Mean, median and mode: which average to use, and when the mean lies

"The average salary is £45,000" and "the median salary is £34,000" can both be true of the same company, and the gap between them is the boss. Average is three different things, and choosing the wrong one is the easiest way to mislead with a single number. The average calculator gives all three; this guide says which to trust.

Three averages

The mean is the sum divided by the count. The median is the middle value when the data is sorted (or the mean of the two middle values for an even count). The mode is the most frequent value. For 2, 3, 3, 4, 8 the mean is 4, the median 3 and the mode 3. All three describe the "centre" of a set of numbers; they differ in what they're sensitive to.

The mean and its weakness

The mean uses every value, which makes it the right basis for totals and rates: mean revenue per customer times customers is revenue; mean speed over a journey is distance over time. Its weakness is the same property — every value pulls on it, so a few extreme ones drag it away from where most of the data sits. Nine people earning £30,000 and one earning £300,000 have a mean of £57,000, which describes nobody. Skewed data — incomes, house prices, wait times, file sizes, anything bounded at zero with a long tail — always has a mean above the median, and reporting the mean flatters the tail.

The median: robust to outliers

The median ignores how extreme the extremes are; it only cares that half the values are above and half below. Change the top earner from £300,000 to £3 million and the median doesn't move. That's why house prices, salaries, household incomes and response times are reported as medians: they answer "what does a typical one look like?" The cost is that the median throws away information about the tail — and for some questions the tail is the point (total tax revenue, total cost, the risk of the worst case).

The mode: for categories

The mode is the only average that works for categories: the most common shoe size, the most-ordered dish, the most frequent answer. For continuous measurements it's nearly useless (every value appears once) unless the data is binned, and a set can have two modes (bimodal — often a sign of two populations mixed together, such as men's and women's heights) or none. The word frequency counter is a mode finder for text.

Choosing, and a worked example

• Symmetric data, no outliers (heights, test scores): mean, and the median agrees.
• Skewed data or outliers (incomes, prices, durations): median for "typical", mean only when you need the total.
• Categories: mode.
• Rates and totals (average speed, cost per unit): mean — but weight it by the right denominator (How to calculate percentages covers the arithmetic).

Example: a shop's orders one day are £12, £15, £15, £18, £22, £25, £30, £480 (a corporate bulk order). Mean £77 — "the average order is £77" would set a wildly wrong expectation. Median £20 describes a typical customer. Mode £15. Total £617 is the mean times eight, which is the only reason to want the mean here. Report the median for the typical order and the total for the day; report the mean only alongside its spread — Standard deviation explained: what it measures, how it's calculated, and the 68–95–99.7 rule explains how — and never on its own for skewed data. The standard deviation calculator gives the spread; the percentage calculator the comparisons.