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.