7.2.5 Variance and Standard Deviation

(A) Variance and Standard Deviation For Grouped Data (without Class Interval)

Example 2:

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

Number of child	2	3	4	5	6	7	8
Frequency	6	8	5	3	3	3	2

Find the variance and standard deviation of the data.

Solution:

\begin{array}{l} Mean \bar{x} = \frac{\sum f x}{\sum f} \\ = \frac{(6) (2) + (8) (3) + (5) (4) + (3) (5) + (3) (6) + (3) (7) + (2) (8)}{6 + 8 + 5 + 3 + 3 + 3 + 2} \\ = \frac{126}{30} = 4.2 \\ \frac{\sum f x^{2}}{\sum f} \\ = \frac{(6) {(2)}^{2} + (8) {(3)}^{2} + (5) {(4)}^{2} + (3) {(5)}^{2} + (3) {(6)}^{2} + (3) {(7)}^{2} + (2) {(8)}^{2}}{6 + 8 + 5 + 3 + 3 + 3 + 2} \\ = \frac{634}{30} = 21.13 \\ Variance, σ^{2} = \frac{\sum f x^{2}}{\sum f} - {\bar{x}}^{2} \\ σ^{2} = 21.133 - {4.2}^{2} \\ σ^{2} = 3.493 \\ Standard deviation, σ = \sqrt{variance} \\ σ = \sqrt{3.493} \\ σ = 1 .869 \end{array}

(B) Variance and Standard Deviation For Grouped Data (with Class Interval)

Example 3:

Find the mean of daily salary and its standard deviation.

Solution:

\begin{array}{l} Mean \bar{x} = \frac{\sum f x}{\sum f} \\ Mean of daily salary = \frac{1815}{100} = 18.15 \\ Variance, σ^{2} = \frac{\sum f x^{2}}{\sum f} - {\bar{x}}^{2} \\ Standard deviation, σ = \sqrt{variance} \\ σ^{2} = \frac{37185}{100} - {18.15}^{2} \\ σ^{2} = 42.43 \\ σ = \sqrt{42.43} \\ σ = 6 .514 \end{array}