Moments, Skewness and Kurtosis
What the exam tests
Moments summarise shape; skewness measures asymmetry and kurtosis measures peakedness and tail weight. Expect calculation of raw and central moments, β1, β2, γ1, γ2, and interpretation.
Core rules
- Raw moment: μr′ = Σfx^r/N. Central moment: μr = Σf(x − x̄)^r/N.
- μ1 = 0; μ2 = μ2′ − (μ1′)²; μ3 = μ3′ − 3μ2′μ1′ + 2(μ1′)³; μ4 = μ4′ − 4μ3′μ1′ + 6μ2′(μ1′)² − 3(μ1′)⁴.
- β1 = μ3²/μ2³; γ1 = ±√β1, taking the sign of μ3. β2 = μ4/μ2²; γ2 = β2 − 3.
- Symmetric distribution: μ3 = 0 and β1 = 0. Positive skew: μ3 > 0, typically mean > median > mode. Negative skew: μ3 < 0.
- Kurtosis: β2 = 3 is mesokurtic; β2 > 3 is leptokurtic (sharp peak, heavy tails); β2 < 3 is platykurtic (flat peak, light tails).
- Pearson's coefficient of skewness Sk = (mean − mode)/SD = 3(mean − median)/SD. Bowley's Sk = (Q3 + Q1 − 2 median)/(Q3 − Q1).
Worked example
For 1, 2, 3, 4, 5: mean = 3; deviations are −2, −1, 0, 1, 2. μ2 = (4+1+0+1+4)/5 = 2. μ3 = (−8−1+0+1+8)/5 = 0. μ4 = (16+1+0+1+16)/5 = 6.8. Hence β1 = 0 and β2 = 6.8/4 = 1.7 < 3, so the distribution is platykurtic. The zero third moment confirms symmetry.
Traps
Reporting β1 as negative - it is a square and cannot be negative; the sign belongs to γ1. Forgetting the −3 in γ2. Using raw moments where central moments are required. Confusing skewness with kurtosis. Applying Pearson's formula when the mode is ill-defined.