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 (1α). The k factor can be obtained from the appropriate table contained in ISO162696 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 tvalue 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 outofspecification 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, steven@statisticaloutsourcingservices.com
