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.

1 分で読めます2026年10月18日更新執筆：Sumzo teamレビュー：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.

解き方付き例題

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.

解き方を表示

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. $\text{mode} = 4,\quad \text{range} = 7$

答え： 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.

解き方付き例題

Standard deviation of the same data

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

解き方を表示

The squared distances from the mean 5 add up to 32. $9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32$
Divide by 8 for the variance, then take the square root. $\sigma^2 = \frac{32}{8} = 4,\quad \sigma = 2$

答え： 2