The standard deviation from a frequency table

Statistics and probability, grade 10 (4 units) Lesson 11 of 20Learning

The standard deviation from a frequency table

In a frequency table each value comes up several times, so each squared deviation goes into the sum several times: it is multiplied by the value’s frequency, and the sum is divided by the sum of the frequencies.

\displaystyle \sigma^2=\frac{(x_1-\bar{x})^2f_1+\dots+(x_k-\bar{x})^2f_k}{f_1+\dots+f_k}

The number of fouls of 40 high jumpers in a competition:

Number of fouls $\displaystyle x$	0	1	2	3	4	Sum
Frequency $\displaystyle f$	4	16	9	6	5	40
$\displaystyle x\cdot f$	0	16	18	18	20	72
$\displaystyle x-\bar{x}$	$\displaystyle -1.8$	$\displaystyle -0.8$	$\displaystyle 0.2$	$\displaystyle 1.2$	$\displaystyle 2.2$
$\displaystyle (x-\bar{x})^2\cdot f$	12.96	10.24	0.36	8.64	24.2	56.4

The mean: $\displaystyle {\bar{x}}={\frac{72}{40}}={1.8}$ fouls.
The variance: $\displaystyle {\sigma^2}={\frac{56.4}{40}}={1.41}$ .
The standard deviation: $\displaystyle {\sigma}={\sqrt{1.41}\approx1.19}$ fouls.

The standard deviation of data in classes

For data in classes the values themselves are not known. As for the mean, all the data in a class are taken to be at the class’s middle, and the work is done with the middles. The result is an estimate of the standard deviation.

The heights of 30 students, in cm:

Height	140–150	150–160	160–170	170–180	180–190	Sum
The class’s middle $\displaystyle m$	145	155	165	175	185
Frequency $\displaystyle f$	2	8	10	8	2	30
$\displaystyle m-\bar{x}$	$\displaystyle -20$	$\displaystyle -10$	$\displaystyle 0$	$\displaystyle 10$	$\displaystyle 20$
$\displaystyle (m-\bar{x})^2\cdot f$	800	800	0	800	800	3200

The estimated mean: $\displaystyle {\bar{x}\approx\frac{145\cdot2+155\cdot8+165\cdot10+175\cdot8+185\cdot2}{30}}={\frac{4950}{30}}={165}$ cm.
The variance: $\displaystyle \sigma^2\approx\frac{3200}{30}\approx106.67$ .
The standard deviation: $\displaystyle \sigma\approx\sqrt{106.67}\approx10.33$ cm.

When one of the end classes is open (“190 and over”), it has no middle, and neither the mean nor the standard deviation can be worked out.

Comparing the spread in two classes

Two bar charts on the same axes show how the marks are distributed in two classes of 20 students:

Both distributions are symmetric about 7, so in both classes the mean is 7, and so are the median and the mode.
In class 10-1 the marks are gathered around 7: from 5 to 9.
In class 10-2 they are spread from 4 to 10.

The standard deviation in class 10-1: the deviations of 5, 6, 7, 8, 9 from the mean are $\displaystyle -2, -1, 0, 1, 2$ , and the frequencies $\displaystyle 2, 4, 8, 4, 2$ .

\sigma^2=\frac{4\cdot2+1\cdot4+0\cdot8+1\cdot4+4\cdot2}{20}=\frac{24}{20}=1.2,\qquad\sigma=\sqrt{1.2}\approx1.10

Exercise 1

20 students were asked how many brothers and sisters they have:

Number of siblings	0	1	2	3
Frequency	5	10	3	2

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