Normality Test

Q: What should I do if my data are not normal?

Options include: using a nonparametric test, transforming the data (log, square root), or invoking the Central Limit Theorem if n > 30 per group. StatRyx recommends the best course of action based on your sample size and degree of non-normality.

Verify whether your data follow a normal distribution before applying parametric statistical tests.

Normality testing assesses whether a variable's distribution is consistent with the normal (Gaussian) distribution — a key assumption underlying t-tests, ANOVA, Pearson correlation, linear regression, and many other parametric tests. While parametric tests are robust to mild non-normality with large samples, normality testing is important for small samples (n < 30) where assumption violations can distort p-values and confidence intervals.

StatRyx applies the Shapiro-Wilk test (most powerful for n < 50), the Kolmogorov-Smirnov test, and computes skewness and excess kurtosis with their standard errors. Visual diagnostics include a histogram with fitted normal curve and a Q-Q plot.

Important: a non-significant normality test does not prove normality; it only indicates that the data are consistent with normality given the sample size. With very large samples, even trivial departures from normality will be significant. StatRyx provides guidance on interpreting results in context.

A normality test asks whether your data could plausibly come from a normal distribution: p > .05 means no significant departure, p < .05 means a significant one. Sample size changes what that means. In small samples the tests have little power and miss real departures; in large samples they flag deviations too small to matter. That is why normality is best judged from the test together with a Q-Q plot, a histogram, and skewness and kurtosis.

When to use the Normality Test

Before applying a parametric test (t-test, ANOVA, correlation) to check whether normality holds
When a reviewer or supervisor requires formal normality testing alongside parametric results
When your sample is small (n < 30–50) and you are uncertain whether normality can be assumed
You must justify the choice between a parametric test and its nonparametric alternative in your methods section
You want to check the residuals of a regression or ANOVA model, which is what the normality assumption is actually about

When not to use it

Your sample is very large (n > 200) — all normality tests will flag trivial deviations — use Rely on histograms and Q-Q plots as visual diagnostics instead instead
Your data are on an ordinal scale — use Use nonparametric tests regardless of normality instead
You want to test the assumption for a t-test or ANOVA in each group separately with tiny groups — use Inspect Q-Q plots per group, or use a nonparametric test instead

Worked example

Test	Statistic	p	Skewness	Kurtosis
Shapiro-Wilk	W = .978	.342	0.23	−0.41
Kolmogorov-Smirnov	D = .052	.420

How to report it (APA 7)

A Shapiro-Wilk test indicated that anxiety scores were approximately normally distributed, W = .978, p = .342. Skewness (0.23, SE = 0.31) and excess kurtosis (−0.41, SE = 0.62) were within acceptable limits. The Q-Q plot confirmed approximate normality.

Frequently asked questions

Is the normality test free on StatRyx?

Yes. You can run it without an account (3 free analyses), and a free account adds 10 more — a one-time allowance, no payment needed. Every result includes the APA 7 write-up. Paid plans (Student from $9/month) add more analyses and Word, PDF and Excel export.

Which normality test is best?

Shapiro-Wilk is the most powerful test for small-to-medium samples (n < 50) and is recommended by most authorities. Kolmogorov-Smirnov is less powerful but can handle larger samples.

What should I do if my data are not normal?

Options include: using a nonparametric test, transforming the data (log, square root), or invoking the Central Limit Theorem if n > 30 per group. StatRyx recommends the best course of action based on your sample size and degree of non-normality.

How do I report a Shapiro-Wilk test in APA 7?

Give W, the degrees of freedom (n) and the p-value: "A Shapiro-Wilk test showed that scores did not deviate significantly from normality, W(48) = .978, p = .342." Add skewness and kurtosis if a reviewer asks.

Shapiro-Wilk or Kolmogorov-Smirnov?

Shapiro-Wilk has more power at most sample sizes and is the usual recommendation. The Kolmogorov-Smirnov test with the Lilliefors correction is an alternative for large samples. Without that correction, Kolmogorov-Smirnov is too lenient when the mean and SD are estimated from the data.

Run a Normality Test on your data →