Minimizing the Problem of OOS - Out-of-specification situations can be reduced using confidence and tolerance intervals. - BioPharm International
Minimizing the Problem of OOS
Out-of-specification situations can be reduced using confidence and tolerance intervals.
 Sep 1, 2008 BioPharm International

TOLERANCE INTERVAL

A confidence interval covers a population parameter with a stated confidence, that is, a certain proportion of the time. There is also a way to cover a fixed proportion of the population with a stated confidence. Such an interval is called a tolerance interval. Statistical tolerance intervals are limits in which we expect a stated proportion of the individual values of the population to lie. For a sample mean (X mean) and sample standard deviation (s), the general formula for a tolerance interval is:

The value of k is based on the sample size (n) and the confidence level (1-α). The k factor can be obtained from the appropriate table contained in ISO-16269-6 Statistical Interpretation of Data reference or calculated by statistical software.

 Table 2. Tolerance interval calculations for 95% confidence and 95% coverage
Revisiting the data from Table 1, the tolerance interval calculations for a 95% confidence and 95% coverage can be found in Table 2.

The tolerance interval means we are 95% confident that 95% of the individual values in the population whose mean is 31.7 with a standard deviation of 3.07 will be in the range of 18.12 to 45.28.

The assumption for confidence intervals and tolerance intervals is that the data are an independent random sample from a single population that is normally distributed. Typically, the true mean and standard deviation from the population are not known, and are therefore estimated from the sample.

CAPABILITY INDICES

Process capability compares the output of a process to the specification limits. The comparison is made by forming the ratio of the spread between the process specifications (the specification "width") to the spread of the process values, as measured by process standard deviation. A capable process is one in which almost all the measurements fall inside the specification limits. Using confidence intervals or tolerance intervals, the percent of the values that would fall outside the specification can be calculated. This is best explained using an example.

Assume we have a process whose specification is that the mean of 5 samples must be between 40 and 50. If we sampled from the process and observed a mean of 46 and standard deviation of 2 based on a sample of 5 units, the 95% confidence interval would be 42.9 to 49.1. The 99% confidence interval would be 41.0 to 51.0. Based on the 99% confidence interval, we can see that we would expect the mean to be outside the upper specification more than 0.5% of the time (because the entire probability of exceeding the interval is 1%, then the probability of exceeding just the upper interval is 0.5%).

 Table 3. Analysis of 99% confidence interval
The exact value for the percent of the means of a sample size equal to 5 that would exceed the specification based on the observed mean and standard deviation can be calculated. This is accomplished by solving for the t-value in the confidence interval equation by setting the specification equal to the confidence interval limits. Because the mean is not centered in the specification, it will be necessary to break the problem into two parts, solving for the lower and upper limits individually, and then adding the probabilities together. Table 3 shows the results of these calculations.

SUMMARY

Confidence limits are limits in which we expect a given population parameter to fall. Statistical tolerance limits are limits in which we expect a stated proportion of the population to lie. Using the confidence interval and tolerance interval, specifications can be set that minimize the number of out-of-specification situations. A combination of tolerance intervals and confidence limits defines the overall process parameter (mean) and the distribution of the individuals.

Steven Walfish is the president of Statistical Outsourcing Services, Olney, MD, 301.325.3129,