Skewness and Kurtosis
What the domain covers
Skewness measures asymmetry; kurtosis measures peakedness and tail weight. The exam tests signs and interpretation of Karl Pearson's and Bowley's coefficients, moments, and the classification mesokurtic, leptokurtic and platykurtic.
Core rules
- Symmetric distribution: mean = median = mode; positive skew has a longer right tail (mean > median > mode); negative skew has a longer left tail.
- Karl Pearson's coefficient: Sk = (mean − mode)/SD, or Sk = 3(mean − median)/SD when mode is ill-defined.
- Bowley's coefficient: Sk = (Q3 + Q1 − 2Q2)/(Q3 − Q1). It lies between −1 and 1.
- Moments about mean: μr = Σf(x − x̄)^r/Σf. β1 = μ3²/μ2³ measures skewness; β2 = μ4/μ2² measures kurtosis.
- β2 = 3 mesokurtic; β2 > 3 leptokurtic (sharper peak, heavier tails); β2 < 3 platykurtic.
Worked example
A distribution has mean 50, median 48 and SD 10. Using Sk = 3(50 − 48)/10 = 0.6. The positive sign means a right-skewed distribution; the mean exceeds the median. If β2 were 4.2, the distribution would be leptokurtic relative to the normal curve.
Traps
Do not reverse the sign: a long right tail gives positive skew. Bowley's measure can be computed without mode but needs quartiles. Kurtosis is not simply 'height of peak'; tail behaviour matters. A zero skew coefficient does not prove normality, only symmetry. Do not confuse β1 with its square root.