The standard deviation

Finance and economics, grade 11 (3 units) Lesson 9 of 14Learning

The standard deviation

Two funds with the same mean

The mean alone does not tell how widely the data are spread round it. For that there is a measure of spread.

The return of a savings fund in a year is the percent by which the money in the fund grew that year (a negative return – the money shrank).
The returns of two funds over the last five years, in percent:

The mean of fund A: $\displaystyle {\frac{4+5+6+5+5}{5}}={5}$ .
The mean of fund B: $\displaystyle {\frac{-2+12+3+10+2}{5}}={5}$ .

With the same mean, fund A’s return hardly changes from year to year, and fund B’s jumps – from a year of loss to a year of big gain.

The range

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

\displaystyle x_{\max}-x_{\min}

Fund A: $\displaystyle {6-4}={2}$ .
Fund B: $\displaystyle {12-(-2)}={14}$ .

The range depends on two values alone, the extreme ones. One outlier makes it much larger, even when all the others are close together.

The deviation from the mean

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

The deviations in fund A: $\displaystyle -1,\ 0,\ 1,\ 0,\ 0$ .

The deviations from the mean always add up to 0 – those above and those below cancel each other out. So each deviation is squared: a square is always positive (or 0), and a large deviation gives a very large square.

The variance and the standard deviation

The variance is the mean of the squared deviations, and the standard deviation $\displaystyle \sigma$ is its square root. The standard deviation is about the usual distance of a value from the mean, in the same units as the data.

\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}}

The work for fund B (the mean is 5):

The return $\displaystyle x$	The deviation $\displaystyle x-\bar{x}$	The deviation squared
$\displaystyle -2$	$\displaystyle -7$	49
12	7	49
3	$\displaystyle -2$	4
10	5	25
2	$\displaystyle -3$	9
	Sum: 0	Sum: 136

The variance: $\displaystyle {\frac{136}{5}}={27.2}$ .
The standard deviation: $\displaystyle {\sigma}={\sqrt{27.2}\approx5.22}$ percent.

In fund A the squared deviations are $\displaystyle 1, 0, 1, 0, 0$ : the variance is $\displaystyle {\frac{2}{5}}={0.4}$ , and the standard deviation $\displaystyle \sqrt{0.4}\approx0.63$ percent. Fund B’s standard deviation is about 8 times as large: its returns are far more spread out.

The standard deviation from a frequency table

In a frequency table each squared deviation is counted as many times as its value appears – it is multiplied by the frequency.

The number of shifts a week of 10 workers in a restaurant:

Shifts	3	4	5
Workers	2	5	3

The mean: $\displaystyle {\frac{3\cdot2+4\cdot5+5\cdot3}{10}}={\frac{41}{10}}={4.1}$ .
The deviations: $\displaystyle -1.1$ , $\displaystyle -0.1$ , $\displaystyle 0.9$ .
The variance: $\displaystyle {\frac{1.21\cdot2+0.01\cdot5+0.81\cdot3}{10}}={\frac{4.9}{10}}={0.49}$ .
The standard deviation: $\displaystyle {\sigma}={\sqrt{0.49}}={0.7}$ shifts.

Exercise 1

The price of the same packet of coffee in five shops, in shekels:
$\displaystyle 10,\ 12,\ 11,\ 9,\ 13$

Work out the mean price.

/ – 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