# 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.