Why Is the Average Wrong for Everybody? There Are Two Groups, Not One
By the BrainSnail editorial team. How these articles are written and checked, and how to tell us when one is wrong.
Data with two separate peaks has an average that falls between them and describes nobody. Reporting a single number for such data is one of the commonest ways statistics mislead.
What the shape looks like
Plotting how often each value occurs usually gives a single hump, with most observations near the middle and fewer towards each end, and a single average summarises that reasonably. Some data instead gives two separate humps with a dip between them, which means the observations cluster around two different values rather than one. The average of such data falls in the dip, so it names a value that is not merely unrepresentative but actively uncommon, and quoting it describes a typical case that hardly exists.
Why the shape appears
Two peaks nearly always mean two populations have been combined:
- •A measurement taken across two groups that genuinely differ
- •A process with two distinct modes of operation
- •A threshold effect, where cases fall to one side or the other
- •Two time periods combined, with a change between them
- •Two measuring instruments or observers calibrated differently
- •Very occasionally, a single process that genuinely behaves this way
What to do about it
The response is to separate rather than to summarise, which is the whole practical point. If the two groups can be identified by some other variable in the data, splitting on it and describing each separately gives two useful summaries in place of one useless one, and that separation is frequently the most important finding rather than a preliminary step. If they cannot be identified, the shape itself is evidence that something is dividing the data and worth chasing. What should not be done is to report the mean, quote a standard deviation that assumes a single hump, or run tests that assume the same, all of which produce confident nonsense.
How to spot it before summarising
The shape is invisible in any summary statistic, so finding it requires looking at the data rather than at numbers describing it. Plotting a histogram is the direct method and should precede any summary, which is advice repeated in every statistics course and skipped constantly. The choice of bin width matters, since bins that are too wide merge the two peaks into one and bins that are too narrow break the data into noise, so several widths should be tried. A density plot avoids the bin problem. And a dataset whose mean and median differ substantially is worth plotting immediately, since that gap indicates the shape is not a simple hump.
Where it turns up
Real examples are easy to find once the shape is recognised. Marks on an examination that most students either understood or did not produce two clusters rather than a spread. Times taken to complete a task split between people who knew a shortcut and people who did not. Body size in species where males and females differ markedly. Traffic on a road across a full day, peaking twice. Eruption intervals of certain geysers, which fall into two families. And the classic teaching example, a dataset that appears to describe a typical case while actually describing two quite different things averaged together.
The takeaway
Two separate peaks mean the observations cluster around two values, so the average falls in the dip between them and describes something uncommon rather than typical. The shape nearly always indicates that two different populations have been combined, and the useful response is to identify and separate them rather than to summarise. Tests assuming a single hump produce confident nonsense on such data.