Sample size determination for Cp comparisons

S. M. Chen, J. T. Liaw, Y. S. Hsu

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


Comparison of quality for products (supplies and goods) is extremely important for manufacturers and consumers. Based on correct comparisons, manufacturers and consumers can find better suppliers to cooperate and better merchandise to purchase, respectively. Quality is often measured and compared by process capability indices, among which Cp is very efiective, simple to apply, and particularly useful for the first round of comparison. In practice, Cp is unknown and should be estimated from observations. Let dCpi denote the maximum likelihood estimator obtained from normal process, Xi, with index value Cpi; i = 1; 2. If dCp1 > (<)d Cp2 is observed, we will conclude that Cp1 > (<) Cp2 and decide that X1 is better (worse) than X2. Given a small and positive number, , there is no need to make comparison when (1-€) Cp2 < Cp1 < (1+€) Cp2 since Cp1 is close to Cp2. It is desirable to observe dCp1 <d Cp2 with high probability when (1 + €) Cp2 > Cp1 and with low probability when (1-€) Cp2 < Cp1. Given 0 < 1, 2 < 1, based on the table constructed from P(dCp1 >d Cp2), we demonstrate how to find the smallest sample size needed to ensure observing dCp1 d Cp2 with probability greater than 1 €1 when (1 + €) Cp2 < Cp1 and smaller than 2 when (1 €) Cp2 < Cp1.

Original languageEnglish
Pages (from-to)3072-3085
Number of pages14
JournalScientia Iranica
Issue number6
StatePublished - 2016


  • Biased estimator
  • C
  • C
  • Maximum likelihood estimator
  • Process capability Index
  • Unbiased estimator


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