The standard deviation

Science and society, grade 11 (3 units) Lesson 9 of 16Learning

The standard deviation

The standard deviation in a table, step by step

The standard deviation measures how far the data are from the mean, on average. It is worked out in five steps, conveniently written in a table.

For eight days Maya wrote down how many minutes late her bus was:
2, 4, 4, 4, 5, 5, 7, 9.

The mean: $\displaystyle {\bar{x}}={\frac{2+4+4+4+5+5+7+9}{8}}={\frac{40}{8}}={5}$ .
The deviation of each datum from the mean: $\displaystyle x-\bar{x}$ .
Each deviation squared: $\displaystyle (x-\bar{x})^2$ – all the squares are positive or 0.
The mean of the squares: $\displaystyle {\frac{32}{8}}={4}$ . This number is called the variance.
The square root of the variance: $\displaystyle {\sigma}={\sqrt{4}}={2}$ .

$\displaystyle x$	2	4	4	4	5	5	7	9	Sum
$\displaystyle x-\bar{x}$	$\displaystyle -3$	$\displaystyle -1$	$\displaystyle -1$	$\displaystyle -1$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 2$	$\displaystyle 4$	$\displaystyle 0$
$\displaystyle (x-\bar{x})^2$	9	1	1	1	0	0	4	16	32

The standard deviation of the delays is 2 minutes.

The formula of the standard deviation

The five steps are written as one formula. The standard deviation is written with the Greek letter $\displaystyle \sigma$ (sigma):

\displaystyle \sigma=\sqrt{\frac{(x_1-\bar{x})^2+(x_2-\bar{x})^2+\dots+(x_n-\bar{x})^2}{n}}

In the numerator – the sum of the squared deviations;
$\displaystyle n$ – the number of data;
what is under the root is the variance.

Noa’s points in the last lesson: 14, 16, 12, 18, 15, with a mean of 15 and deviations $\displaystyle -1, 1, -3, 3, 0$ :

\sigma=\sqrt{\frac{1+1+9+9+0}{5}}=\sqrt{\frac{20}{5}}=\sqrt{4}=2

A rounded standard deviation

Most of the time the variance is not the square of a whole number, and the standard deviation comes out as a long decimal. It is rounded, usually to two decimal places, and written with $\displaystyle \approx$ .

The morning temperature, in degrees, on four days:
18, 20, 21, 25.

The mean: $\displaystyle {\frac{84}{4}}={21}$ .
The deviations: $\displaystyle -3, -1, 0, 4$ ; the squares: $\displaystyle 9, 1, 0, 16$ .

\sigma=\sqrt{\frac{9+1+0+16}{4}}=\sqrt{6.5}\approx2.55

The standard deviation is about 2.55 degrees.

The units of the standard deviation

The standard deviation is measured in the same units as the data: minutes, degrees, points.

Squaring turns minutes into “minutes squared” – that is why the variance is hard to interpret.
The square root takes us back to the data’s units.
The standard deviation is always 0 or a positive number. It is 0 only when all the data are equal.

A scientific calculator has a statistics mode (STAT): type in the data, and it gives the mean $\displaystyle \bar{x}$ and the standard deviation – named $\displaystyle \sigma_x$ or $\displaystyle \sigma_n$ (not $\displaystyle s_x$ , which is worked out differently).

Exercise 1

Noam’s daily screen time, in hours, on four days:
1, 3, 5, 7.

Work out the mean.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power