Normal Tolerance Interval Calculator
A normal tolerance interval gives limits of the form mean − k·s and mean + k·s that are expected to contain at least a proportion p of a normally distributed population, with confidence C. This calculator works from summary statistics only: the sample size N, the sample mean and the sample standard deviation s.
For a two-sided interval it returns the factor k2 from the Howe (1969) approximation, which is weak for small N. For a one-sided bound it shows two factors side by side: the Natrella approximation and the exact factor from the non-central t distribution. The limits are mean − k·s and mean + k·s for each factor shown.
The data are assumed normal, and the result is a statistical factor only. Below the results, every formula is repeated with your values substituted, so each step can be checked.
Tolerance factor and limits
Results
| One-sided result | Natrella | Exact (non-central t) |
|---|---|---|
| k (dimensionless) | — | — |
| Upper limit = mean + k·s (unit of your measurements) | — | — |
| Lower limit = mean − k·s (unit of your measurements) | — | — |
The two-sided factor is the Howe approximation, weak for small N. Data are assumed normal. The result is a statistical factor only. k is given to 4 decimals and limits to 4 significant figures.
Formulas with your values
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How to use the calculator
- Enter the sample size N, a whole number from 3 to 1000.
- Enter the sample mean and the sample standard deviation s, both in the unit of your measurements; s must be above 0.
- Enter coverage p and confidence C as fractions above 0 and below 1, for example 0.90 and 0.99.
- Choose two-sided or one-sided.
- Read the k factor and the limits. Results update as you type; the Calculate button runs the same computation.
- Check the formulas with your values substituted, shown below the results.
Method and formulas
- z(q) is the standard normal quantile and ν = N − 1. All computation runs in double precision with no library.
- Two-sided, Howe approximation: k2 = z((1 + p)/2) × √(ν(1 + 1/N) / X(1 − C, ν)). X(a, ν) is the value x with P(χ² with ν degrees of freedom ≤ x) = a, found by 200 bisection steps on the regularized lower incomplete gamma function P(ν/2, x/2), evaluated by its power series.
- One-sided, Natrella: zp = z(p), zC = z(C), a = 1 − zC²/(2ν), b = zp² − zC²/N, k1 = (zp + √(zp² − a·b))/a. It is not defined when a ≤ 0 or zp² − a·b < 0.
- One-sided, exact: δ = zp√N; t is the C quantile of the non-central t distribution with ν degrees of freedom and parameter δ; k = t/√N. The cumulative probability P(T ≤ t) is the integral over v from 0 to ν + 40√(2ν) + 60 of Φ(t√(v/ν) − δ) times the chi-square density with ν degrees of freedom, by the Simpson rule with 4000 steps; t is found by 100 bisection steps.
- Limits: two-sided mean − k2·s and mean + k2·s; one-sided upper limit mean + k·s and lower limit mean − k·s for each k shown.
- Worked example: N = 43, mean 50, s = 2, p = 0.90, C = 0.99 gives a chi-square quantile of 23.650, k2 = 2.2173 and two-sided limits 45.57 and 54.43. One-sided, Natrella gives 1.8752 and the exact factor is 1.8740 (δ = 8.4037, t = 12.28834).
Frequently asked questions
Which method gives the two-sided factor k2?
The Howe (1969) approximation: k2 = z((1 + p)/2) × √(ν(1 + 1/N) / X(1 − C, ν)), where ν = N − 1, z is the standard normal quantile and X(1 − C, ν) is the chi-square quantile with lower-tail probability 1 − C and ν degrees of freedom. The approximation is weak for small N.
Why does the one-sided result show two k values?
The calculator shows the Natrella approximation and the exact factor side by side. Natrella uses a = 1 − zC²/(2ν), b = zp² − zC²/N and k = (zp + √(zp² − a·b))/a, and is not defined when a ≤ 0 or zp² − a·b < 0. The exact factor is k = t/√N, where t is the C quantile of the non-central t distribution with ν degrees of freedom and noncentrality δ = zp√N. For N = 3, p = 0.90 and C = 0.99, Natrella is not defined and the exact factor is 13.7112.
What values does the calculator accept?
The sample size N must be a whole number from 3 to 1000, the sample mean can be any real number, the sample standard deviation s must be above 0, and coverage p and confidence C must be above 0 and below 1. An empty or invalid field gives no result, only dashes and a message naming the field.
Does the tolerance interval prove that my product meets its specification?
No. The data are assumed to be normally distributed, and the result is a statistical factor only. It is an estimate and not a substitute for a qualified engineer.
Disclaimer
This calculator gives an estimate only and is not a substitute for a qualified engineer. Data are assumed normal and the result is a statistical factor only. Source: NIST/SEMATECH e-Handbook of Statistical Methods, section 7.2.6.3, with Howe (1969) and Natrella.