Math · Introductory statistics · Worked example

Find the mean, median and standard deviation

Eight students report how many hours they slept: 4, 8, 6, 5, 3, 7, 9 and 6. Find the mean, the median and the sample standard deviation.

Find the mean

The eight values add to 48.

\frac{4 + 8 + 6 + 5 + 3 + 7 + 9 + 6}{8} = 6

Find the median

Sorted: 3, 4, 5, 6, 6, 7, 8, 9. With eight values, the median is the average of the 4th and 5th, both 6.

\frac{6 + 6}{2} = 6

Square the deviations

Subtracting 6 from each value gives −2, 2, 0, −1, −3, 1, 3 and 0. They add to 0, as deviations from the mean always do. Their squares add to 28.

4 + 4 + 0 + 1 + 9 + 1 + 9 + 0 = 28

Divide by n − 1 and take the root

\sqrt{\frac{28}{8 - 1}} = 2

Result

Mean 6 hours, median 6 hours, sample standard deviation 2 hours.

Your turn

Find the mean and the sample standard deviation of 10, 12, 14, 16 and 18.

Show the answer and explanation

Mean 14; s = √10 ≈ 3.16.

The deviations are −4, −2, 0, 2 and 4, whose squares add to 40. Divide by n − 1 = 4 to get 10, then take the square root.

\begin{array}{l} \frac{10 + 12 + 14 + 16 + 18}{5} = 14 \\ \sqrt{\frac{40}{4}} \approx 3.16 \end{array}

Keep exploring

In Statistics, the Summary shows the mean 6, the median 6 and the sample standard deviation 2. The population standard deviation beside it, about 1.87, divides by 8 instead of 7.

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Summarize the data in Statistics Check the arithmetic in Math Open worked example on a board Sample standard deviation in Math Reference

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