An average is not an outcome

§ 01The arithmetic is honest; the implication is not

Take ten people. Nine of them earn £30,000 a year. One earns £930,000. The mean income for the group is £120,000 — a figure that no individual in the room actually receives, and that is four times the wage familiar to nine out of ten members. The arithmetic is faultless. The implication — that £120,000 is somehow typical — is wrong.

This is the structural problem with averages in skewed distributions. The mean is pulled toward the tail by extreme values, and in many real-world distributions those extreme values are large and rare. The result is a measure of central tendency that sits well above the bulk of the data, describing almost nobody while technically describing the whole.

The distinction matters mechanically, not morally. Income, asset prices, company revenues, waiting times, and financial losses all tend to follow distributions with long right tails — a mass of ordinary observations on the left and a thin but powerful strand of very large ones on the right. In each case, the long tail drags the mean upward and away from the median, which is the middle value: the point where exactly half the observations fall above and half below.

§ 02The median sits where the people are

In the ten-person example above, the median income is £30,000. That is the value that best represents where the mass of the group lives. The mean and the median diverge as soon as the distribution becomes asymmetric — and they diverge dramatically when the tail is heavy.

The mean is pulled toward the tail by extreme values, and in many real-world distributions those extreme values are large and rare.

Understanding this difference is not a statistical nicety. It changes what a summary number actually tells you. If you are asking what a randomly selected member of a group is likely to experience, the median is almost always the more informative figure for a skewed distribution. The mean tells you the total divided by the count; the median tells you where the middle of the distribution sits. These are different questions with different answers.

The confusion between the two is easy to produce and slow to correct, because both are legitimately called "averages" in everyday speech. A politician citing mean household wealth and a statistician citing median household wealth are both being accurate. They are answering different questions, and the gap between their figures can be very large indeed.

§ 03Outcomes cluster; the mean floats above them

The practical error that follows from treating a mean as a typical outcome is the expectation that most members of a group will land somewhere near it. In a symmetric distribution — a normal, bell-shaped one — that is roughly true. Most values cluster around the middle. The mean and median are close together. "Average" and "typical" mean something similar.

But financial and economic distributions are rarely symmetric. More companies earn small profits than large ones. More trades are small than large. The distribution of returns across a large portfolio of individual assets is heavily right-skewed when winners are allowed to compound — which is why multiply, do not add governs growth arithmetic. In all these cases, the majority of observations fall below the mean.

That sentence is worth pausing on: in a right-skewed distribution, most outcomes are below average. This is not a failure of the group; it is a feature of the shape. It means that citing the mean as a benchmark against which individuals should measure themselves is, in a skewed world, a systematic way of making most people feel they are underperforming even when nothing is wrong.

The correct tool depends on the question. The mean is the right figure when you want to scale — when you need to know total output or total cost from a per-unit figure. The median is the right figure when you want to know what a typical member of the distribution actually experiences. Getting these confused is not an innocent rounding error. It is a category mistake, and one that recurs because the mean is easier to compute and the word "average" is rarely defined before it is deployed.