How to use it
Paste the numbers, for example 2, 4, 4, 4, 5, 5, 7, 9. The sample skewness and excess kurtosis are at the top; the population versions, standard errors and z-values are below.
Formulas
With the mean x̄ and the central moments m₂ = Σ(x − x̄)²/n, m₃ = Σ(x − x̄)³/n, m₄ = Σ(x − x̄)⁴/n:
- Population skewness g₁ = m₃ / m₂^(3/2); sample skewness G₁ = g₁ · √(n(n − 1)) / (n − 2).
- Population excess kurtosis g₂ = m₄ / m₂² − 3; sample excess kurtosis G₂ = ((n + 1)g₂ + 6)(n − 1) / ((n − 2)(n − 3)).
- Standard errors: SES = √(6n(n − 1) / ((n − 2)(n + 1)(n + 3))), SEK = 2 · SES · √((n² − 1) / ((n − 3)(n + 5))).
Worked example
Data 2, 4, 4, 4, 5, 5, 7, 9: n = 8, mean 5. The deviations −3, −1, −1, −1, 0, 0, 2, 4 give m₂ = 32/8 = 4, m₃ = 42/8 = 5.25 and m₄ = 356/8 = 44.5.
- g₁ = 5.25 / 4^1.5 = 0.65625 and G₁ = 0.65625 · √56 / 6 = 0.818488.
- g₂ = 44.5/16 − 3 = −0.21875 and G₂ = (9 · (−0.21875) + 6) · 7 / 30 = 0.940625.
- SES = 0.752101, so the skewness ratio is 1.08827: the right skew is not significant for only eight values.
Tips and common mistakes
- Check which kurtosis a program reports: some give g₂ + 3 (normal = 3), most give excess kurtosis (normal = 0).
- Skewness and kurtosis are very sensitive to outliers; one extreme value can dominate m₃ and m₄.
- With small samples the standard errors are large, so a visible skew may not be meaningful.