Averages and spread: mean, median, mode, range and standard deviation

Three ways to describe a typical value, two ways to describe how spread out data is, and when to use each.

阅读约 1 分钟更新于 2026年10月18日作者：Sumzo team审核：Sumzo team

Mean: add the values and divide by how many there are.
Median: the middle value once the data is in order (the mean of the two middle values if the count is even).
Mode: the value that appears most often.
Range: largest value minus smallest value.

例题

Summarise 2, 4, 4, 4, 5, 5, 7, 9

Find the mean, median, mode and range of 2, 4, 4, 4, 5, 5, 7, 9.

显示解题过程

There are 8 values adding to 40. $\bar{x} = \frac{40}{8} = 5$
The middle two values are 4 and 5. $\text{median} = \frac{4 + 5}{2} = 4.5$
4 appears most often; the range is 9 − 2. $\text{mode} = 4,\quad \text{range} = 7$

答案： Mean 5, median 4.5, mode 4, range 7

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Standard deviation

The standard deviation measures how far values typically are from the mean. Square each distance from the mean, average those squares (the variance), then take the square root.

\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n}}

For a sample, divide by n − 1 instead of n.

例题

Standard deviation of the same data

Find the population standard deviation of 2, 4, 4, 4, 5, 5, 7, 9.