Shapiro–Wilk Test

he Shapiro–Wilk test is a statistical test for normality — it checks whether a given dataset is drawn from a normal (Gaussian) distribution.

It’s one of the most powerful and widely used normality tests, especially for small to medium sample sizes (n < 5000).

In simple terms:

“The Shapiro–Wilk test checks if your data follow a bell-shaped normal distribution.”

When to Use It

• Continuous data - Works on numerical values
• Small to medium samples - Best for n < 5000
• Need to choose test type - Helps decide between parametric and non-parametric tests

When Not to Use It

• Categorical data - Not suitable
• Large samples - Even tiny deviations appear significant (use visual + other tests too)

Hypotheses

• H₀ (Null Hypothesis) - The data are normally distributed
• H₁ (Alternative Hypothesis) - The data are not normally distributed

Test Statistic

The Shapiro–Wilk test computes a W statistic, which measures how close your sample’s distribution is to a normal one.

Where:

• x_(i)​: ordered sample values (sorted smallest → largest)
• a_i​: constants from expected normal distribution
• X̅: sample mean

If W ≈ 1, data are close to normal.
If W is much smaller, data deviate from normality.

Decision Rule

• p > 0.05 - Fail to reject H₀ → data look normal
• p ≤ 0.05 - Reject H₀ → data not normal

Example in Python

Let’s check if a dataset follows a normal distribution.

import numpy as npfrom scipy.stats import shapiro​# Example datadata = np.array([10, 12, 11, 14, 13, 15, 12, 11, 13, 14])​# Perform Shapiro–Wilk Teststat, p = shapiro(data)print(f"Shapiro–Wilk Statistic: {stat:.3f}")print(f"P-value: {p:.4f}")​# Interpretif p > 0.05:    print("Data look normally distributed (fail to reject H₀).")else:    print("Data are not normally distributed (reject H₀).")

Example Output:

Shapiro–Wilk Statistic: 0.967P-value: 0.8234Data look normally distributed (fail to reject H₀).

Example with Non-Normal Data

# Create skewed dataskewed = np.random.exponential(scale=2, size=100)​stat, p = shapiro(skewed)print(f"W={stat:.3f}, p={p:.4f}")​if p > 0.05:    print("Normal")else:    print("Not normal")

Example output:

W=0.812, p=0.0001Not normal

Visual Check:

Histogram and KDE

QQ Plot

If the points follow a straight line in the Q–Q plot → roughly normal, Curved or S-shaped patterns → not normal