Normality Tests

Normality tests help you answer a simple but crucial question:

“Does this data look like it came from a normal distribution?”

Why this matters:

• Parametric tests (t-tests, ANOVA) assume normality
• Non-parametric tests don’t
• Many statistical decisions begin here

SciPy provides several ways to test normality, each with its strengths.

Practical Interpretation of Normality Tests

All normality tests return:

• test statistic
• p-value

Interpretation is the same for all:

If p ≥ 0.05

• Data is consistent with normal distribution
• You can reasonably use parametric tests

If p < 0.05

• Data is NOT normally distributed
• Consider non-parametric tests (Chapter 3)

Important: Normality tests are very sensitive to sample size:

• Large samples → even tiny deviations become “significant”
• Small samples → tests often have low power

Always pair normality tests with a plot (histogram, Q-Q plot).

Shapiro–Wilk Test (Recommended)

The go-to normality test for small to medium samples.

Works best for:

• Sample sizes up to ~5000
• General-purpose normality checking
• Pre-checking assumptions for t-tests or ANOVA

Example

from scipy import stats​data = [12.3, 11.5, 12.1, 12.6, 11.8, 12.0]​stat, p = stats.shapiro(data)print(stat, p)

Interpretation

• p < 0.05 → NOT normal
• p ≥ 0.05 → normal enough

Advantages

• Most powerful normality test
• Works well for small samples

Disadvantages

• Too sensitive for huge datasets

Kolmogorov–Smirnov (K–S Test)

Compares data to a specified distribution.

For normality, you must supply:

• mean
• standard deviation

Example

data = np.array([4.8, 5.0, 5.1, 4.9, 5.2])mu, sigma = np.mean(data), np.std(data)​stat, p = stats.kstest(data, 'norm', args=(mu, sigma))print(stat, p)​

Interpretation

Same general rule:

• p < 0.05 → not normal
• p ≥ 0.05 → consistent with normality

Notes

• Less sensitive than Shapiro–Wilk
• Not recommended for small samples
• Good for checking against any distribution, not just normal

Anderson–Darling Test

This test always returns a decision threshold, not a simple p-value.

Example

result = stats.anderson(data, dist='norm')print(result.statistic)print(result.critical_values)print(result.significance_level)​

Interpretation

If the test statistic is:

• > critical value → reject normality
• <= critical value → fail to reject

Advantages

• More sensitive in the tails
• Good for moderate to large samples

Disadvantages

• Slightly more complicated interpretation
• No simple p-value

D’Agostino and Pearson’s Test (K2 Test)

Combines skewness and kurtosis to test normality.

Example

stat, p = stats.normaltest(data)print(stat, p)​

Use when:

• Sample size ≥ 20
• You want a test sensitive to deviations in skewness and kurtosis

Avoid when:

• Very small samples (< 20)

Visual Normality Checks (Highly Recommended)

Don’t rely solely on p-values — always look at the distribution.

Histogram

import matplotlib.pyplot as pltplt.hist(data, bins=10)plt.show()​

Q-Q Plot

import scipy.stats as statsimport matplotlib.pyplot as plt​stats.probplot(data, dist="norm", plot=plt)plt.show()

• If points ≈ straight line → data is approx. normal
• If points bend or curve → non-normal

Choosing the Right Normality Test