﻿ Choosing a normality test

# Choosing a normality test

Prism offers three normality tests.

We recommend the D'Agostino-Pearson normality test. It first computes the skewness and kurtosis to quantify how far the distribution is from Gaussian in terms of asymmetry and shape. It then calculates how far each of these values differs from the value expected with a Gaussian distribution, and computes a single P value from the sum of these discrepancies. It is a versatile and powerful normality test, and is recommended. Note that D'Agostino developed several normality tests. The one used by Prism is the "omnibus K2" test.

An alternative is the Shapiro-Wilk normality test. We prefer the D'Agostino-Pearson test for two reasons. One reason is that, while the Shapiro-Wilk test works very well if every value is unique, it does not work as well when several values are identical. The other reason is that the basis of the test is hard to understand.

Earlier versions of Prism offered only the Kolmogorov-Smirnov test. We still offer this test (for consistency) but no longer recommend it. It computes a P value from a single value: the largest discrepancy between the cumulative distribution of the data and a cumulative Gaussian distribution. This is not a very sensitive way to assess normality, and we now agree with this statement1: "The Kolmogorov-Smirnov test is only a historical curiosity. It should never be used."1

The Kolmogorov-Smirnov method as originally published assumes that you know the mean and SD of the overall population (perhaps from prior work). When analyzing data, you rarely know the overall population mean and SD. You only know the mean and SD of your sample. To compute the P value, therefore, Prism uses the Dallal and Wilkinson approximation to Lilliefors' method (2). Since that method is only accurate with small P values, Prism simply reports “P>0.10” for large P values. In case you encounter any discrepancies, you should know that we fixed a bug in this test many years ago in Prism 4.01 and 4.0b.

## Reference

1. RB D'Agostino, "Tests for Normal Distribution" in Goodness-Of-Fit Techniques edited by RB D'Agostino and MA Stephens, Macel Dekker, 1986.

2. Dallal GE and Wilkinson L (1986), "An Analytic Approximation to the Distribution of Lilliefors's Test Statistic for Normality," The American Statistician, 40, 294-296.