"Is my data normally distributed?" is one of the most searched questions in statistics, and one of the most misunderstood. Many students run a normality test on every variable, panic when one is significant, and switch everything to nonparametric tests. Often that isn't necessary.
This guide explains what actually needs to be normal, how to check it with plots and tests, and what to do when the assumption is violated.
In this guide
- What actually needs to be normal?
- The central limit theorem: why sample size matters
- Method 1: Look at the data (always do this)
- Method 2: Skewness and kurtosis
- Method 3: Formal normality tests
- Worked example
- What to do if your data aren't normal
- How to report normality checks in APA 7
- Frequently asked questions
What actually needs to be normal?
Parametric tests do not require every raw variable to be normal:
| Analysis | What should be approximately normal |
|---|---|
| Independent t-test, one-way ANOVA | The outcome within each group (equivalently, the residuals) |
| Paired t-test | The differences between the paired scores |
| Regression | The residuals, not the predictors or the outcome |
| Pearson correlation (for significance tests) | Roughly bivariate normal; linearity matters more |
| Chi-square, Mann-Whitney, Kruskal-Wallis | Nothing; no normality assumption |
Predictors in a regression can be skewed, binary or anything else. Checking them for normality is a common wasted effort.
The central limit theorem: why sample size matters
t-tests, ANOVA and regression make inferences about means and coefficients, and by the central limit theorem the sampling distribution of a mean becomes approximately normal as the sample grows, even when the raw data are not. With roughly 30 or more observations per group, moderate skewness rarely causes trouble. Normality matters most in small samples, and there your tests have the least power to detect non-normality. That is why plots matter more than p-values.
Method 1: Look at the data (always do this)
- Histogram: roughly bell-shaped and symmetric? Watch for long tails, floor or ceiling effects and two peaks (which may mean two subgroups).
- Q-Q plot: plots your data's quantiles against those of a normal distribution. Points close to the diagonal line indicate normality. An S-shape suggests heavy or light tails; a curve suggests skew.
- Box plot: shows skew and outliers at a glance.
The Shapiro-Wilk calculator draws the histogram and Q-Q plot for you, and the descriptive statistics calculator and box plot maker are useful too.
Method 2: Skewness and kurtosis
- Skewness measures asymmetry (0 = symmetric; positive = long right tail).
- Excess kurtosis measures tail heaviness (0 = normal; positive = heavier tails).
Common rules of thumb:
| Approach | Rule |
|---|---|
| Absolute values (common in social science) | Skewness between −2 and +2 and kurtosis between −7 and +7 for SEM (West et al., 1995); stricter authors use ±1 |
| z-scores (statistic ÷ SE) | |z| < 1.96 in small samples (n < 50); |z| < 3.29 in medium samples (50–300) (Kim, 2013) |
| Large samples (n > 300) | Judge the absolute values and the plots, not the z-scores |
Method 3: Formal normality tests
Shapiro-Wilk test
The most powerful general-purpose normality test, especially for small to medium samples.
- H₀: the data come from a normal distribution.
- p > .05: no evidence against normality.
- p < .05: the data deviate significantly from normality.
Kolmogorov-Smirnov (Lilliefors) test
An older test with lower power than Shapiro-Wilk. If you use it, use the Lilliefors-corrected version (the calculator reports both).
The sample-size paradox
Formal tests are underpowered in small samples, where normality matters most, and overpowered in large samples, where it matters least. With n = 1,000, a trivial deviation that has no effect on a t-test will be "significant". So:
- Small samples: use the test together with the plots, and be cautious.
- Large samples: rely on plots and skewness and kurtosis; don't let a significant p-value alone push you to nonparametric tests.
Worked example
A researcher measures time on task (seconds) for 45 participants.
| Statistic | Value |
|---|---|
| Skewness | 1.42 (z = 4.01) |
| Excess kurtosis | 2.10 (z = 3.02) |
| Shapiro-Wilk | W = .89, p < .001 |
The histogram shows a long right tail, typical of time data. Both the test and the skewness z-score indicate clear non-normality, and n = 45 is too small to rely fully on the central limit theorem. Options: log-transform the times, or use a nonparametric test.
What to do if your data aren't normal
- Check for errors and outliers first. A typo (450 instead of 45) can create apparent non-normality. Use the outlier calculator.
- Consider whether it matters. With large, similar-sized groups, t-tests and ANOVA are robust. Unequal variances are a bigger problem; use Welch's tests.
- Transform the variable. Log, square-root or reciprocal transformations often fix right skew (time, income, counts). Interpret the results on the transformed scale.
- Use a nonparametric test:
| Parametric test | Nonparametric alternative |
|---|---|
| Independent t-test | Mann-Whitney U test |
| Paired t-test | Wilcoxon signed-rank test |
| One-way ANOVA | Kruskal-Wallis test |
| Repeated measures ANOVA | Friedman test |
| Pearson correlation | Spearman correlation |
- Use robust methods: bootstrapped confidence intervals or robust standard errors in regression.
- Use a model suited to the data: logistic regression for binary outcomes, Poisson or negative binomial regression for counts.
How to report normality checks in APA 7
Normality was assessed with Shapiro-Wilk tests and inspection of Q-Q plots. Time on task was positively skewed (skewness = 1.42, SE = 0.35; Shapiro-Wilk W = .89, p < .001), so it was log-transformed before analysis. After transformation, the distribution was approximately normal (W = .98, p = .62).
Frequently asked questions
Which normality test is best?
The Shapiro-Wilk test is generally the most powerful for small and medium samples. Always combine it with a histogram and a Q-Q plot.
What does a significant Shapiro-Wilk test mean?
A p-value below .05 means the data deviate significantly from a normal distribution. With large samples, even trivial deviations become significant, so check the plots before deciding what to do.
Do my independent variables need to be normally distributed?
No. In regression, only the residuals should be approximately normal. Predictors can be skewed or categorical.
What is an acceptable skewness and kurtosis?
Common guidelines treat skewness between −2 and +2 as acceptable, with kurtosis below about 7 for SEM. Stricter authors use ±1. In small samples, z-scores (statistic divided by its standard error) below 1.96 are often used.
Can I use a t-test if my data are not normal?
Often yes, if each group has about 30 or more observations, the groups are of similar size and there are no extreme outliers. For small skewed samples, use the Mann-Whitney U test or transform the data.