The Assumption of Normality: How to Test It and What to Do If It Fails

April 18, 2026 4 min read By Angel Reyes

The assumption of normality is the one your committee will ask about first. It's the most discussed, most tested, and most frequently violated assumption in parametric statistics. But here's the good news: it's rarely as catastrophic as students fear.

What the Assumption Actually Requires

First, let's clarify what needs to be normal. It depends on the test:

  • t-tests and ANOVA: The dependent variable should be approximately normally distributed within each group. With large samples, it's actually the sampling distribution of the mean that needs to be normal — and the Central Limit Theorem handles that for you.
  • Regression: The residuals (prediction errors) should be approximately normal, not the raw variables themselves.
  • Correlation: Both variables should be roughly bivariate normal.

This distinction matters. Many students test the wrong thing — checking whether the entire dataset is normal rather than checking within groups or examining residuals.

How to Test Normality

Visual Methods (Do These First)

Histograms: Create a histogram of your variable (or residuals). Does it look roughly bell-shaped? Small departures from perfection are fine. You're looking for severe skewness or obvious bimodality.

Q-Q Plots (Quantile-Quantile Plots): These plot your data's quantiles against what you'd expect from a normal distribution. If the points fall roughly along the diagonal line, your data is approximately normal. Points that curve away from the line at the tails indicate skewness or heavy tails.

Statistical Tests

Shapiro-Wilk Test: The most commonly recommended formal test. If p > .05, you fail to reject the null hypothesis of normality (your data is "normal enough"). If p < .05, the data significantly departs from normality.

In SPSS: Analyze → Descriptive Statistics → Explore → Plots → check "Normality plots with tests"

Skewness and Kurtosis Values

Check the skewness and kurtosis statistics in your descriptive output. As a rule of thumb:

  • Skewness between -1 and +1 is generally acceptable
  • Kurtosis between -2 and +2 is generally acceptable

Some sources use stricter or more lenient thresholds — check your methodology textbook.

The Big Caveat About Shapiro-Wilk

Here's something most textbooks underemphasize: with large samples (n > 100), the Shapiro-Wilk test will flag trivial departures from normality as significant. The test becomes too sensitive. Your histogram might look perfectly fine, but Shapiro-Wilk says p = .003.

This is why visual methods matter. If your histogram and Q-Q plot look reasonable and your sample is large, a significant Shapiro-Wilk test doesn't necessarily mean you have a problem. Many methodologists prioritize visual inspection over formal tests for this reason.

What to Do When Normality Fails

If your data is genuinely non-normal, you have several options. See our comprehensive guide on handling violated assumptions for the full treatment. Here's a summary:

Option 1: Proceed Anyway (Often Legitimate)

Many parametric tests are robust to normality violations, especially with:

  • Sample sizes of 30+ per group
  • Roughly equal group sizes
  • Mild-to-moderate skewness

The Central Limit Theorem means that with sufficient sample sizes, the sampling distribution of the mean approaches normality regardless of the population distribution.

Option 2: Transform Your Data

Common transformations:

  • Log transformation: For right-skewed data
  • Square root transformation: For moderately skewed data or count data
  • Reflect and transform: For left-skewed data (reverse the scores, then transform)

Option 3: Use Non-Parametric Tests

Non-parametric alternatives don't assume normality:

  • Mann-Whitney U instead of independent t-test
  • Wilcoxon Signed-Rank instead of paired t-test
  • Kruskal-Wallis instead of one-way ANOVA

Option 4: Bootstrapping

Bootstrapping uses resampling to generate confidence intervals and p-values without assuming normality. It's available in SPSS (look for the Bootstrap button in analysis dialogs) and is straightforward in R.

How to Report Normality Testing

In your results chapter, include a brief statement:

"The assumption of normality was assessed using Shapiro-Wilk tests and visual inspection of Q-Q plots. Test scores were approximately normally distributed for both the control group (W = 0.98, p = .42) and treatment group (W = 0.97, p = .31)."

If normality was violated:

"Shapiro-Wilk tests indicated that reading scores were significantly non-normal for the treatment group (W = 0.91, p = .003). However, visual inspection of the histogram revealed only mild negative skewness (skewness = -0.72), and given the sample size (n = 65), the independent samples t-test was deemed robust to this violation (Norman, 2010)."

The Bottom Line

Test normality, document what you find, and take appropriate action. But don't panic over minor violations in reasonably sized samples. The assumption of normality is important, but it's also forgiving — especially when your sample is large enough to let the Central Limit Theorem work in your favor.