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Question
Find variance and S.D. for the following set of numbers:
65, 77, 81, 98, 100, 80, 129
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Solution 1
We construct the following table to compute variance and S.D.:
| xi | `x_"i" - bar(x)` | `(x_"i" - bar(x))^2` |
| 65 | – 25 | 625 |
| 77 | – 13 | 169 |
| 81 | – 9 | 81 |
| 98 | 8 | 64 |
| 100 | 10 | 100 |
| 80 | – 10 | 100 |
| 129 | 39 | 1521 |
| `sumx_"i"` = 630 | `sum(x_"i" - barx)^2` = 2660 |
From the table, `sumx_"i"` = 630, n = 7
∴ `barx = (sumx_"i")/"n" = 630/7` = 90
Var (X) = `(sum(x_"i" - barx)^2)/"n" = 2660/7` = 380
∴ S.D. = `sqrt("Var(X)")`
= `sqrt(380)`
= 19.49
Solution 2
| xi |
ui = xi – A |
ui2 |
| 65 | – 33 | 1089 |
| 77 | – 21 | 441 |
| 81 | – 17 | 289 |
| 98 | 0 | 0 |
| 100 | 2 | 4 |
| 80 | – 18 | 324 |
| 129 | 31 | 961 |
| `sumu_"i"` = – 56 | `sumu_"i"^2` = 3108 |
From the table, `sumu_"i"` = – 56, `sumu_"i"^2` = 3108, n = 7
∴ Var(x) = Var(u) = `(sumu_"i"^2)/"n" - ((sumu_"i")/"n")^2`
= `3108/7 - ((-56)/7)^2`
= 444 – 64
= 380
S.D.(x) = `sqrt("Var(x)")`
= `sqrt(380)`
= 19.49
Hence, variance = 380, S.D. = 19.49
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