Goodness of Fit > Anderson-Darling Goodness of Fit

## What is the Anderson-Darling Test?

The Anderson-Darling Goodness of Fit Test (AD-Test) is a measure of **how well your data fits a specified distribution**. It’s commonly used as a test for normality.

## Performing the AD-Test by Hand

The hypotheses for the AD-test are:

H_{0}: The data comes from a specified distribution.

H_{1}: The data does not come from a specified distribution.

The formula is:

Where:

n = the sample size,

F(x) = CDF for the specified distribution,

i = the *i*th sample, calculated when the data is sorted in ascending order.

As you can probably see, the test statistic is cumbersome to calculate by hand. The general steps are:

Step 1: Calculate the AD Statistic for each distribution, using the formula above.

Step 2: Find the statistic’s p-value (probability value). The formula for the p-value depends on the value for the AD statistic from Step 1. The following formulas are taken from Agostino and Stephen’s Goodness of Fit Techniques.

AD statistic | P-Value Formula |

AD ≥ 0.60 | p = exp(1.2937 – 5.709(AD)+ 0.0186(AD)^{2} |

0.34 < AD* < .60 | p = exp(0.9177 – 4.279(AD) – 1.38(AD)^{2} |

0.20 < AD* < .34 | p = 1 – exp(-8.318 + 42.796(AD)- 59.938(AD)^{2} |

AD≤ 0.20 < AD* < .34 | p = 1 – exp(-13.436 + 101.14(AD)- 223.73(AD)^{2} |

Small p-values (less than your chosen alpha level) means that you can reject the null hypothesis. In other words, the data does not come from the named distribution.

**If you are comparing several distributions, choose the one that gives the largest p-value; this is the closest match to your data.**

## Using Technology

The steps are basically the same, except that software will do the legwork for you and calculate the AD statistic and the p-value. All you have to do is Step 2 above: compare your AD-test p-values to your alpha levels.

- SPSS: The test is not available in SPSS at the time of writing.
- R: use the ‘nortest’ package outlined here.

## References

D’Agostino, R. & Stephens, M. (1986). Goodness-of-fit-techniques (Statistics: a Series of Textbooks and Monographs, Vol. 68) 1st Edition. Dekker.