Home / Shared Math Details / Shared Estimators and Tests / Anderson-Darling Test

Anderson-Darling Test¶

This page defines the Anderson-Darling goodness-of-fit test for normality that Quantum XL uses. Several tools report its statistic and p-value as a normality check.

Notation¶

Term	Description
\(n\)	number of data values
\(x_{(1)} \le x_{(2)} \le \cdots \le x_{(n)}\)	the data sorted in ascending order
\(\bar{x}\)	sample mean
\(s\)	sample standard deviation (n − 1 denominator)
\(\Phi\)	standard normal cumulative distribution function
\(F(x)\)	fitted normal CDF, \(F(x) = \Phi\!\left(\dfrac{x - \bar{x}}{s}\right)\)
\(A^2\)	Anderson-Darling statistic
\(A^{*2}\)	finite-sample adjusted statistic

Test statistic¶

\[ A^2 = -n - \frac{1}{n}\sum_{i=1}^{n}(2i - 1)\left[\ln F\!\left(x_{(i)}\right) + \ln\!\left(1 - F\!\left(x_{(n+1-i)}\right)\right)\right] \]

\(F\) is the standard normal CDF applied to the standardized value. When Quantum XL tests a sample for normality, \(\bar{x}\) and \(s\) are the sample mean and sample standard deviation (n − 1 denominator). When it tests a fitted distribution (for example in distribution fitting), \(F\) uses the fitted parameters instead of re-estimating them.

Used by: the normality check reported by the tools listed below.

Finite-sample adjustment¶

The statistic is adjusted for sample size using the Stephens correction:

\[ A^{*2} = A^2\left(1 + \frac{0.75}{n} + \frac{2.25}{n^2}\right) \]

p-value¶

The p-value is a piecewise function of the adjusted statistic:

\[ p = \begin{cases} 0 & A^{*2} > 13 \\[4pt] \exp\!\left(1.2937 - 5.709\,A^{*2} + 0.0186\,(A^{*2})^2\right) & 0.6 \le A^{*2} \le 13 \\[4pt] \exp\!\left(0.9177 - 4.279\,A^{*2} - 1.38\,(A^{*2})^2\right) & 0.34 \le A^{*2} < 0.6 \\[4pt] 1 - \exp\!\left(-8.318 + 42.796\,A^{*2} - 59.938\,(A^{*2})^2\right) & 0.2 \le A^{*2} < 0.34 \\[4pt] 1 - \exp\!\left(-13.436 + 101.14\,A^{*2} - 223.73\,(A^{*2})^2\right) & A^{*2} < 0.2 \end{cases} \]

The reported p-value is clamped to a minimum of \(0.005\).

Used by¶

Summary Statistics: the normality-test section.
Johnson Transformation: goodness-of-fit selection.
Tolerance Interval: the advisory normality check.

It also underlies the standalone Normality Test, the distribution-fitting goodness-of-fit results, and the capability analyses, which do not have separate math pages.

References¶

Anderson, T. W., and Darling, D. A. (1954). A test of goodness of fit. Journal of the American Statistical Association, 49(268), 765-769.
Stephens, M. A. (1974). EDF statistics for goodness of fit and some comparisons. Journal of the American Statistical Association, 69(347), 730-737.
D'Agostino, R. B., and Stephens, M. A. (1986). Goodness-of-Fit Techniques. Marcel Dekker.