To check the normality of your data, combine a visual method (a histogram and a Q-Q plot) with a formal test (the Shapiro-Wilk test for samples under about 50, or looking at skewness and kurtosis for larger samples) — if the Shapiro-Wilk p-value is above .05 and the Q-Q plot points fall roughly along the diagonal line, your data are approximately normal. If you're staring at your dataset unsure whether you can run a t-test or ANOVA, this is the exact gate you need to clear first — because those tests assume your data (or your residuals) follow a normal distribution.
Key Takeaways
- Normality means your data follow a bell-shaped distribution — checking it tells you whether to use a parametric test (t-test, ANOVA) or a non-parametric alternative (Mann-Whitney U, Kruskal-Wallis).
- Always use both a visual check and a formal test. A histogram plus a Q-Q plot catches problems that a single number can miss.
- The Shapiro-Wilk test is the most powerful normality test for small-to-moderate samples (roughly n < 50), where a p > .05 suggests your data do not significantly deviate from normal.
- In large samples the Shapiro-Wilk test flags trivial deviations, so for n > 300 rely on skewness/kurtosis (both roughly between −2 and +2) and Q-Q plots instead.
- Many tests only require the residuals to be normal, not the raw data — a common source of confusion for beginners.
What does "normality" actually mean?
Normality means your data are distributed in a symmetric, bell-shaped curve where most values cluster around the mean and taper off evenly on both sides. It's the classic "normal distribution" (or Gaussian distribution) you've seen drawn as a smooth hill.
You check for it because the most common statistical tests — the independent-samples t-test, paired t-test, one-way ANOVA, and Pearson correlation — assume normality. If your data are heavily skewed or have extreme outliers, these tests can give you a misleadingly small (or large) p-value, and your conclusions become unreliable.
One important nuance: for t-tests and ANOVA, the assumption is really about the residuals (the leftover error after accounting for group differences), not the raw variable. For a simple one-sample comparison, checking the variable itself is fine.
What are the three ways to check normality?
There are three practical approaches, and the strongest strategy uses all three together.
1. The histogram (eyeball the shape)
Plot a histogram of your variable. You're looking for a roughly symmetric hump in the middle. A long tail to the right means positive skew (common with reaction times, income, or symptom counts); a long tail to the left means negative skew. Histograms are quick but subjective — a small sample can look lumpy even when the underlying data are normal.
2. The Q-Q plot (the most reliable visual)
A quantile-quantile (Q-Q) plot compares your data's quantiles against the quantiles expected from a perfect normal distribution. If your data are normal, the points fall along a straight diagonal line. Systematic curving at the ends signals skew; an S-shape signals heavy or light tails. Most statisticians trust the Q-Q plot more than the histogram because deviations are easier to spot.
3. Formal normality tests (get a number)
The two most common formal tests are:
- Shapiro-Wilk test — the most powerful and widely recommended for samples up to about 2,000, and especially strong under n = 50.
- Kolmogorov-Smirnov test — older and generally less powerful than Shapiro-Wilk; still reported in some fields but no longer the first choice.
For both, the null hypothesis is that your data are normal. So a p-value above .05 is the "good" result — it means you have no significant evidence of non-normality.
How do I interpret the Shapiro-Wilk test?
A Shapiro-Wilk p-value greater than .05 means your data do not significantly differ from a normal distribution, so you can proceed with parametric tests; a p-value of .05 or below means the data deviate significantly from normal. Report it as W (the test statistic) and its p-value.
The counterintuitive part: because the test's power grows with sample size, a large n will flag even microscopic, harmless deviations as "significant." This is why you should never rely on the p-value alone in big datasets — pair it with the Q-Q plot and skewness/kurtosis.
Worked example: checking normality with real numbers
Suppose you collected anxiety scores from 30 participants and want to run an independent-samples t-test. First, check normality.
Step 1 — Histogram: The distribution shows a slight right tail but a clear central hump.
Step 2 — Skewness and kurtosis: You calculate skewness = 0.42 and kurtosis = −0.31. Both fall comfortably within the ±2 rule of thumb, suggesting acceptable normality.
Step 3 — Shapiro-Wilk: The test returns W = .965, p = .41. Because p > .05, you fail to reject the null — there's no significant departure from normality.
Step 4 — Q-Q plot: The points hug the diagonal line with only minor scatter at the tails.
Conclusion: All four checks agree. Your data are approximately normal, so the independent-samples t-test is appropriate. In APA 7 you'd write:
A Shapiro-Wilk test indicated that anxiety scores did not deviate significantly from normality, W = .965, p = .41.
If the test had instead returned W = .88, p = .003 with a curved Q-Q plot, you'd switch to the non-parametric Mann-Whitney U test instead — see our guide on Mann-Whitney vs the t-test to decide.
Which normality check should I use for my sample size?
| Sample size | Best formal test | Best visual | Watch out for |
|---|---|---|---|
| n < 50 | Shapiro-Wilk | Q-Q plot | Small samples look noisy even when normal |
| 50 ≤ n ≤ 300 | Shapiro-Wilk | Q-Q plot + histogram | Balanced — trust the combination |
| n > 300 | Skewness/kurtosis (±2) | Q-Q plot | Shapiro-Wilk over-rejects trivial deviations |
The core principle: the smaller your sample, the more you lean on the formal test; the larger your sample, the more you trust the visuals and effect-size-style measures like skewness.
What if my data are not normal?
You have three good options, in rough order of preference:
- Use a non-parametric test. The Mann-Whitney U replaces the independent t-test; the Wilcoxon signed-rank replaces the paired t-test; the Kruskal-Wallis replaces one-way ANOVA. These make no normality assumption.
- Transform the variable. A log or square-root transform can pull a right-skewed distribution back toward normal — useful for reaction times and income.
- Rely on the Central Limit Theorem. With large samples (often n > 30 per group), the sampling distribution of the mean is approximately normal even when the raw data aren't, so a t-test remains fairly robust.
Manually running Shapiro-Wilk, generating Q-Q plots, computing skewness, and then choosing the right follow-up test is exactly the fiddly workflow that trips up non-statisticians. StatRyx checks normality automatically the moment you upload your data, flags violations in plain English, and recommends the correct parametric or non-parametric test — then