Four Numbers, Four Questions
| Measure | Definition | Best used when |
|---|---|---|
| Mean | Sum divided by count | Data is symmetric with no extreme values |
| Median | The middle value once sorted | Data is skewed or contains outliers |
| Mode | The most frequent value | Data is categorical, or you want the typical case |
| Range | Maximum minus minimum | A quick sense of spread is enough |
Why the Median Often Wins
Consider nine salaries: eight around $50,000 and one at $2,000,000. The mean is $267,000 — a figure nobody earns. The median stays near $50,000, which describes the group accurately.
This is precisely why official statistics report median household income and median house prices rather than means. A single extreme value moves the mean without limit; moving the median requires moving half the data.
The Mode Is the Odd One Out
It is the only measure that works on categories — the most common blood type, the best-selling colour, the modal response on a survey. It is also the only one that may not exist (when every value appears once) or may be multiple (when several tie). A distribution with two peaks is bimodal, and often a sign that two different populations have been mixed together.
Range Is the Weakest Spread Measure
It uses only two values and discards everything between them, so a single extreme observation determines it entirely. Two data sets with the same range can have completely different distributions. Standard deviation or interquartile range convey far more, which is why range appears mainly as a first glance.
Skew and the Gap Between Them
- Mean > median — skewed right, a long tail of high values. Income, house prices, wait times.
- Mean < median — skewed left, a long tail of low values. Exam scores near a ceiling, age at death.
- Mean ≈ median — roughly symmetric. Heights, measurement errors.
Frequently Asked Questions
What if there are two middle values?
With an even count, the median is the average of the two middle values once sorted. It need not be a value that actually appears in the data.
Can a data set have no mode?
Yes, when every value appears exactly once. Some textbooks say the mode is then "all values"; the more useful convention is that there is no mode.
Which average do people mean by "average"?
Almost always the mean. The ambiguity is exploited often enough — quoting whichever of the three best supports a claim — that it is worth asking which one is being used.