Report the mean when the values are roughly symmetric and every one of them should count. Report the median when a few extreme values would drag the mean somewhere unrepresentative. Report the mode when the data is categories rather than quantities, or when the most common value is the thing being asked about. For the list 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5, the median is 4.5 and the mode is 4 — three defensible answers to the same question.
The gap between them is not a problem to resolve. It is a description of the shape of the data, and it is usually more useful than any one of the three.
What does each one actually measure?
A different notion of the middle.
| Average | Definition | Moved by an outlier |
|---|---|---|
| Mean | Sum divided by count | Strongly |
| Median | Middle value once sorted | Barely |
| Mode | Most frequent value | Not at all |
Take that eight-number list and change the 9 to 90. The mean jumps from 5 to 15.125. The median stays at 4.5. The mode stays at 4. One value moved and only one statistic noticed.
When does the mean mislead?
Whenever the distribution has a long tail, which in practice means income, wealth, response times, file sizes, session lengths and almost anything where a small number of cases are very large.
Salary is the standard example. A team of nine people earning £30,000 and one earning £300,000 has a mean salary of £57,000 and a median of £30,000. The mean is arithmetically correct and describes nobody in the room.
The general rule: if the mean and the median differ noticeably, the mean is being pulled by a tail, and the direction of the difference tells you which way the tail runs.
Can a list have more than one mode?
Yes. A list where two values tie for most frequent is bimodal, and that is often the most important fact about it — two peaks usually mean two populations mixed together.
A list where every value appears once has no mode at all. That is a legitimate result rather than an error, and it is why the mode is rarely useful for continuous measurements: with enough decimal places, everything is unique.
Categories are the case where the mode is the only option available. Asking for the average colour of a car in a car park is not a question the mean or the median can answer, because the values cannot be ordered or added — the most frequent value is the only middle there is.
Which standard deviation should you use?
The sample version if your numbers are a sample of something larger, and the population version only if you have measured everything there is. For the eight-number list the sample deviation is 2.1381 and the population deviation is 2 — a difference of about 7 per cent, which is enough to matter in a report.
The difference is the denominator: the sample version divides by one less than the count, which corrects for the fact that a sample tends to underestimate the spread of the population it came from. With eight values the correction is 7 per cent; with eighty it is 0.6 per cent; with eight hundred it is negligible.
Choose on what the numbers are, not on which figure looks better. Almost all real data is a sample.
There is a fourth measure worth knowing about for skewed data: the trimmed mean, which discards a fixed percentage from each end before averaging. It sits between the mean and the median in how much it resists outliers, and it is what several sports scoring systems and price indices use for exactly that reason.
What should actually go in the report?
More than one number, because a single average conceals whatever it was chosen to conceal.
- The count. An average of four values and an average of four thousand are different claims.
- The median and the mean together where they differ, since the gap is the finding.
- A measure of spread — the standard deviation, or the range, or both.
- The extremes, if any single value is doing significant work.
That is four numbers instead of one and it takes the same amount of space as a sentence. It also makes the result checkable, which a bare average never is.
Questions people ask
Why is the median 4.5 when no value is 4.5? With an even count there is no single middle value, so the median is the midpoint of the two central ones — here 4 and 5. It is normal for a median not to appear in the data.
Does it handle negatives and decimals? Yes. Sorting and summing work identically, and the only operation that cares about sign is none of them.
What is the range for? The distance between the smallest and largest value. It is the crudest spread measure, it is entirely determined by two data points, and it is still worth reporting because it bounds everything else.
Should I ever remove an outlier? Only with a reason you can state — a known measurement error, a test row, a duplicate. Removing a value because it is inconvenient is how a mean becomes a median without anyone saying so.
Pick the average that matches the shape, then show enough of the others that a reader can check you. The average calculator reports all three alongside both deviations and the range, the percentage calculator handles the comparisons that come next, and the scientific calculator is there for whatever the analysis turns into.