Skewness and Kurtosis
Core rules
- Skewness measures asymmetry. Symmetric: mean = median = mode; positive or right skew: mean > median > mode; negative or left skew: mean < median < mode.
- Pearson coefficient: Sk = (mean − mode)/SD, or 3(mean − median)/SD.
- Bowley coefficient: (Q3 + Q1 − 2Q2)/(Q3 − Q1).
- Kurtosis measures peakedness and tail heaviness. Mesokurtic β2 = 3; leptokurtic > 3 (sharp peak, heavy tails); platykurtic < 3 (flat).
- Excess kurtosis = β2 − 3.
Worked example
Mean = 45, median = 42, SD = 10. Pearson Sk = 3(45−42)/10 = 0.9. Positive, so the distribution is right-skewed. Check with mode: mode = 3(42)−2(45) = 126−90 = 36; (45−36)/10 = 0.9. The mean is pulled to the right by high values.
Traps
- Reporting only the size 0.9 and omitting direction.
- Using the mean−mode formula when mode is ill-defined; use the median version.
- Saying negative skew means small values are extreme; it means a left tail.
- Confusing kurtosis with skewness.
- Forgetting Bowley uses quartiles and lies between −1 and 1.