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Question
Compute the variance and S.D.
| X | 31 | 32 | 33 | 34 | 35 | 36 | 37 |
| Frequency | 15 | 12 | 10 | 8 | 9 | 10 | 6 |
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Solution
We prepare the following table to compute variance and S.D.:
| xi | fi | ui = xi – A A = 34 |
fiui | fiui2 = fiui × ui |
| 31 | 15 | – 3 | – 45 | 135 |
| 32 | 12 | – 2 | – 24 | 48 |
| 33 | 10 | – 1 | – 10 | 10 |
| 34 | 8 | 0 | 0 | 0 |
| 35 | 9 | 1 | 9 | 9 |
| 36 | 10 | 2 | 20 | 40 |
| 37 | 6 | 3 | 18 | 54 |
| `sumf_"i"` = 70 | `sumf_"i"u_"i"` = – 32 | `sumf_"i"u_"i"^2` = 296 |
From the table, `sumf_"i"u_"i" = -32, sumf_"i" = 70, sumf_"i"u_"i"^2` = 296
∴ `bar(u) = (sumf_"i"u_"i")/(sumf_"i") = (-32)/70` = – 0.4571
`sigma_u^2 = (sumf_"i"u_"i"^2)/(sumf_"i") - (baru)^2 = 296/70 - (-0.4571)^2`
= 4.2286 – 0.2089 = 4.0197
∴ Var (x) = `sigma_x^2 = sigma_u^2` = 4.0197
S.D. = `sigma_x = sqrt(4.0197)` = 2.005
Hence, variance = 4.0197 and S.D. = 2.005
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