Skewness (Pearson's Coefficient)

Karl Pearson introduced two coefficients. The first uses the mode: Sk₁ = (x̄ − Mode) / s. The second uses the median: Sk₂ = 3·(x̄ − Median) / s. The mode-based version is theoretically appealing (the mode is the peak of the distribution), but the mode is unstable and often ambiguous in practice: real datasets may have multiple local peaks, no clear peak, or a peak at an arbitrary binned value. The median is robust and unambiguous, and for a wide range of distributions Sk₂ ≈ Sk₁ due to the empirical relationship Mode ≈ 3M − 2x̄. The second coefficient has therefore become the standard.

A common rule of thumb is: |Sk| < 0.5 — approximately symmetric; 0.5 ≤ |Sk| < 1 — moderately skewed; |Sk| ≥ 1 — highly skewed. Some sources use |Sk| < 1 as "slight" or "negligible" skewness and |Sk| > 1 as "substantial." For normality testing, |Sk| > 2 is often considered a strong departure from normality. The practical interpretation depends on context: for a small sample from a naturally skewed population (like income), Sk of 1.5 might be perfectly normal; for residuals in a regression model, Sk of 0.5 might signal a problem. Always interpret skewness alongside the data context and other diagnostics.

Pearson's coefficient as defined here (using the median) is bounded by approximately ±3 for most distributions, though there is no strict mathematical upper bound in general. For symmetric distributions, Sk = 0. For moderately skewed distributions, |Sk| rarely exceeds 1. For heavily skewed distributions like the exponential or log-normal, |Sk| can reach 1.5–3. Fisher's moment-based coefficient of skewness (g₁ = m₃ / m₂^(3/2)) has no upper bound and can exceed 5 for very heavy-tailed distributions like the Cauchy or Pareto with small shape parameters.

Skewness is used across statistics, finance, and engineering. In finance, return distributions with negative skewness are considered riskier because they have frequent small gains and rare large losses (like selling options). In quality control, skewed measurement distributions may signal a non-normal process that requires transformation or non-parametric methods. In machine learning, skewed features are often log-transformed before modeling to reduce the influence of extreme values. In risk analysis, skewness informs Value-at-Risk and Expected Shortfall calculations. Reporting skewness alongside mean and standard deviation gives a fuller picture of a distribution's shape.

Bowley's coefficient of skewness (also called quartile skewness) uses quartiles rather than mean and median: Sk_B = (Q₃ − 2Q₂ + Q₁) / (Q₃ − Q₁). It ranges from −1 to +1 and is highly robust to outliers because it depends only on quartiles, not on extreme values or the mean. Bowley skewness is preferred when the data contain outliers or heavy tails that distort the mean. Pearson's coefficient, in contrast, uses the mean and is more sensitive to extreme values — which can be an advantage (it captures more information) or a disadvantage (it can be dominated by a single outlier). Both are reported in practice, and they usually agree in sign when skewness is genuine.

Yes, sample size affects the reliability of skewness estimates. Small samples (n < 30) can show spuriously high or low skewness due to sampling variability, especially if the population is symmetric but the sample happens to contain a few extreme values. As n grows, the sample skewness converges to the population value. For formal inference, the standard error of skewness for normally distributed data is approximately √(6/n), so a sample of n = 100 has SE ≈ 0.245. A common normality test compares the sample skewness to this SE — a value larger than about 2 SE suggests significant departure from normality. With small samples, always interpret skewness cautiously.