Normality Tests¶
Many statistical methods — t-tests, ANOVA, linear regression — assume that the underlying data are normally distributed. Before applying these methods, normality tests provide a formal way to check whether this assumption is reasonable. A failed normality test signals that non-parametric alternatives or data transformations may be necessary.
The general hypotheses for all normality tests are
where \(\mu\) and \(\sigma^2\) are typically estimated from the data.
Shapiro-Wilk Test¶
The Shapiro-Wilk test is one of the most powerful normality tests, particularly effective for small to moderate sample sizes (\(n \leq 5000\)). It assesses how well the order statistics of the sample match the expected order statistics of a normal distribution.
The test statistic is
where \(X_{(1)} \leq X_{(2)} \leq \cdots \leq X_{(n)}\) are the order statistics, and the weights \(a_i\) are derived from the means, variances, and covariances of the order statistics of a standard normal sample of size \(n\).
The statistic \(W\) takes values in \((0, 1]\), where \(W = 1\) indicates a perfect match with normality. Reject \(H_0\) when \(W\) is significantly smaller than 1.
from scipy import stats
import numpy as np
np.random.seed(42)
data = np.random.normal(loc=0, scale=1, size=100)
w_stat, p_value = stats.shapiro(data)
print(f"Shapiro-Wilk statistic: {w_stat:.4f}")
print(f"p-value: {p_value:.4f}")
Anderson-Darling Test¶
The Anderson-Darling test compares the empirical CDF to the normal CDF, with extra weight given to discrepancies in the tails. This makes it more sensitive than the Kolmogorov-Smirnov test to departures from normality in the extreme values.
The test statistic is
where \(Z_{(i)} = (X_{(i)} - \bar{X}) / s\) are the standardized order statistics and \(\Phi\) is the standard normal CDF.
The anderson function returns critical values at multiple significance levels rather than a single p-value:
from scipy import stats
import numpy as np
np.random.seed(42)
data = np.random.normal(loc=0, scale=1, size=100)
result = stats.anderson(data, dist='norm')
print(f"Anderson-Darling statistic: {result.statistic:.4f}")
for sl, cv in zip(result.significance_level, result.critical_values):
decision = "Reject H0" if result.statistic > cv else "Fail to reject H0"
print(f" {sl}% significance: critical value = {cv:.4f} -> {decision}")
D'Agostino-Pearson Test¶
The D'Agostino-Pearson test combines two separate tests — one for skewness and one for kurtosis — into a single omnibus test. This approach detects departures from normality due to either asymmetry or heavy/light tails.
The test first computes a skewness statistic \(Z_1\) based on the sample skewness \(\sqrt{b_1}\) and a kurtosis statistic \(Z_2\) based on the sample kurtosis \(b_2\). The combined test statistic is
Under \(H_0\), \(K^2\) approximately follows a \(\chi^2_2\) distribution. This test requires \(n \geq 20\) to produce reliable results.
from scipy import stats
import numpy as np
np.random.seed(42)
data = np.random.normal(loc=0, scale=1, size=100)
k2_stat, p_value = stats.normaltest(data)
print(f"K-squared statistic: {k2_stat:.4f}")
print(f"p-value: {p_value:.4f}")
Sample Size Sensitivity
Normality tests become increasingly sensitive with large sample sizes. A statistically significant departure from normality (small p-value) may be practically insignificant if the sample is large enough. Always supplement formal tests with visual diagnostics such as QQ plots.
Comparison of Normality Tests¶
| Test | SciPy Function | Best For | Minimum \(n\) |
|---|---|---|---|
| Shapiro-Wilk | stats.shapiro |
Small to moderate samples | 3 |
| Anderson-Darling | stats.anderson |
Tail departures | 8 |
| D'Agostino-Pearson | stats.normaltest |
Skewness and kurtosis departures | 20 |
Summary¶
Normality tests check whether data plausibly come from a normal distribution, which is an assumption underlying many parametric methods. The Shapiro-Wilk test offers the best overall power for small samples, the Anderson-Darling test is most sensitive to tail departures, and the D'Agostino-Pearson test specifically targets asymmetry and tail weight. In practice, combine formal tests with QQ plots for a complete assessment.