How to interpret a normality test
The Shapiro-Wilk test inverts the logic of the tests you are used to, and that is exactly where most people go wrong. The null hypothesis is "the data come from a normally distributed population". Therefore:
- p > .05 — normality was not rejected. You may proceed with parametric tests. Here, a high p value is the desirable outcome.
- p < .05 — normality was rejected. Use non-parametric tests, transform the variable, or invoke the central limit theorem if your sample is large.
Note that the test never "proves" the data are normal — it only indicates there is insufficient evidence to claim otherwise. That logical distinction is worth stating correctly in your write-up.
Why you should test within each group
If you have two groups with different means and you test the pooled sample, the resulting distribution will be bimodal and will fail Shapiro-Wilk even when each group is perfectly normal. The assumption behind the t-test is normality within each group, not of the combined sample. Use the "split by group" field in this tool.
The large-sample paradox
Above roughly n = 300, Shapiro-Wilk becomes hypersensitive: it detects minute departures with no practical consequence and rejects normality for data that are, for all working purposes, normal. In those cases, prioritise:
- The Q-Q plot — visual inspection is more informative than the p value.
- Skewness between −2 and +2 and kurtosis between −7 and +7, ranges widely accepted as indicating adequate approximate normality.
- The central limit theorem: with a large n, the sampling distribution of the mean approaches normality regardless of the underlying distribution, which makes the t-test robust.
Conversely, with a very small n (below 15) the test has little power and rarely rejects normality — which is not evidence that normality holds. Small samples favour the conservative choice of a non-parametric test.
What to do when normality is violated
1. Non-parametric tests. The most direct and safest route. Mann-Whitney replaces the independent-samples t-test, Wilcoxon the paired t-test, Kruskal-Wallis the ANOVA, and Spearman the Pearson correlation. The cost is a slight loss of statistical power and the need to report medians rather than means.
2. Transforming the variable. For positively skewed data, a log transformation often normalises the distribution. Square root works for count data; the reciprocal for severe skew. The drawback is interpretive: coefficients now live on the transformed scale, which complicates reporting.
3. Removing outliers. Legitimate only with documented technical justification — a data-entry error, equipment failure, a participant who did not meet inclusion criteria. Dropping points merely because they spoil the test is data manipulation and a research integrity problem.
4. Assuming robustness. With groups of similar size and a reasonable n, the t-test and ANOVA are notoriously robust to moderate departures from normality. This decision is defensible, but it requires an explicit justification and a citation.
Shapiro-Wilk or Kolmogorov-Smirnov?
Shapiro-Wilk, in the large majority of cases. It has greater statistical power to detect
departures from normality, particularly in small and medium samples, and it is the test recommended
by contemporary methodological literature. Kolmogorov-Smirnov with the Lilliefors correction
survives in some SPSS handbooks by tradition, but it is less sensitive. This tool implements
Royston's (1995) AS R94 algorithm — the same one behind R's shapiro.test.
Frequently asked questions
Do skewness and kurtosis replace Shapiro-Wilk?
They complement it, they do not replace it. They are especially useful in large samples, where the p value loses practical value. Zero skewness indicates perfect symmetry; zero kurtosis (in the excess definition used here) indicates the same peakedness as the normal distribution.
Do I need to test normality for linear regression?
Yes, but of the residuals, not of the original variables. This is a very common mistake. Fit the regression first, then test the normality of the residuals it produces.
How do I cite this test in my paper?
Cite the source of the method, not the tool: Shapiro, S. S., & Wilk, M. B. (1965). An analysis of variance test for normality (complete samples). Biometrika, 52(3–4), 591–611. The text generated above already follows the expected wording.
Once you have settled on an approach, move on to the statistics calculator, which applies the appropriate test automatically, or read the guide on which statistical test to use.