Normality Tests with SciPy

Normality tests are useful when you need to check whether data are approximately normally distributed. This matters because many statistical methods assume normality, especially for small samples.

### Testing a Sample that is Approximately Normal

import numpy as np
import warnings
from scipy import stats

np.random.seed(11)
warnings.filterwarnings("ignore", category=FutureWarning)

normal_data = np.random.normal(loc=0, scale=1, size=200)

shapiro = stats.shapiro(normal_data)
anderson = stats.anderson(normal_data, dist="norm")

print(f"Shapiro-Wilk statistic: {shapiro.statistic:.3f}")
print(f"Shapiro-Wilk p-value: {shapiro.pvalue:.6f}")
print(f"Anderson-Darling statistic: {anderson.statistic:.3f}")
print("Critical values:", anderson.critical_values)

Shapiro-Wilk statistic: 0.995
Shapiro-Wilk p-value: 0.676268
Anderson-Darling statistic: 0.201
Critical values: [0.559 0.629 0.749 0.87  1.031]

### Testing a Clearly Skewed Sample

import numpy as np
import warnings
from scipy import stats

np.random.seed(11)
warnings.filterwarnings("ignore", category=FutureWarning)

skewed_data = np.random.exponential(scale=1.0, size=200)

shapiro = stats.shapiro(skewed_data)
anderson = stats.anderson(skewed_data, dist="norm")

print(f"Shapiro-Wilk statistic: {shapiro.statistic:.3f}")
print(f"Shapiro-Wilk p-value: {shapiro.pvalue:.6f}")
print(f"Anderson-Darling statistic: {anderson.statistic:.3f}")
print("Critical values:", anderson.critical_values)

Shapiro-Wilk statistic: 0.865
Shapiro-Wilk p-value: 0.000000
Anderson-Darling statistic: 8.173
Critical values: [0.559 0.629 0.749 0.87  1.031]

### Histograms of the Two Samples

import numpy as np
import matplotlib.pyplot as plt

np.random.seed(11)

normal_data = np.random.normal(loc=0, scale=1, size=200)
skewed_data = np.random.exponential(scale=1.0, size=200)

fig, axes = plt.subplots(1, 2, figsize=(10, 4.5))
axes[0].hist(normal_data, bins=20, alpha=0.75)
axes[0].set_title("Approximately Normal")
axes[1].hist(skewed_data, bins=20, alpha=0.75)
axes[1].set_title("Skewed")
plt.tight_layout()
plt.show()

### Q-Q Plots

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

np.random.seed(11)

normal_data = np.random.normal(loc=0, scale=1, size=200)
skewed_data = np.random.exponential(scale=1.0, size=200)

fig, axes = plt.subplots(1, 2, figsize=(10, 4.5))
stats.probplot(normal_data, dist="norm", plot=axes[0])
axes[0].set_title("Q-Q Plot: Normal Data")
stats.probplot(skewed_data, dist="norm", plot=axes[1])
axes[1].set_title("Q-Q Plot: Skewed Data")
plt.tight_layout()
plt.show()

### Practical Example: Choosing a Test Based on Normality

import numpy as np
import warnings
from scipy import stats

np.random.seed(31)
warnings.filterwarnings("ignore", category=FutureWarning)

sample = np.random.exponential(scale=1.2, size=120)
shapiro = stats.shapiro(sample)

print(f"Shapiro-Wilk p-value: {shapiro.pvalue:.6f}")
if shapiro.pvalue < 0.05:
    print("Conclusion: the sample is not well modeled as normal; consider a nonparametric method.")
else:
    print("Conclusion: the sample does not show a strong departure from normality.")

Shapiro-Wilk p-value: 0.000000
Conclusion: the sample is not well modeled as normal; consider a nonparametric method.

### Conclusion

Normality tests work best when combined with visual checks like histograms and Q-Q plots. SciPy gives you both formal tests and the tools to support a more defensible interpretation.

Normality Tests with SciPy

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