Tests for Normality

There are many types of tests for normality.

Tests for normality are part of a broader category of tests for Goodness of Fit.

Tests for normality are often used alongside a normal probability plot or a Q-Q plot.

General Information

HYPOTHESIS

The null hypothesis is (generally) that the data comes from a population that is normally distributed. Based on the p-value from a test for normality, we either reject or fail to reject the null hypothesis.

STATISTICAL POWER

Tests for normality can be statistically underpowered, so just because a test fails to reject the null hypothesis, this does not guarantee that the data comes from a population that is normally distributed.

Types of Normality Tests

Normality tests include:

• Shapiro-Wilk

• Anderson-Darling

• Lillifors

• Kolmogorov-Smirnov (K-S)

• Cramer von Mises

• Basic checks for skewness and kurtosis (3rd and 4th moments of the normal distribution, respectively)

What to do if a software package reports more than one measure of normality?

• First, it is best to consider using only one statistical method. When more than one methods are applied for the same analysis goal, then it is possible to inflate the Type 1 error.

• However, some software packages calculate and report many similar measures without being asked to do so. If this is encountered, consider what each specific statistical method is designed to do. For assessing normality, the various methods check different aspects of the data.

• For checking normality, these statistical tests tend to be statistically underpowered (meaning that the data may come from a population that is not normally distributed, but that the test missed it due to sampling error). If the p-value of at least one test statistic (K-S, Lillifors, etc.) is less than alpha, then we should consider rejecting H0.

• We should also consider the robustness of the conclusions against the assumption of normality. A non-parametric method might be used to double-check the conclusions if the assumption of normality is rejected.

When is normality important?

Normality and Transformations: A few thoughts from Jeremy Anglim's Blog: Psychology and Statistics

What about data transformations?

Notes on the use of data transformations by Jason W. Osborne, Ph.D North Carolina State University

Box-Cox transform

Transforms on a regression response variable

Regression variance stabilizing transforms