Why Averages Can Lie: Outliers, Skew and Simpson's Paradox

One Extreme Value Can Double the Answer

Add 32,000, 34,000, 35,000, 36,000, 37,000 and 250,000 and divide by six and you get roughly 70,667. Nobody in the list earns that. A single value 12 times the next largest pulled the mean far above every real observation, which is why the mean of an income figure is a poor description of typical income.

Skew Pushes the Mean in One Direction

Many real world distributions are not symmetric. Salaries, house prices, wait times and claim sizes all have a long right tail. In a right skewed distribution the mean always sits above the median, because the tail drags it up. In a left skewed distribution, such as most scores on an easy test, the mean sits below the median.

The Range and Standard Deviation Add Context

A mean with no measure of spread tells you very little. Reporting the range alongside the mean shows the full width of the data, and reveals immediately whether a large mean is the result of a genuinely high set of values or of one outlier doing all the work.

Simpson's Paradox: Averages That Reverse

Simpson's paradox is the case where every subgroup shows one trend and the pooled average shows the opposite. The classic example is the 1975 analysis of 12,763 applications to UC Berkeley graduate school. Pooled across the whole university, male applicants appeared 1.8 times more likely to be admitted than female applicants.

Why the Berkeley Result Reversed

Separating the data by department changed the conclusion. Of the departments examined, about as many showed a bias favouring women as favouring men, and no pattern of discrimination by the admissions committees emerged. The pooled figure had been distorted because men and women applied in different proportions to departments with very different admission rates, and because the easier departments required more mathematics.

What Simpson's Paradox Teaches

An average computed across unlike groups describes nobody in particular. Before trusting any single pooled figure, ask whether the groups underneath it are comparable. When they are not, the correct move is to compute the average within each group first and compare those, which is exactly what the Berkeley researchers did across 85 departments.

Practical Habits for Avoiding Misleading Averages

Always report the count alongside the mean. Check the median against the mean and investigate any large gap. Look at the range before quoting a figure. Avoid pooling across groups with different base rates. And when one value is doing the work, name it as an outlier rather than quietly leaving it in.

When the Average Is Still the Right Answer

Averages are not inherently deceptive. They are the correct summary when every value belongs to the same comparable group and the distribution is reasonably symmetric. Resisting the impulse to distrust every number is just as important as catching the cases that genuinely mislead, because over correcting leads people to ignore a perfectly good figure.