---
title: "Anderson-Darling (A-D) Statistic"
canonical: "https://modelassist.epixanalytics.com/space/EA/26575335/Anderson-Darling%20(A-D)%20Statistic"
format: markdown
---
The A-D statistic *A**2**n* is defined as:
> Macro (mathblock)
where
> Macro (mathinline)
*n* = total number of data points
*F(x)* = distribution function of the fitted distribution
*f(x)* = density function of the fitted distribution
*F**n**(x)* = *i*/*n*
*i* = the cumulative rank of the data point
The Anderson-Darling statistic is a sophisticated version of the [Kolmogorov-Smirnoff statistic](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575334/). It is more powerful for the following reasons:
1. *Y(x)* compensates for the [increased variance](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575334/) of the vertical distances between distributions' *s**K-S*2
2. *f(x)* weights the observed distances by the probability that a value will be generated at that *x*-values
3. The vertical distances are integrated over *all* values of *x* to make maximum use of the observed data (the K-S statistic only looks at the maximum vertical distance).
The A-D statistic is therefore a generally more useful measure of fit than the K-S statistic, especially where it is important to place equal emphasis on fitting a distribution at the tails as well as the main body. On the other hand, it requires a lot more number-crunching and is not therefore very practical to calculate manually. Crystal Ball's distribution fitting feature performs an Anderson-Darling test automatically.
An alternative equation for the Anderson-Darling statistic is:
> Macro (mathblock)
The principle behind the Anderson-Darling statistic is that one is fitting a distribution with known parameter values to the data. When the parameters are being estimated from the data a [correction](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575332/) needs to be applied, especially for small samples.
---