Variance and Standard Deviation – Problems

Variance and Standard Deviation – Problems

1. Find the variance for the discrete data given below 6, 7, 10, 12, 13, 4, 8, 12.

Solution: 6, 7, 10, 12, 13, 4, 8, 12

Mean (x̄) = \(\frac{6+7+10+12+13+4+8+12}{8}=\frac{72}{8}\)= 9,

xᵢ	xᵢ – x̄	(xᵢ – x̄)²
6	6 – 9 = -3	9
7	7 – 9 = -2	4
10	10 – 9 = 1	1
12	12 – 9 = 3	9
13	13 – 9 = 4	16
4	4 – 9 = – 5	25
8	8 – 9 = -1	1
12	12 – 9 = 3	9
		∑(xᵢ – x̄)² = 74

Variance \(({{\sigma }^{2}})=\frac{\sum\limits_{i=1}^{8}{{{({{x}_{i}}-\overline{x})}^{2}}}}{n}=\frac{74}{8}=9.25\).

2. Find the variance and standard deviation of the following frequency distribution.

xᵢ	6	10	14	18	24	28	30
fᵢ	2	4	7	12	8	4	3

Solutions:

xᵢ	fᵢ	xᵢ fᵢ	xᵢ – x̄	(xᵢ – x̄)²	fᵢ (xᵢ – x̄)²
6	2	12	-13	169	338
10	4	40	-9	81	324
14	7	98	-5	25	175
18	12	216	-1	1	12
24	8	192	5	25	200
28	4	112	9	81	324
30	3	90	11	121	363
	N = 40	760			1736

Mean (x̄) = 760/ 40 = 19

Variance \(({{\sigma }^{2}})=\frac{\sum\limits_{i=1}^{8}{{{({{x}_{i}}-\overline{x})}^{2}}}}{N}\)= 1736/ 40 = 43.4,

Standard deviation (σ) = √(43.4) = 6.59.