Tolerance interval basics

In This Topic

What is a tolerance interval?
How do tolerance intervals differ from confidence intervals and prediction intervals?
Parametric and nonparametric methods

What is a tolerance interval?

Use tolerance intervals to compute a range of values for a product's characteristic that likely covers a specified proportion of future product output. A tolerance interval defines the upper and/or lower bounds within which a certain percent of the process output falls with a stated confidence.

To generate tolerance intervals, you must specify both a minimum percentage of the population and a confidence level. Traditionally, both values are close to 100. The percentage is the minimum percentage of the population that you want the interval to cover. The confidence level is the likelihood that an interval will actually cover the minimum percentage.

For example, a parts manufacturer wants to determine the limits that define where 99% of the parts lengths with 95% confidence will be, and compare this range to the customer specifications. Analysts randomly sample 30 parts and record the width in millimeters (mm). The tolerance interval states with 95% confidence that 99% of the population have widths that fall within the interval [5, 8]. The manufacturer is 95% confident that 99% of all parts will have widths that are between 5 and 8 mm. If this range is wider than the clients' requirements, then the process may produce excessive waste.

Note

Minitab uses defaults of 95% for both confidence level and minimum percentage of population in the interval.

How do tolerance intervals differ from confidence intervals and prediction intervals?

Confidence intervals (CI), prediction intervals (PI) and tolerance intervals are commonly used intervals derived from sample statistics.

Confidence interval: A range of values that is likely to contain the value of an unknown population parameter, such as the mean, with a specified degree of confidence.; For example, if the 95% CI of the average fill volume of 375 ml bottles is 368–372 ml, you can be 95% confident that the true value of the process mean is within this interval.
Prediction interval: A range of values for a product's characteristic that represents where the value of a single new observation is likely to fall with a specified degree of confidence.; For example, if the 95% PI of the average fill volume of 375 ml bottles is 360–379 ml, you can be 95% confident that the next sampled bottle will have a fill volume that is within this interval.
Tolerance interval: A range of values for a product's characteristic that likely covers where a specified proportion of the population lies with a specified degree of confidence.; For example, if the 95% tolerance interval for 99% of the population for the fill volume of 375 ml bottles is 358–381 ml, you can be 95% confident that 99% of the bottles to be filled in the future will have volumes that are within this interval.

Parametric and nonparametric methods

Minitab can calculate tolerance intervals using a parametric method, like the method that uses the normal distribution, or a nonparametric method. Use the intervals that match your situation, as follows:

Parametric method

If your data follow a distribution, then a parametric method is more precise and economical than the nonparametric method. A parametric method allows you to achieve smaller margins of error with fewer observations, as long as the chosen distribution is appropriate for your data. Use the parametric method if you know from prior experience or analysis that your population follows a known distribution. A goodness-of-fit test, such as the one that Minitab includes with Stat > Quality Tools > Individual Distribution Identification, can help you decide if your data follow a distribution. Use Tolerance intervals (Normal distribution) if your data follow a normal distribution. Use Tolerance intervals (Nonnormal distribution) if your data follow one of the following distributions:

Lognormal
Gamma
Exponential
Smallest extreme value
Weibull
Largest extreme value
Logistic
Loglogistic

Minitab includes a specific goodness-of-fit test with any tolerance interval so that you can assess the distribution.

Nonparametric method

Parametric methods are not robust to severe departures from the distribution. If you are unsure of the parent distribution, or you know that the parent distribution is not in Minitab, then use the nonparametric method. The nonparametric method requires only that the data are continuous.

The nonparametric method usually requires larger sample sizes than the parametric method. For example, if the minimum percentage of the population in the interval is 95%, the sample size should be approximately 90 or more for the tolerance interval to be accurate. Larger percentages of the population in the interval require larger sample sizes. For example, if the minimum percentage of the population in the interval is 99%, the sample size should be approximately 500 or more to obtain an accurate two-sided 95% tolerance interval. To have an accurate tolerance interval, the achieved confidence level must be close to your target confidence level. If your sample size is not large enough, the nonparametric interval is a non-informative interval that ranges from negative infinity to infinity. In this case, Minitab displays a finite interval based on the range of your data. As a result, the achieved confidence level is much lower than the target confidence level.

To determine an appropriate sample size for a tolerance interval that meets your accuracy and precision objectives, go to Stat > Power and Sample Size > Sample Size for Tolerance Intervals .