Testing Normality: How to Tell If Your Data Is Normal and What to Do If It Isn't

When the normality assumption matters and when it doesn't, Shapiro-Wilk and Kolmogorov-Smirnov, histograms and Q-Q plots in SPSS, skewness and kurtosis, and what to do when the assumption fails.

Part of the series: Statistical Analysis for a Seminar Paper or Thesis: The Complete Guide

t-tests, ANOVA, Pearson correlation and regression all assume normality. Supervisors ask about it and graders look for it in the results chapter. But many students check the wrong thing, or panic over a result that changes nothing.

What exactly has to be normal?

• t-test and ANOVA: the dependent variable, within each group separately.
• Paired t-test: the differences between the two measurements.
• Regression: the residuals, not the variables themselves.

A common mistake is testing the whole sample at once, or the independent variables.

When it matters less than you think

Thanks to the central limit theorem, with about 30+ participants per group t-tests and ANOVA are robust to moderate departures from normality. The assumption matters most in small samples. And in very large samples normality tests "fail" almost always, even for trivial deviations.

Three ways to check

1. Tests. In SPSS: Analyze → Descriptive Statistics → Explore; outcome in Dependent List, group in Factor List, and under Plots tick Normality plots with tests. - Shapiro-Wilk: preferred for samples up to a few hundred. - Kolmogorov-Smirnov: for large samples. - p > .05 = no evidence against normality; p < .05 = a departure (but see above on large samples).

2. Plots. A histogram (bell-shaped?) and a Q-Q plot (points close to the diagonal?). Experienced reviewers trust the plots as much as the test.

3. Skewness and kurtosis in the Descriptives table. A common rule of thumb: values between -2 and 2 (stricter: -1 to 1) suggest a roughly normal distribution.

What to do if the data isn't normal

1. Look for outliers. Sometimes one value (a typo, a confused participant) is the whole problem.
2. Large sample? You can usually keep the parametric test and note its robustness.
3. Small sample and a clear departure? Switch to a non-parametric test: Mann-Whitney for independent t, Wilcoxon for paired t, Kruskal-Wallis for ANOVA, Spearman for Pearson.
4. Transformation (e.g. a log for right-skewed data) - possible, but it complicates interpretation. Check with your supervisor.
5. Bootstrapping - available for most tests in recent SPSS versions, giving confidence intervals without the normality assumption.

How to report it

One sentence in the method section or at the start of the results:

"Normality was assessed with the Shapiro-Wilk test and skewness and kurtosis values. Anxiety scores were approximately normal in both groups (W = 0.97, p = .21; W = 0.96, p = .14), so t-tests were used."

If violated: "...response times were right-skewed (skewness = 2.84), so a Mann-Whitney test was used."

FAQ

Do I check normality before chi-square? No. Chi-square works on categorical variables and has no normality assumption.

Can a Likert questionnaire be normal? A single item, not really. The mean of several items is usually close enough.

Where does it fit? Step 5 of 7 - Statistical analysis for a seminar paper: the complete guide. Data looks strange and you're not sure what to do? Tell us.

Last updated: 2026-10-01