Measures of Dispersion (Part 3)

Measures of Dispersion (Part 3)
7.3 Variance and Standard Deviation

1. The variance is a measure of the mean for the square of the deviations from the mean.

2. The standard deviation refers to the square root for the variance.

Example 1:

Solution:
$\begin{array}{l}\text{Variance, }{\sigma }^2=\frac{\sum x^2}{N}-{\overline{x}}^2\\ {\sigma }^2=\frac{{15}^2+{17}^2+{21}^2+{24}^2+{31}^2}{5}\\ -{\left(\frac{15+17+21+24+31}{5}\right)}^2\\ {\sigma }^2=\frac{2492}{5}-{21.6}^2\\ {\sigma }^2=31.84\\ \\ \text{Standard deviation, }\sigma \text{ = }\sqrt{\text{variance}}\\ \sigma \text{ = }\sqrt{31.84}\\ \sigma \text{ = }5.642\end{array}$

The data below shows the numbers of children of 30 families:

Solution:
$\begin{array}{l}\text{Mean }\overline{x}=\frac{\sum fx}{\sum f}\\ =\frac{\left(6\right)\left(2\right)+\left(8\right)\left(3\right)+\left(5\right)\left(4\right)+\left(3\right)\left(5\right)+\left(3\right)\left(6\right)+\left(3\right)\left(7\right)+\left(2\right)\left(8\right)}{6+8+5+3+3+3+2}\\ =\frac{126}{30}=4.2\\ \\ \frac{\sum f{x}^2}{\sum f}\\ =\frac{\left(6\right){\left(2\right)}^2+\left(8\right){\left(3\right)}^2+\left(5\right){\left(4\right)}^2+\left(3\right){\left(5\right)}^2+\left(3\right){\left(6\right)}^2+\left(3\right){\left(7\right)}^2+\left(2\right){\left(8\right)}^2}{6+8+5+3+3+3+2}\\ =\frac{634}{30}=21.13\\ \\ \text{Variance, }{\sigma }^2=\frac{\sum f{x}^2}{\sum f}-{\overline{x}}^2\\ {\sigma }^2=21.133-{4.2}^2\\ {\sigma }^2=3.493\\ \\ \text{Standard deviation, }\sigma \text{ = }\sqrt{\text{variance}}\\ \sigma \text{ = }\sqrt{3.493}\\ \sigma \text{ = 1}\text{.869}\end{array}$

Find the mean of daily salary and its standard deviation.

Daily Salary (RM)	Number of workers, f	Midpoint, x	fx	fx2
10 – 14	40	12	480	5760
15 – 19	25	17	425	7225
20 – 24	15	22	330	7260
25 – 29	12	27	324	8748
30 – 34	8	32	256	8192
Total	100		1815	37185