Sample size for confidence intervals for the Cp and Cpk index

The BIS.Net Team BIS.Net Team

The CP and Cpk Index has been used for many decades in Quality Assurance as a means of summarizing process capability into a simple index. The Cp Index assumes that the process is centered whereas the CPK index is based on the current process average. Although these indexes are used predominantly in manufacturing, they can be used in other areas, such as finance, marketing, health and wellbeing.

The confidence interval for the CP Index, assuming a Normal Distribution is

LCICp = Cp * Sqrt(CHI_Square_Value(1 – alpha/2, N - 1) / (N - 1))

UCICp = Cp * Sqrt(CHI_Square_Value(alpha/2, N - 1) / (N - 1))

Where alpha is the level of significance

Solving these expressions for sample size to meet a specified margin-of-error has three problems.

  • The margin-of-error is not symmetrical about the sample cp and hence +/- margin of error has no meaning
  • Sample size (N) is also part of the Chi Squared value and hence there is no closed form equation that can be easily solved.
  • The confidence interval depends on the sample Cp index which is not available at the time of calculating the sample size

BIS.Net Sample Size overcomes these limitations by

  • Specifying a confidence interval range, instead of margin of error
  • Use of machine powered algorithms to estimate the sample size.
  • Specifying the confidence interval range as a percentage of the sample Cp.

i.e. 100*Confidence Interval Range/CP

Since 100*(Confidence Interval Range)/Cp =100*(UCICP-LCICP)/Cp= 100*(Cp * Sqrt(CHI_Square_Value(1 – alpha/2, N - 1) / (N - 1)) – Cp*Sqrt(CHI_Square_Value(1 – alpha/2, N - 1) / (N - 1)))/Cp=100*(Sqrt(CHI_Square_Value(1 – alpha/2, N - 1) / (N - 1)) – Sqrt(CHI_Square_Value(1 – alpha/2, N - 1) / (N - 1)))

Cp cancels out meaning Cp does not have to be known in advance.

The analyst must accept that it is not currently possible to specify an interval as a percentage of the population Cp index. However, an interval as a percentage of the sample cp, does provide reasonable control over the accuracy of the estimate and is much better than simply selecting an arbitrary sample size. A confidence interval range that is 10% of the sample Cp is better than a confidence interval that is 50% of the sample Cp.

If this is unsatisfactory then sample size needs to be estimated from the Hypothesis testing section.

The Cpk is more complicated.

The confidence limits are

LCICpk = Cpk * (1 - Z_Value(alpha/2) * Sqrt(1 / (9 * N * Cpk ^ 2) + 1 / (2 * (N - 1))))

UCICpk = Cpk * (1 + Z_Value(alpha) * Sqrt(1 / (9 * N * Cpk ^ 2) + 1 / (2 * (N - 1))))

It is not possible to eliminate the sample Cpk as was done with the CP index. Hence it is no possible to directly calculate sample size for simple confidence intervals. There are only two alternatives. One alternative is to use the Hypothesis Testing section noting that hypothesis testing is related to confidence intervals. The other alternative is to use the sample size determined for the Cp index and accept that the confidence interval width for the Cpk index will be up to 1.3 times larger.

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