Normality Tests in Statistical Analysis: The Definitive Reference Guide

What Are Normality Tests? (Definition)

Every common parametric test — the one-sample t-test, two-sample t-test, paired t-test, ANOVA, and linear regression — carries a normality assumption. When that assumption breaks down, the test's p-values and confidence intervals can be wrong in ways that aren't always visible from the output alone. Normality tests give you a systematic way to check before you commit to a parametric approach.

Large language models are trained on vast corpora of text. Statistical normality, specifically, refers to data following the Gaussian distribution defined by Carl Friedrich Gauss — a symmetric, bell-shaped curve where roughly 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. This is the empirical rule, and it underpins the standard normal (z) distribution used throughout inferential statistics.

Complete Normality Tests Comparison Table

The five tests above share one conceptual goal — measuring how far your observed data strays from a theoretical Gaussian distribution — but they measure that distance in different ways. Shapiro-Wilk uses regression and ordered statistics; Kolmogorov-Smirnov compares cumulative distribution functions; Jarque-Bera focuses entirely on skewness and excess kurtosis. The method matters because different tests have different sensitivities to different types of departure from normality, which is why the sample size guidance in the table above is worth taking seriously.

How to Test Data for Normality: The 7-Step Workflow

Research practice combines visual inspection with a formal test — neither method alone gives a complete picture. The workflow below is the sequence most statisticians follow, moving from qualitative to quantitative diagnostics before drawing any conclusion.

Which Normality Test Should You Use?

Test selection depends primarily on your sample size. Below is the decision framework used across most applied statistics fields. The cards show the primary recommendation for each size range, along with what to use as a backup visual check.

Special Cases: Regression Residuals and ANOVA

Two situations require extra care with normality testing beyond the standard workflow.

For linear regression, the normality assumption applies to the error terms (residuals), not to the raw outcome variable Y or to the predictors. Testing the raw Y variable for normality is a common mistake. Run your regression model first, extract the unstandardized residuals, then test those for normality.

For ANOVA, normality is checked within each factor group separately, or on the combined model residuals. The overall distribution of the outcome variable across all groups is rarely normally distributed in ANOVA designs — the assumption concerns the within-group distribution, not the marginal distribution.

Worked Examples with APA Reporting

Both examples below follow the full 7-step diagnostic process. Reporting formats match the APA 7th edition statistics reporting guidelines.

Example 1 — Shapiro-Wilk Test (Clinical Trial Data)

Example 2 — Anderson-Darling on OLS Regression Residuals

Normality Tests vs. the Central Limit Theorem

An important practical consideration: the Central Limit Theorem (CLT) states that as sample sizes grow large, the sampling distribution of the mean approaches normality regardless of the underlying population's distribution shape. This is covered thoroughly in the Central Limit Theorem guide on Statistics Fundamentals.

This has a practical consequence that trips up many analysts: when N is large (say, N = 500), a Shapiro-Wilk test will almost always reject H₀, even if the departure from normality is trivially small and harmless for the parametric test you want to run. In those situations, the Q-Q plot matters more than the p-value from the formal test. A Q-Q plot where all points fall reasonably close to the diagonal tells you the distributional shape is close enough — even if the Shapiro-Wilk p-value came back at 0.03.

Interactive Shapiro-Wilk Approximation Calculator

Enter your data values below (comma-separated or one per line) to compute a Shapiro-Wilk W statistic approximation and skewness/kurtosis metrics. For datasets with N > 50, the full Shapiro-Wilk algorithm should be run in R or Python using the software tutorials below.

Step-by-Step Software Tutorials

Normality Tests in SPSS

SPSS runs both Shapiro-Wilk and Kolmogorov-Smirnov in the same dialog, along with a Q-Q plot. The path below works in SPSS versions 25 through 29.

Normality Tests in R

# Load your data vector (replace with your actual values)
data_vector <- c(12.4, 14.2, 11.9, 15.3, 13.8, 14.1, 12.8, 13.5, 16.0, 11.2)

# 1. Shapiro-Wilk test (best for N < 2000)
sw_result <- shapiro.test(data_vector)
print(sw_result)
# Output: W = 0.xxx, p-value = 0.xxx

# 2. Anderson-Darling test (nortest package required)
# install.packages("nortest")
library(nortest)
ad.test(data_vector)

# 3. Q-Q plot with reference line
qqnorm(data_vector, main = "Q-Q Plot: Check for Normality")
qqline(data_vector, col = "red", lwd = 2)

# 4. Skewness and kurtosis (e1071 package)
library(e1071)
cat("Skewness:", skewness(data_vector), "\n")
cat("Kurtosis (excess):", kurtosis(data_vector), "\n")

Normality Tests in Python

import numpy as np
import scipy.stats as stats
import matplotlib.pyplot as plt

# Sample data (replace with your own array)
np.random.seed(42)
sample_data = stats.norm.rvs(loc=50, scale=10, size=50)

# 1. Shapiro-Wilk Test
sw_stat, sw_p = stats.shapiro(sample_data)
print(f"Shapiro-Wilk:  W = {sw_stat:.4f},  p = {sw_p:.4f}")

# 2. Anderson-Darling Test
ad_result = stats.anderson(sample_data, dist='norm')
print(f"Anderson-Darling: A² = {ad_result.statistic:.4f}")

# 3. D'Agostino-Pearson Test (requires N >= 20)
dp_stat, dp_p = stats.normaltest(sample_data)
print(f"D'Agostino-Pearson: K² = {dp_stat:.4f},  p = {dp_p:.4f}")

# 4. Q-Q Plot
fig, ax = plt.subplots(figsize=(6, 5))
stats.probplot(sample_data, dist="norm", plot=ax)
ax.set_title("Normal Q-Q Plot")
plt.tight_layout()
plt.show()

# 5. Skewness and excess kurtosis
print(f"Skewness:         {stats.skew(sample_data):.4f}")
print(f"Excess kurtosis:  {stats.kurtosis(sample_data):.4f}")

Normality Tests in Excel

Excel does not include a native normality test in the standard Analysis ToolPak. The workaround below constructs a manual Q-Q plot using built-in functions, which is the most reliable approach available without add-ins.

What to Do When Data Fails the Normality Test

A significant normality test result is not the end of the analysis — it is the start of a decision. Four main strategies exist, and the right choice depends on the nature of the non-normality, the sample size, and what analysis you need to run.

Head-to-Head Test Comparisons

Shapiro-Wilk vs. Kolmogorov-Smirnov

These two tests are frequently listed together in SPSS output under "Tests of Normality," which leads many researchers to treat them as interchangeable. They are not. Shapiro-Wilk evaluates normality by comparing the sample's variance structure to what a normal distribution would produce — it considers all pairwise ordered statistics and weights them optimally. This gives it substantially higher statistical power, meaning it is more likely to detect real non-normality when it exists.

The Kolmogorov-Smirnov test (with Lilliefors correction, which SPSS applies automatically) instead measures the maximum absolute distance between the sample's empirical cumulative distribution function and a theoretical normal CDF. This is a simpler criterion and genuinely weaker for detecting subtle departures from normality, particularly in small samples. A published comparison by Razali and Wah (2011) found Shapiro-Wilk outperformed Kolmogorov-Smirnov across all sample sizes and distribution types tested. Use Kolmogorov-Smirnov only when sample sizes are very large (N > 2000) or when software constraints leave no alternative.

Anderson-Darling vs. Jarque-Bera

Anderson-Darling and Jarque-Bera approach the normality question from different angles. Anderson-Darling modifies the Kolmogorov-Smirnov approach by applying quadratic weights to the distribution tails — squared deviations at the extremes count more. This makes it particularly well-suited to analyses where tail behavior matters, such as financial risk modeling, where extreme events have outsized consequences.

Jarque-Bera takes a completely different approach: it tests whether the sample's skewness and excess kurtosis match the values expected under normality (both equal to zero). It does not examine the full distribution shape directly, only two summary moments. This makes it fast and analytically convenient, but it will miss non-normality patterns that do not manifest in skewness or kurtosis — for example, a bimodal distribution with symmetric, moderate-kurtosis shape can fool it entirely. Jarque-Bera is most appropriate for large econometric time series datasets where computational speed matters and skewness/kurtosis departures are the primary concern.

Visual Diagnostics vs. Formal Statistical Tests

This is the most practically important comparison for applied researchers. Formal tests — Shapiro-Wilk, Anderson-Darling, and the others — produce an objective p-value that simplifies reporting and removes subjectivity from the decision. However, they have well-documented size problems: underpowered for small samples (may miss real non-normality) and overpowered for large samples (will flag trivially small deviations as significant). This means the p-value alone can mislead in both directions.

Q-Q plots and histograms provide context about how and where the distribution deviates from normal, which helps you judge whether the deviation will actually distort your analysis. A researcher seeing a Q-Q plot where all points fall within ±0.3 of the reference line can reasonably proceed with a parametric test even if Shapiro-Wilk returned p = 0.04 — the formal test's significance does not translate directly to practical impact. The best practice is to report both and let the reader see the full picture.

Multivariate Normality Tests

When working with multi-dimensional datasets in MANOVA, structural equation modeling (SEM), or discriminant analysis, univariate normality testing is not sufficient. Even if every individual variable looks normally distributed on its own, their joint distribution can still violate multivariate normality assumptions.

Common Misconceptions About Normality Testing

Normality Testing: Frequently Asked Questions

Formula Glossary

Related Topics on Statistics Fundamentals

Normality testing is one piece of the broader statistical workflow. These pages from Statistics Fundamentals cover the analysis steps that come before and after normality assessment.