Skewed Distributions
A distribution is skewed when one tail stretches farther than the other — the bell curve with a limp. The direction of the tail, not the hump, names the skew.
The two directions
| Type | Shape | Real examples | Mean vs median |
|---|---|---|---|
| Right (positive) skew | Long tail to the right | Incomes, house prices, hospital stays | Mean > median — the tail drags the mean up |
| Left (negative) skew | Long tail to the left | Age at death in rich countries, easy-exam scores | Mean < median |
The practical rule
Mean-vs-median is the instant skew detector: when a country’s average income sits far above its median income, you’re looking at right skew — a few enormous values pulling the average away from the typical person. For skewed data, the median is the honest "typical" number, which is why house prices are reported as medians.
Why it matters for z-scores
The percentile meaning of a z-score comes from the normal distribution — on skewed data, "z = 2 → top 2.3%" is simply false. Options: report medians and percentiles directly, transform the data (log transforms tame right skew beautifully), or use nonparametric methods. Averages of skewed data, though, still tend toward normal as samples grow — the Central Limit Theorem’s gift — which is why the z table stays useful for testing means even when raw data is lopsided.