Skip to content

Home / Tolerance Interval / Normal Method

Normal Method

The normal method assumes the sample comes from a normal distribution. The bounds are built from the sample mean and standard deviation, scaled by a tolerance factor \(k\) that accounts for both the requested population coverage \(P\) and the confidence level \(C\). (The results output also includes an Anderson-Darling goodness-of-fit check for assessing whether the normality assumption is reasonable.)

See Tolerance Interval for the notation used on this page.

Sample statistics

\[ \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i \qquad\qquad s = \sqrt{\frac{\sum_{i=1}^{n}\left(x_i - \bar{x}\right)^2}{n-1}} \]

Bounds

Interval type Bounds
Two-sided \(\left[\ \bar{x} - k_2\, s,\ \ \bar{x} + k_2\, s\ \right]\)
One-sided lower bound \(\left[\ \bar{x} - k_1\, s,\ \ \infty\ \right)\)
One-sided upper bound \(\left(\ -\infty,\ \ \bar{x} + k_1\, s\ \right]\)

The one-sided and two-sided tolerance factors \(k_1\) and \(k_2\) are computed differently, as follows.

One-sided tolerance factor

The one-sided factor has an exact closed form based on the noncentral t distribution:

\[ k_1 = \frac{t_{\,n-1,\ C}(\delta)}{\sqrt{n}} \qquad \text{with noncentrality} \quad \delta = z_P\,\sqrt{n} \]

where:

  • \(z_P = \Phi^{-1}(P)\) is the \(P\)-quantile of the standard normal distribution, and
  • \(t_{\,n-1,\ C}(\delta)\) is the \(C\)-quantile of the noncentral \(t\) distribution with \(n-1\) degrees of freedom and noncentrality parameter \(\delta\).

The same \(k_1\) is used for the lower and upper one-sided bounds.

Two-sided tolerance factor

The two-sided factor \(k_2\) is computed exactly (not from the common Howe approximation): it is the value at which the probability that the interval \(\bar{x} \pm k_2\, s\) captures at least the proportion \(P\) of the population equals exactly the requested confidence \(C\):

\[ \Pr\!\left[\,\text{coverage} \geq P\,\right] \;=\; 2\int_{0}^{\infty} \left[\, 1 - F_{\chi^2_{n-1}}\!\left( \frac{(n-1)\;\chi^{2}_{1,\,P}\!\left(z^{2}/n\right)}{k_2^{\,2}} \right) \right]\varphi(z)\; dz \;=\; C \]

where:

  • \(\chi^{2}_{1,\,P}(\lambda)\) is the \(P\)-quantile of the noncentral chi-square distribution with 1 degree of freedom and noncentrality parameter \(\lambda\),
  • \(F_{\chi^2_{n-1}}\) is the CDF of the central chi-square distribution with \(n-1\) degrees of freedom, and
  • \(\varphi(z)\) is the standard normal density.

This equation has no closed-form solution for \(k_2\), so it is solved numerically with a root-finding algorithm. The search starts from the Krishnamoorthy–Mathew approximation:

\[ k_2 \approx \sqrt{\frac{(n-1)\;\chi^{2}_{1,\,P}\!\left(1/n\right)}{\chi^{2}_{\,n-1,\ \alpha}}} \]

where \(\chi^{2}_{\,n-1,\ \alpha}\) is the \(\alpha\)-quantile of the central chi-square distribution with \(n-1\) degrees of freedom.

Note

Because \(k_2\) is solved exactly, the two-sided normal interval attains the requested confidence \(C\) precisely, rather than approximately as with tabulated or approximated \(k\) factors.

References

  1. Krishnamoorthy, K. and Mathew, T. (2009). Statistical Tolerance Regions: Theory, Applications, and Computation. Wiley. https://onlinelibrary.wiley.com/doi/book/10.1002/9780470473900
  2. NIST/SEMATECH e-Handbook of Statistical Methods, "Tolerance intervals for a normal distribution." https://www.itl.nist.gov/div898/handbook/prc/section2/prc263.htm
  3. Hahn, G. J., Meeker, W. Q., and Escobar, L. A. (2017). Statistical Intervals: A Guide for Practitioners and Researchers, 2nd ed. Wiley.