Definitions
The mean is the sum of the values divided by how many there are. The median is the middle value once the data are sorted; with an even count, it is the average of the two middle values. The mode is the most frequent value.
Worked example
Data: 4, 7, 7, 9, 13. The sum is 40 and there are 5 values, so the mean is 40 / 5 = 8. Sorted, the middle value is 7, so the median is 7. The value 7 appears twice and every other value once, so the mode is 7.
With an even count, say 4, 7, 9, 13, the median is (7 + 9) / 2 = 8.
Multiple modes or none
A data set can have more than one mode. The data 1, 1, 2, 3, 3 have two modes, 1 and 3. When every value appears once, as in 2, 5, 8, there is no mode. The mode is also the only one of the three that works for categories, such as the most common shirt size.
What an outlier does
Take five salaries in thousands of dollars: 40, 42, 45, 48, 50. The mean is 225 / 5 = 45 and the median is 45. Now add one salary of 250. The sum becomes 475, so the mean is 475 / 6 ≈ 79.2, while the median is (45 + 48) / 2 = 46.5.
One extreme value moved the mean by more than 34 while the median barely changed. OpenStax notes that the median is generally a better measure of the center when there are extreme values or outliers.
Which one to report
Match the measure to the shape of the data and the question you are answering.
- Roughly symmetric data without outliers: the mean uses every value and is the usual choice.
- Skewed data or outliers, such as income or home prices: the median describes a typical value better.
- Categories or counts of the most common result: the mode.
- When in doubt, report the mean and median together. A large gap between them signals skew or outliers.
Sources and further reading
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