But this calculation, being very complex, is beyond the scope of this article.

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In Part II we saw how a prediction interval can be calculated to define a region within which a single future observation or a multiple number of single future values is likely to be located with a given confidence.

In this, the third and final statistical interval to be discussed, we will look at an interval to cover a specified proportion of a population distribution with a given confidence.

This type of interval is called a tolerance interval and is closely related to measures of process capability.

Larger ratios indicate a more capable process while smaller ratios indicate a less capable process.

The capabilities of two different processes are depicted in Figure 1.

The distribution on the right, on the other hand, exhibits poor capability; the tail areas of this distribution fall outside of specification range.

Notice also that the capability of a process is determined not just by the location of the sample mean, but the tail areas of the distribution as well.

Instead, a small sample of a total population is inspected or tested and the data used to gauge how well the entire production batch conforms to specifications.

This is because capability is usually calculated to cover a fixed multiple of sample standard deviations (usually 3 representing 99.73 of the data population).

But this percentage only holds true for larger sample sizes; that is, greater than 50.

Therefore, In order to maintain a reasonably accurate estimate of the capability of a process for smaller sample sizes, we need to adjust the number of multiple sample standard deviations used to define the region covering the desired proportion of the population distribution with a given confidence.

Notice in figure 2 that the interval range extends beyond the tail areas of the actual population distribution (solid line).

This uncertainty is represented by the confidence level associated with the interval.

The confidence level indicates the likelihood that the interval covers the desired proportion of the population.

The solid line represents the actual population distribution.

The dotted line distributions result from the uncertainty of knowing the true location of the population mean.

The difference in location of the mean due to this uncertainty is defined by the confidence interval.

These values of k 2 were calculated iteratively using a numerical integration process described by Garaj and Janiga in 2002.

Statistical Tolerance Interval