Work through an example
For the sorted data 2, 3, 3, 4, 18:
- Add the five values: 2 + 3 + 3 + 4 + 18 = 30. Divide by 5 to get the mean, 6.
- The third value is in the middle, so the median is 3.
- The value 3 appears twice and every other value appears once, so the mode is 3.
Now replace 18 with 100. The mean rises from 6 to 22.4, but the median remains 3 and the mode remains 3. This is why the mean is sensitive to extreme values, while the median is resistant: one very large or small observation usually moves the mean more. Resistance does not mean the median is always more accurate; it means it responds differently to extremes.
Which one should I use?
| Summary | What it tells you | Often useful when | Main limitation |
|---|
| Mean | The arithmetic balance point; every value contributes | Values are numerical and a typical deviation from the centre matters, especially for roughly symmetric data | A few extreme values can move it substantially |
| Median | The halfway point in the ordered data | Numerical data are skewed or contain extreme observations; it is also useful for ordinal rankings | It does not use the exact distances between all values |
| Mode | The value or category that occurs most often | You need the most common category, size, response, or repeated value | A dataset can have several modes or no informative mode |
For a roughly symmetric set of measurements without strong outliers, the mean often gives a useful centre and works naturally with variance and many statistical models. For a strongly right-skewed quantity such as income, the median often better describes the middle person because a few very large values pull up the mean. For nominal categories such as transport type, the mean and median may not make sense, but the mode identifies the most frequent category. For ordered ratings, a median can be meaningful even if the gaps between rating levels are not equal.
Do not choose by a slogan. Ask what “typical” should mean for the decision, whether distances between values are meaningful, and whether the distribution has a long tail or separate clusters. A mean and median can both hide multiple groups; inspect a plot or report the groups when that structure matters.
Modes, ties, and small samples
There can be no mode that is useful when every value occurs once, or several modes when values tie for highest frequency. Some books call such a dataset “no mode”; others report all tied modes. State the convention if the distinction matters. A category may be modal even though its numeric code has no arithmetic meaning—for example, the mode of labels 1, 2, and 3 can be reported, but averaging those labels usually cannot.
With two middle values, the median need not be an observed value. For 2, 3, 8, 10, the median is (3 + 8) / 2 = 5.5. It marks the midpoint of the two central observations, not a claim that anyone scored 5.5.
Try this above
Add a very large value and watch which measure changes most. Then make a second category as frequent as the current mode and check how the displayed mode handles a tie. Compare the result with quantiles and percentiles, which generalise the median to other positions, and mean, variance, and standard deviation, which pair the mean with measures of spread.