Tolerance Intervals for Normal Distribution

Revised: 09/08/2005

Introduction. Loosely speaking, a tolerance interval for a measured quantity is the interval in which there is some "likelihood" (or, of which you feel a some "level of confidence") that a specified fraction of the population's values lie, based on a sample that you measured from this population. Tolerance intervals have been widely used in statistical process control. This page will calculate tolerance intervals for any specified population fraction, and for any specified level of confidence, from the mean and standard deviation of a finite sample, under the assumption that the population is normally distributed. One-sided (upper and lower) intervals, as well as the two-sided interval, are calculated.

Reference: NIST/Sematech Engineering Statistics Handbook, Section

The fields below are pre-loaded with sample data taken from one of the examples in this Handbook.

Note: The very last line on the Handbook page:
"The upper (one-sided) tolerance limit is therefore 97.07 + 1.8752*2.68 = 102.096."
is wrong. The standard deviation in their example is 0.0268, not 2.68, so the answer should be 97.12, not 102.096 .

These calculations are also available in a downloadable Excel spreadsheet: tolintvl.xls .

Instructions: Enter the five parameters (N, mean, SD, confidence level, and coverage fraction) into the five cells below. Use the Tab key after entering each value. The calculations will be updated whenever you "leave" one of these five fields, either by tabbing out of it, or by clicking in another field.

Before using this page for the first time, make sure you read the JavaStat user interface guidelines.

If I measure a sample consisting of


and get a mean value of

and a standard deviation of

then I can be

% certain


% of the population

lies within the interval from:


(a Two-sided Tolerance Interval)


lies below the value:

(an Upper One-sided Tolerance Interval)


lies above the value:

(a Lower One-sided Tolerance Interval)

You can ignore the following intermediate quantities used in the calculations. They are displayed so that you can compare them with the values shown in the Sematech handbook.









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