When the Bell Curve Leans
Not all data distributions are symmetric like the normal curve. Skewness describes the direction and degree of asymmetry in a distribution. In a positively (right) skewed distribution, most values cluster at the lower end with a long tail stretching toward higher values — household income is a classic real-world example, since a small number of very high earners stretch the tail rightward. In a negatively (left) skewed distribution, most values cluster at the higher end with a long tail stretching toward lower values — age at retirement in a population with mandatory retirement policies can show this pattern, with a long tail of early retirees.
Skew directly affects the relationship between the mean and median: in a positively skewed distribution, the mean is pulled toward the tail, so mean > median; in a negatively skewed distribution, the mean is pulled the other way, so mean < median. In a perfectly symmetric distribution, mean = median.
| Skew direction | Tail points | Mean vs. median |
|---|---|---|
| Positive (right) skew | Toward higher values | Mean > median |
| Negative (left) skew | Toward lower values | Mean < median |
| No skew (symmetric) | No tail | Mean = median |
This is precisely why the choice of central tendency measure, discussed earlier in this course, matters so much: for skewed data, the median better represents the "typical" case, since the mean gets dragged toward the direction of the long tail by relatively few extreme values. Kurtosis is a related, less commonly discussed property describing how sharply PEAKED or flat a distribution is, independent of its skew.
Common pitfall: confusing which direction of skew is named for the TAIL versus the BULK of the data. Skew is always named for where the TAIL points, not where most of the data sits — a positively (right) skewed distribution has most of its data on the LEFT (lower) side, with the defining long tail stretching to the right.
Three small histograms: a right-skewed shape with mean-marker right of median-marker; a symmetric bell with both markers overlapping.