper specification limit, and e is the natural logarithm base. In an analysis of twenty randomly selected unilateral-plus tolerance characteristics, sixteen failed the Chi-square goodness of fit test for normality at a = 0.05, and therefore, Cp and Cpk gave improper results for these processes. The goodness of fit test indicated that Cplog was a better fit for ten of the twenty data sets; that for four data sets both Cplog and the normal distributions both were acceptable, though the latter gave erroneous negative values in their histograms; and that for the other six data sets, both normal and lognormal distributions gave inadequate fits. Of these last six, three data sets were severely overestimated by Cplog, indicating that this new index has some limitations.

Keywords

Process capability index (Cpk),Normal distribution,Non-normality,Manufacturing,Lognormal distribution


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