Principles and concepts underlying higher-order statistics
Principles and concepts underlying higher-order statistics
This domain tests distribution shape - the third and fourth standardized moments - and the consequences of skewness and kurtosis for summary statistics and inference. Questions typically ask you to classify a distribution from its mean and median, interpret a reported skewness or kurtosis value, or choose a procedure that is robust when shape is non-normal.
Standardized moments
- The k-th standardized central moment is E[(X − μ)^k] / σ^k. The first two give location and spread; the third and fourth describe shape.
- Skewness γ1 = E[(X − μ)^3] / σ^3. γ1 > 0 means a longer right tail, so the mean exceeds the median; γ1 < 0 means a longer left tail and mean below median; γ1 = 0 indicates symmetry (not necessarily normality).
- Kurtosis γ2 = E[(X − μ)^4] / σ^4. Software usually reports excess kurtosis, γ2 − 3. Excess > 0 is leptokurtic (heavier tails and more extreme outliers than the normal); excess < 0 is platykurtic (lighter tails); 0 is mesokurtic.
Consequences for inference
- The mean is not resistant to skew; the median and IQR are. Report the median for strongly skewed data.
- Skewness inflates the standard error of the mean in small samples, so t-tests and z-intervals can mislead. Common remedies are a log or square-root transformation, or a rank-based test such as Wilcoxon or Kruskal–Wallis.
- Kurtosis affects the validity of variance-based tests and the coverage of confidence intervals; heavy tails make nominal 95% intervals too narrow in small samples.
- Pearson's second skewness coefficient is 3(mean − median)/s, a quick sign check.
Worked example
Data: 2, 3, 3, 4, 18. Mean = 30/5 = 6; median = 3. Because mean > median, skew is positive. Deviations from 6 are −4, −3, −3, −2, 12. Cubed deviations sum to −64 −27 −27 −8 +1728 = 1602; population third moment = 1602/5 = 320.4. Variance = (16+9+9+4+144)/5 = 36.4, so σ = 6.03 and σ^3 = 219.6. Skewness = 320.4/219.6 ≈ 1.46, confirming a strong right skew driven by 18.
Traps that cost marks
- Reversing skew direction: a long right tail gives positive skew, and the mean is pulled toward that tail.
- Treating kurtosis as 'peakedness' only; it primarily measures tail weight.
- Forgetting to subtract 3 when the item asks for excess kurtosis.
- Assuming the central limit theorem rescues a heavily skewed sample of n = 8.
- Reporting the mean for ordinal or badly skewed data.