The range, the variance and the standard deviation

Statistics and probability, grade 10 (4 units) Lesson 10 of 20Learning

The range, the variance and the standard deviation

Measures of spread

A measure of centre says around which value the data lie; a measure of spread says how spread out they are around it – close to each other or far apart.

Two groups of five students sat the same test.
Group A: 68, 69, 70, 71, 72.
Group B: 50, 60, 70, 80, 90.

In both groups the mean is 70 and the median is 70.
In group A all the marks are close to 70; in group B they are up to 20 points away from it.
The measures of centre do not show this difference. The measures of spread – the range, the variance and the standard deviation – show it.

The range

The range is the difference between the greatest value and the least:

\displaystyle x_{max}-x_{min}

Group A: $\displaystyle {72-68}={4}$ points.
Group B: $\displaystyle {90-50}={40}$ points.

The range’s advantages and limits:

It is simple and quick to work out, and it fits our feeling for what spread is.
It rests on two values only, the extreme ones: one outlier changes it a lot, and the rest of the data do not affect it.
It cannot be worked out for a qualitative variable, or when an end class is open (“80 and over”).

The deviations from the mean

The deviation of a datum from the mean is $\displaystyle x-\bar{x}$ : positive for a datum above the mean, negative for one below it.

In group B the mean is 70, and the deviations are:

-20,\ -10,\ 0,\ 10,\ 20

The deviations from the mean always add up to 0: the positive and negative ones cancel out. So the mean of the deviations does not measure spread, and each deviation is squared – the square of a number is always 0 or positive.

The variance and the standard deviation

The variance is the mean of the squared deviations from the mean. The standard deviation is the square root of the variance. It is written with the Greek letter $\displaystyle \sigma$ (sigma), and the variance $\displaystyle \sigma^2$ .

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

\displaystyle \sigma=\sqrt{\sigma^2}

In group B:

$\displaystyle x$	50	60	70	80	90	Sum
$\displaystyle x-\bar{x}$	$\displaystyle -20$	$\displaystyle -10$	$\displaystyle 0$	$\displaystyle 10$	$\displaystyle 20$	$\displaystyle 0$
$\displaystyle (x-\bar{x})^2$	400	100	0	100	400	1000

\sigma^2=\frac{1000}{5}=200,\qquad\sigma=\sqrt{200}\approx14.14

In group A the deviations are $\displaystyle -2, -1, 0, 1, 2$ , and the squares add up to 10:

\sigma^2=\frac{10}{5}=2,\qquad\sigma=\sqrt{2}\approx1.41

Group B’s standard deviation is about 10 times as large: its marks are far more spread out.

The properties of the standard deviation

The standard deviation rests on all the data, which is why it is the usual measure of spread.

The standard deviation is measured in the same units as the data: points, minutes, shekels. The variance’s units are their square – “points squared” – which makes it hard to interpret.
$\displaystyle \sigma\ge0$ , and $\displaystyle {\sigma}={0}$ only when all the data are equal.
Data far from the mean count for more than their weight: a deviation of 20 adds 400 to the sum, a deviation of 2 only 4.
On a scientific calculator, in statistics mode (STAT), this standard deviation is called $\displaystyle \sigma_x$ or $\displaystyle \sigma_n$ .

Exercise 1

The noon temperature, in degrees, on five days:
14, 18, 20, 22, 26.

Work out the range.

/ – 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