# Methods and formulas for observed performance in Nonparametric Capability Analysis

## PPM < LSL for Observed Performance

The number of parts out of one million that have measurements that are less than the lower specification limit is calculated as follows:

### Notation

TermDescription
LSLLower specification limit
NTotal number of observations

## PPM > USL for Observed Performance

The number of parts out of one million that have measurements that are greater than the upper specification limit is calculated as follows:

### Notation

TermDescription
USLUpper specification limit
NTotal number of observations

## PPM Total for Observed Performance

The total number of parts out of a million that are outside the specification limits is calculated as follows:

### Notation

TermDescription
LSLLower specification limit
USLUpper specification limit
NTotal number of observations

## Confidence intervals for PPM and % out of specification

The calculation of the confidence interval is the same for PPM and percent except for a step at the end to convert the result to the correct units. The following formulas are for the two-sided 100(1 – )% confidence intervals. To find one-sided confidence intervals, replace with (1 – ).

To convert the result of the formula to PPM, multiply by 1,000,000.

To convert the result of the formula to percentage, multiply by 100.

If no observed units are outside of a specification limit, then the lower confidence bound is 0. If all the observed units are outside of a specification limit, then the upper confidence bound is 1.

### Notation

TermDescription
Lower confidence bound
Upper confidence bound
nTotal number of units
Empirical probability that a unit is out of specification: defectives / total.
(1 – )/2 percentile from a standard normal distribution
1 – the confidence level

### References

Newcombe, R. G. (1998). Two-sided confidence intervals for the single proportion: comparison of seven methods. Statistics in Medicine, 17(8), 857-872.

Tong, L. I., & Chen, J. P. (1998). Lower confidence limits of process capability indices for nonnormal process distributions. International Journal of Quality & Reliability Management, 15(8/9), 907-919.