When the mean is the wrong average, and what to quote instead

Paste the column into FIGURES. It gives the mean and the median, and then says in words whether the mean is a fair summary of this particular column and why not: the skew, the gap between the two, the share held by the largest values, and whether the shape has two humps. Nothing is uploaded.

What the mean actually is

The arithmetic mean is the balance point of the data. Imagine the values as weights on a beam: the mean is where you would put the fulcrum. That definition explains everything else about it, including its failure mode.

A weight far out on one end shifts the fulcrum a long way, and it does not matter how many weights are clustered at the other end. One value can move the mean arbitrarily far. The median, the middle value when they are sorted, cannot be moved at all by changing the largest value, only by moving values across the middle.

That is the whole difference, and it is why the two disagree exactly when it matters.

A worked example

Fifteen salaries. Fourteen of them are between £27,800 and £34,500. One is £412,000.

The mean is not wrong. It is the correct arithmetic mean of those fifteen numbers. It is simply not an answer to the question “what does somebody here earn?”, and that is the question it will be read as answering.

The tell is the gap. When the mean is nearly twice the median, something is dragging it, and a single sentence naming what is dragging it is worth more than either number on its own.

Four signs the mean is not the summary you want

1. The mean and the median disagree

The simplest diagnostic and the one to run first. If they are within a few per cent, the distribution is roughly symmetric and the mean is fine. If they differ by ten per cent or more, quoting only the mean is a choice, and somebody is entitled to ask why you made it.

2. The distribution is skewed

Skew measures the lopsidedness. Positive skew means a long tail to the right: most values low, a few very high. Incomes, house prices, waiting times, file sizes, claim amounts and company sizes are all positively skewed, essentially always. Negative skew (a long tail to the left) turns up in things with a ceiling, like exam marks on an easy paper or time-to-completion against a deadline.

A skew above about 1 in magnitude is enough to make the mean unrepresentative. Above 2 it is describing the tail rather than the body.

3. A few values hold most of the total

Work out what share of the sum the largest tenth of the values accounts for. If it is a third or more, any average over the column describes almost none of it. This is a blunter and often more persuasive statement than skew, because it is in units anybody understands: “the top three of these forty claims are half the money.”

4. The shape has two humps

This is the one that makes a single average meaningless rather than merely unrepresentative. If a column has two clusters (two teams, two products, two shifts, full-time and part-time), then the mean sits in the valley between them and describes a value that almost nobody has.

Bimodality is nearly always two populations that have been added together. The fix is not a better summary statistic; it is to separate them and describe each.

The one thing only the mean can do

The mean has a property the median does not, and it is the reason the mean is not simply the worse average.

The mean multiplied by the count gives the total. If you know the mean claim is £55,437 and there were fifteen claims, you know the claims cost £831,555. The median tells you nothing about the total whatsoever.

So the right summary depends on the question:

Budgets need means. Descriptions of people need medians. A report that uses the mean for both is wrong half the time and a report that uses the median for both is wrong the other half.

Response times, where this does real damage

Average response time is the standard example of an average that quietly conceals the thing anybody cares about. Suppose a service answers most calls in two seconds and a few in ninety. The mean might be four seconds, which sounds excellent, and the caller who waited ninety seconds is the one who complains, cancels, or writes to somebody.

This is why service levels are written as percentiles rather than averages: “95% answered within ten seconds” is a promise about the experience of almost everybody, and “average ten seconds” is compatible with one in twenty callers waiting three minutes.

What about trimming?

A trimmed mean (discard the top and bottom few per cent, then average) is a reasonable middle course and is used in real measurement work. Two cautions. First, the amount trimmed is a decision, and a decision made after seeing the result is not a method. Decide the trim before you look. Second, trimming discards the very values that may be the finding: in an audit, the outlier is often the point.

The same caution applies to removing outliers. A value outside the usual fences is unusual, not wrong. The £412,000 salary may be the chief executive, which is a fact about the organisation rather than an error in the spreadsheet.

The habit worth having

Quote two numbers, not one. The median with the quartiles is a complete description of a typical case and its spread, in three figures anybody can read. Add the mean when somebody needs a total. And when the two disagree, say so in a sentence; that sentence is usually the most informative thing on the page.