Tolerance Interval Calculator

Work out tolerance interval instantly with clear inputs, formula shown and shareable results.

About Tolerance Interval Calculator

A statistical tolerance interval bounds a stated proportion of the population, not just its mean. The half-width is k x sample standard deviation, where k exceeds the normal z-score for the coverage because the mean and standard deviation are themselves estimated from a finite sample. k shrinks towards z as sample size grows.

Formula

Tolerance interval

Limits = mean +/- k x s, with k inflated above the normal z-score for finite sample size

Worked example

With the default values below, Tolerance Interval Calculator returns Tolerance Interval Width of 0.24512. Change any input above and every figure here is recalculated in your browser.

Inputs used

Sample Meanminimum -1,000,000,000: 50.02
Sample Standard Deviationminimum 0: 0.045
Sample Sizeminimum 2: 30
Proportion To Coveraccepts 50–99.99 %: 99 %

Results

Tolerance Interval Width: 0.24512
Lower Tolerance Limit: 49.89744
Upper Tolerance Limit: 50.14256
k Factor Used: 2.7235

Step by step

1Values used
2Tolerance interval
3Tolerance Interval Width= 0.24512
4Lower Tolerance Limit= 49.89744
5Upper Tolerance Limit= 50.14256
6k Factor Used= 2.7235

Assumptions and limits

The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 5 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

How does this differ from a confidence interval?

A confidence interval bounds the population mean. A tolerance interval bounds where individual future observations will fall, which is what matters for specification setting.

Why is k so large for small samples?

With 10 observations the standard deviation estimate is itself uncertain, so the interval must be widened to retain coverage. Published k-factor tables should be used where the interval has contractual weight.