Standard deviation

Standard deviation as a measure of spread about the mean, calculating it by hand and with a calculator, and interpreting and comparing standard deviations.

What it measures

The standard deviation ($\sigma$) is roughly the typical distance of values from the mean. A small $\sigma$ means the data is clustered near the mean.

Population standard deviation

$$\sigma = \sqrt{\frac{\sum (x - \bar{x})^2}{n}}$$

Example 1●●

Find the standard deviation of $6, 8, 10, 12, 14$.

Mean $= 10$. Deviations: $-4, -2, 0, 2, 4$. Squares: $16, 4, 0, 4, 16$.

$\sigma = \sqrt{\dfrac{40}{5}} = \sqrt{8} \approx 2.83$

Flashcards

Small standard deviation

Data close to the mean

Calculator key

$\sigma_x$

Add 5 to every value

Mean goes up 5; standard deviation unchanged.

Practice

Standard deviation of $6, 8, 10, 12, 14$ (2 d.p.)?
Standard deviation of $2, 4, 4, 4, 5, 5, 7, 9$ (population)?
Standard deviation of $3, 7, 8, 10, 12$ (2 d.p.)?
Standard deviation of $5, 5, 5, 5$?
Team A heights: $\sigma = 4$ cm. Team B: $\sigma = 9$ cm. Which team's heights vary more?
Every value in a data set is increased by $5$. The standard deviation:
Every value is doubled. The standard deviation:
Standard deviation of $10, 12, 15, 18, 20$ (2 d.p.)?