A normal distribution is a symmetrical, bell-shaped pattern where most of your data clusters around the average, and values become rarer the further they fall from that centre. If you've ever been told your data "should be normal" before running a t-test or ANOVA — and had no idea what that meant or how to check it — this guide is for you.
Key Takeaways
- A normal distribution (or "bell curve") is symmetrical: the mean, median, and mode all sit at the same central point, and the two halves mirror each other.
- The 68-95-99.7 rule says roughly 68% of values fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.
- Normality matters because many common tests — t-tests, ANOVA, Pearson correlation, linear regression — assume your data (or the model's residuals) are approximately normal.
- Real data rarely fits perfectly; "approximately normal" is usually good enough, and mild deviations often don't invalidate your results with a decent sample size.
- You can check normality with a histogram, a Q-Q plot, or a Shapiro-Wilk test — StatRyx runs all three automatically and tells you in plain English whether your assumption holds.
What does a normal distribution actually look like?
Picture the heights of 1,000 adult women. A few are very short, a few are very tall, but the vast majority cluster around the average — say, 163 cm. If you plotted every height as a bar chart, the bars would rise smoothly to a peak in the middle and taper off symmetrically on both sides. That smooth, symmetrical hump is the bell curve, and it's the visual signature of a normal distribution.
Three things define it:
- It's symmetrical. The left half is a mirror image of the right half.
- The centre is the average. The mean, median, and mode all land at the same point — the top of the bell.
- The spread is measured by standard deviation. A small standard deviation makes a tall, narrow curve; a large one makes a short, wide curve.
The normal distribution is so common because of a principle called the central limit theorem: when a value is influenced by many small, independent factors (genetics, diet, environment for height), those influences tend to average out into a bell shape.
What is the 68-95-99.7 rule?
The 68-95-99.7 rule (the "empirical rule") states that in a normal distribution, about 68% of values fall within 1 standard deviation of the mean, about 95% within 2 standard deviations, and about 99.7% within 3. This is the single most useful fact about the normal distribution because it lets you translate a raw score into "how unusual is this?"
Here's the intuition with real numbers. Say adult women's heights are normally distributed with a mean of 163 cm and a standard deviation of 7 cm:
- 68% of women are between 156 cm and 170 cm (163 ± 7).
- 95% are between 149 cm and 177 cm (163 ± 14).
- 99.7% are between 142 cm and 184 cm (163 ± 21).
So a woman who is 184 cm tall sits at the far edge of that 99.7% — genuinely rare. This is exactly how statistical significance works under the hood: a result that would only happen 5% of the time (roughly 2 standard deviations out) is what we usually flag as "significant."
Why does the normal distribution matter for my analysis?
The normal distribution matters because many of the most-used statistical tests assume your data — or the residuals from your model — are approximately normal, and violating that assumption can distort your p-values. If you run the wrong test on non-normal data, you risk reporting a "significant" finding that isn't real, or missing a real one.
Tests that carry a normality assumption include:
- Independent and paired t-tests
- One-way and factorial ANOVA
- Pearson correlation
- Linear regression (specifically, the residuals should be normal)
When your data clearly isn't normal — heavily skewed income data, or a small sample with outliers — you switch to a non-parametric test that makes no normality assumption, such as the Mann-Whitney U test instead of the t-test. If you're weighing that choice, see our guide on Mann-Whitney vs the t-test.
How do I check if my data is normal? (Worked example)
You check normality three ways: look at a histogram, inspect a Q-Q plot, and run a formal test like Shapiro-Wilk. No single method is definitive, so use them together.
Imagine you've collected reaction times (in milliseconds) from 40 participants in a cognition study. Here's the step-by-step check:
Step 1 — Histogram. You plot the 40 reaction times. The bars form a roughly symmetrical hump peaking around 520 ms, with no long tail dragging to one side. Good sign.
Step 2 — Q-Q plot. A Q-Q plot compares your data against a perfect normal distribution. If the points fall close to the diagonal reference line, your data is approximately normal. Your points hug the line with only slight wobble at the extremes — still fine.
Step 3 — Shapiro-Wilk test. This is a formal hypothesis test where the null hypothesis is that your data IS normal. You get W = .97, p = .38. Because p is well above .05, you fail to reject the null — meaning there's no evidence your data departs from normality. You're clear to run a parametric test.
The counter-intuitive part trips people up: with normality tests, you want a non-significant result (p > .05). A significant Shapiro-Wilk (p < .05) is bad news — it says your data is not normal.
What if my data isn't normal?
If your data isn't normal, you have three main options: transform it, use a non-parametric test, or rely on the central limit theorem if your sample is large. Skewed data can sometimes be normalised with a log or square-root transformation. Alternatively, swap to a distribution-free test (Mann-Whitney U, Kruskal-Wallis, Spearman correlation). And with samples above roughly n = 30 per group, the central limit theorem means t-tests and ANOVA are often robust to mild non-normality anyway.
Normal vs. skewed distributions: key differences
| Feature | Normal Distribution | Skewed Distribution |
|---|---|---|
| Shape | Symmetrical bell | Lopsided, one long tail |
| Mean vs. median | Equal | Differ (mean pulled toward the tail) |
| Common examples | Height, IQ, measurement error | Income, reaction times, house prices |
| Best-suited tests | t-test, ANOVA, Pearson | Mann-Whitney, Kruskal-Wallis, Spearman |
| Shapiro-Wilk result | p > .05 (non-significant) | p < .05 (significant) |
How do I report normality in APA 7 style?
In APA 7, report the Shapiro-Wilk statistic with its degrees of freedom and p-value, italicising the test statistic and p. For the example above:
A Shapiro-Wilk test indicated that reaction times did not deviate significantly from normality, W = .97, p = .38.
Note the APA conventions: the test statistic W is italicised, p is italicised, and the leading zero is dropped from both (.97 and .38, not 0.97). If you'd reported a mean, you'd add it with its standard deviation, e.g., M = 521.4, SD = 48.2.
Checking normality by hand — building histograms, generating Q-Q plots, and interpreting Shapiro-Wilk — is exactly the kind of repetitive setup that eats up thesis time. StatRyx runs all