Hypothesis Tests

D'Agostino-Pearson Omnibus Test

Omnibus normality test: Combines Z-scores from skewness and kurtosis into a χ²(2) test statistic. Requires n ≥ 20.

Statusvalidated

Version1.0.0

Minitab equivalentNot directly in Minitab — available in SciPy (scipy.stats.normaltest), R (moments::agostino.test), Stata, EViews

Description

The D'Agostino-Pearson omnibus test transforms sample skewness and kurtosis into approximately standard-normal Z-scores (Zs and Zk). The sum K² = Zs² + Zk² is approximately chi-squared distributed with 2 degrees of freedom under H₀. The test is particularly informative because it reveals whether deviation from normality is due to skewness, kurtosis, or both. The skewness transformation is from D'Agostino (1970), the kurtosis transformation from Anscombe & Glynn (1983). The test requires at least 20 data points for reliable results.

Formulas

K²

K^{2} = Z_{s}^{2} + Z_{k}^{2} \sim χ^{2} (2)

Omnibus test statistic: Sum of squared Z-scores from skewness and kurtosis

Z_{s} = δ \cdot ln \frac{Y}{α} + (\frac{Y}{α})^{2} + 1

Skewness Z-score after D'Agostino (1970) — Y is the scaled skewness, δ and α depend on n

Z_{k} = \frac{\frac{1 - 2/ A}{1 + x _{2} 2/ ( A - 4 )} ^{1/3} \cdot sign - ( 1 - \frac{2}{9 A} )}{2/ ( 9 A )}

Kurtosis Z-score after Anscombe & Glynn (1983) — Cube-root transformation of standardized kurtosis

p-Value

p = e^{- K^{2} /2}

Exact survival function of the χ²(2) distribution

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 ≥ 20 (transformations are unreliable for smaller n)
Continuous data

Limitations

Requires n ≥ 20 — use Shapiro-Wilk for smaller samples
The Z-score transformations are approximations, not exact results
For highly skewed distributions the kurtosis Z-score can become unstable

References

D'Agostino, R. B. (1970), Transformation to normality of the null distribution of g1, Biometrika, 57(3), 679–681
Anscombe, F. J. & Glynn, W. J. (1983), Distribution of the kurtosis statistic b2 for normal samples, Biometrika, 70(1), 227–234
D'Agostino, R. B. & Pearson, E. S. (1973), Tests for departure from normality. Empirical results for the distributions of b2 and √b1, Biometrika, 60(3), 613–622
D'Agostino, R. B., Belanger, A. & D'Agostino Jr., R. B. (1990), A suggestion for using powerful and informative tests of normality, The American Statistician, 44(4), 316–321