The Shapiro-Wilk normality test is a statistical test that determines whether a dataset follows a normal distribution. It is one of the most powerful and commonly used normality tests, especially for small to moderately sized samples.
The Shapiro-Wilk test compares the distribution of your sample data to a perfect normal distribution with the same mean and standard deviation. It produces a W statistic and a corresponding p-value, which help you assess whether the deviations from normality are statistically significant.
When to Use the Shapiro-Wilk Normality Test
- You want to test whether your data follow a normal distribution.
- Your sample size is relatively small (commonly used for n < 50, but valid up to around 2,000 observations).
- You’re planning to use statistical methods that assume normality, such as t-tests or ANOVA.
To perform a Shapiro-Wilk normality test, use my free online Normality Test Calculator!
Interpreting the Results
The null hypothesis of the Shapiro-Wilk normality test is that the data are a random sample that was drawn population that follows a normal distribution.
- High p-value (p > 0.05): Fail to reject the null hypothesis. The data do not significantly deviate from normality.
- Low p-value (p ≤ 0.05): Reject the null hypothesis. The data significantly deviate from normality.
Like all hypothesis tests, the Shapiro-Wilk test is sensitive to sample size. In small samples, it can lack statistical power and miss true deviations from normality. In large samples, even small departures from normality can produce statistically significant results. Always pair the test with visual tools like Q-Q plots and histograms.
The graph below displays the Shapiro-Wilk normality test p-values along with their histograms and Q-Q plots for a normal (left) and non-normal (right) sample.
Shapiro-Wilk Test Example
A researcher collects the test scores of 20 students and wants to check if they follow a normal distribution before performing a t-test.
They run the Shapiro-Wilk normality test and obtain a p-value of 0.42. Because the p-value is greater than 0.05, they conclude that the data are consistent with a normal distribution and proceed with the parametric analysis.
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