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Question
Find variance and S.D. for the following set of numbers:
7, 11, 2, 4, 9, 6, 3, 7, 11, 2, 5, 8, 3, 6, 8, 8, 2, 6
Solution
Tabulated form of the above data is
| xi | fi |
| 2 | 3 |
| 3 | 2 |
| 4 | 1 |
| 5 | 1 |
| 6 | 3 |
| 7 | 2 |
| 8 | 3 |
| 9 | 1 |
| 11 | 2 |
Calculation of variance and S.D.;
| xi | fi | fixi | xi2 | fixi2 |
| 2 | 3 | 6 | 4 | 12 |
| 3 | 2 | 6 | 9 | 18 |
| 4 | 1 | 4 | 16 | 16 |
| 5 | 1 | 5 | 25 | 25 |
| 6 | 3 | 18 | 36 | 108 |
| 7 | 2 | 14 | 49 | 98 |
| 8 | 3 | 24 | 64 | 192 |
| 9 | 1 | 9 | 81 | 81 |
| 11 | 2 | 22 | 121 | 242 |
| Total | N = 18 | `sumf_"i"x_"i"` = 108 | `sumf_"i"x_"i"^2` = 792 |
`bar(x) = (sumf_"i"x_"i")/"N" = 108/18` = 6
Var (X) = `sigma_x^2 = (sumf_"i"x_"i"^2)/"N" - (barx)^2`
= `792/18 - (6)^2`
= 44 – 36
= 8
∴ S.D. = `sigma_x = sqrt("Var"(X)`
= `sqrt(8)`
= 2.82
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