# Process Capability VII – Confidence Limits

## Introduction

In prior articles on process capability, sample statistics and SPC statistics were assumed to be population parameters and ignored sampling variability. This article reviews the analytic methods that can be used to develop confidence bounds on the process capability indices.

## $-P_p-$ Index

The P_{p} index calculation requires an estimate of the parameter σ. The index is calculated as:

$$P_p=\frac{USL-LSL}{6\sigma}$$

(1)

The parameter σ is estimated from the sample standard deviation s, ignoring sampling variability. However, the sample variance s^{2} follows a Chi-square distribution with n-1 degrees of freedom. The $-\chi^2-$ statistic is defined as:

$$\chi^2=\frac{(n-1)s^2}{\sigma^2}$$

(2)

Combining equations 1 and 2 yield:

$$P_p=\frac{USL-LSL}{6s}\sqrt{\frac{\chi^2}{(n-1)}}$$

(3)

Equation 3 is similar to equation 1 with an additional factor $-\sqrt{\chi^2/(n-1)}-$that corrects for sample variation.

Equation 3 is used to calculate P_{p} index confidence limits by using values of $-\chi^2-$ that correspond to percentiles on the Chi-square distribution with n-1 degrees of freedom.

### Example

A part characteristic has the specification 10±0.2 (LSL = 9.8 and USL= 10.2). A critical characteristic was measured for 30 sample parts and a sample standard deviation of 0.065 was calculated. The 0.065 was used for population $-\sigma-$ so equation 1 yields P_{p}=1.026.

When centered between the specification limits, it is possible for the normal µ±3s limits to be contained between the specification limits. However, process center wandering is common in manufacturing. This degrades the actual process capability so the P_{pk} be less than the $-P_p=1.026-$ value. Further degradation occurs as sampling variability has not been considered in the calculation of $-\sigma-$.

A common practice is to calculate 5^{th} and 95^{th} percentile P_{p} confidence limits. For the lower bound, $-\chi_{0.05,29}^2=17.71-$ so the 5^{th} percentiles lower bound of P_{p} is

$$P_p=\frac{(10.2-9.8)}{6(0.065)}\sqrt{\frac{17.71}{29}}=0.801$$

(4)

The upper bound, $-\chi_{0.95,29}^2=42.56-$ so the 95^{th} percentile upper bound of P_{p} is:

$$P_p=\frac{(10.2-9.8)}{6(0.065)}\sqrt{\frac{42.56}{29}}=1.242$$

(5)

These calculations indicates the P_{p} 90% confidence limits on are (0.801, 1.242). The target for short term Pp should be higher than 1. It can be shown that with 30 samples, the Pp should be 1.33 or higher for the $-P_p-$ 90% confidence bounds > 1. Because small sample sizes don’t include long term variation factors, some companies require Pp≥1.67. Meanwhile 6-sigma practitioners often target Pp≥2.0.

## $-C_p-$ Index

SPC data is being used to estimate the population parameters µ and σ. From the R-chart, the grand average $-\bar{R}-$ can be determined. The subgroup size defines factor d_{2}. Then σ may is estimated as $-\frac{\bar{R}}{d_2}-$. Then $-C_p-$ is calculated using equation 1. Generally, a lot of SPC data collected and the population parameters are assumed to match the SPC statistics. I have not found a formula or approach to calculate for lower and upper bounds on Pp from parameters derived from SPC data. If somebody has a recommendation, I would appreciate a reply.

## $-P_{pk}-$ and $-C_{pk}-$ Indices

The tolerance bounds for P_{pk} indices are much more complex as it involves variation in both the sample mean and sample standard deviation.

One approach that is similar to calculating a confidence interval is to use statistical tolerance intervals that contain P% of the population C% of the time, assuming a normal distribution. These are available in “Experimental Statistics, Handbook 91”, United States Department of Commerce, National Bureau of Standards, tables A-6 for 2-sided tolerance intervals and A-7 for 1-sided tolerance intervals.

The tables are constructed using to contain 0.75, 0.9, 0.95, 0.99, and 0.999 of the population (P-values) and 0.75, 0.9, 0.95, and 0.99 confidence limits ($-\lambda-$). In the P_{pk} and C_{pk} calculations, a comparison of the gap between the closest tolerance limit and the 3σ range is made. In a normal distribution the ±3σ range contains 0.9973 of the population. While a small difference in population, the 0.999 P-value is similar to making a tolerance to 6.6σ range comparison.

For now, there isn’t an easy way to obtain confidence limits on P_{pk} and C_{pk}. I recommend that the analyst determine if the tolerance limits contain P=0.999 of the population at C=0.90 confidence.

## Summary

- It is relatively easy to obtain confidence limits to P
_{p}using small sample data using the Chi-square distribution. - There isn’t a simple way to obtain confidence limits on Cp using SPC data.
- Confidence bounds on Ppk and Cpk is difficult. I recommend that the analyst determine if the tolerance limits contain P=0.999 of the population at C=0.90 confidence.

If you want to engage me as a consultant or trainer on this or other topics, please contact me. I have worked in Quality, Reliability, Applied Statistics, and Data Analytics over 30 years in design engineering and manufacturing. In the university, I taught at the graduate level. Also, I provide Minitab seminars to corporate clients, write articles, and have presented and written papers at SAE, ISSAT, and ASQ. I want to assist you.

Dennis Craggs, Consultant

810-964-1529

dlcraggs@me.com

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