Normality Test

A normality test is a statistical procedure that determines whether a random sample was drawn from a normally distributed population. Many statistical methods assume that the underlying population follows a normal distribution. A normality test helps verify whether that assumption is reasonable based on the sample data.

The test evaluates two competing hypotheses:

• Null hypothesis (H₀): The population from which the sample was drawn follows a normal distribution.
• Alternative hypothesis (H₁): The population does not follow a normal distribution.

Unlike most statistical tests, a high p-value is desirable here. A high p-value means that the sample data are consistent with normality. You do not have evidence to reject the null hypothesis. A low p-value indicates you must reject the null hypothesis and conclude the population is not normally distributed.

However, interpret the results cautiously with both small and very large samples. When sample sizes are small, even clearly non-normal data can yield high p-values because the test lacks power to detect deviations from normality. Additionally, very large samples can cause normality tests to flag trivial departures from normality as being statistically significant. In both cases, it’s helpful to combine normality tests with visual tools like Q-Q plots.

Several types of normality tests are commonly used, including:

• Shapiro–Wilk test
• Anderson–Darling test
• Kolmogorov–Smirnov test
• Lilliefors test
• Jarque–Bera test

For example, an analyst tests whether the distribution of reaction times in a sample of 100 participants reflects a normally distributed population. The Shapiro–Wilk test produces a p-value of 0.42. Because the p-value is high, the analyst does not reject the null hypothesis. This result suggests the population of reaction times likely follows a normal distribution.