Hypothesis Tests

Anderson-Darling Test

Tests whether a sample comes from a normally distributed population — especially sensitive in the distribution tails.

Statusvalidated

Version1.0.0

Minitab equivalentStat > Basic Statistics > Normality Test (Anderson-Darling)

Description

The Anderson-Darling test is an EDF-based (empirical distribution function) normality test that measures the discrepancy between the empirical and theoretical distribution functions. Unlike the Kolmogorov-Smirnov test, it places more weight on deviations in the tails, making it especially sensitive to outliers and heavy tails. The test statistic A² is adjusted with a sample-size correction factor to A²*. Small p-values (p < α) indicate evidence against the normality assumption.

Formulas

A²

A^{2} = - n - \frac{1}{n} i = 1 \sum n [(2 i - 1) \cdot (ln Φ (z_{i}) + ln (1 - Φ (z_{n + 1 - i})))]

Test statistic A²: measure of the discrepancy between the empirical and theoretical normal distribution

z_i

z_{i} = \frac{x _{(i)} - x ˉ}{s}

Standardized ordered values (x sorted, using sample mean and standard deviation)

A²*

A^{2 *} = A^{2} \cdot (1 + \frac{0.75}{n} + \frac{2.25}{n ^{2}})

Adjusted test statistic with sample-size correction (D'Agostino & Stephens 1986)

H₀

H_{0} : X_{1}, \dots, X_{n} \sim N (μ, σ^{2})

Null hypothesis: The sample comes from a normal distribution

Assumptions

Data are independent and identically distributed (i.i.d.)
Sample size: n ≥ 3 (recommended: n ≥ 8 for reliable p-values)
Continuous data
Mean and variance are estimated from the sample (composite test)

Limitations

The p-value approximation is based on empirical polynomials (Stephens 1986), not exact tables
For very small samples (n < 8) the p-values are less reliable
Not suitable for discrete data or data with many ties
Tests only for normality with estimated parameters — not for other distributions

References

Anderson, T. W. & Darling, D. A. (1954), A test of goodness of fit, Journal of the American Statistical Association, 49(268), 765–769
Stephens, M. A. (1986), Tests based on EDF statistics, in D'Agostino & Stephens (Eds.), Goodness-of-Fit Techniques, Marcel Dekker, Ch. 4
D'Agostino, R. B. & Stephens, M. A. (1986), Goodness-of-Fit Techniques, Marcel Dekker, New York
NIST/SEMATECH e-Handbook of Statistical Methods, Section 1.3.5.14