The variance of the data is

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Solution

Q.
The variance $σ^{2}$ of the data

$x_{i}$ 0 1 5 6 10 12 17

$f_{i}$ 3 2 3 2 6 3 3

is ________ . [2024]

Ans.
(29)

Mean $(\bar{x})$ $= \frac{\sum f_{i} x_{i}}{\sum f_{i}}$

$= \frac{0 \times 3 + 1 \times 2 + 5 \times 3 + 6 \times 2 + 10 \times 6 + 12 \times 3 + 17 \times 3}{3 + 2 + 3 + 2 + 6 + 3 + 3}$

$= \frac{176}{22} = 8$

$∴$ Variance $σ^{2}$ $= \frac{\sum f_{i} {(x_{i} - \bar{x})}^{2}}{\sum f_{i}}$

$= \frac{3 {(0 - 8)}^{2} + 2 {(1 - 8)}^{2} + 3 {(5 - 8)}^{2} + 2 {(6 - 8)}^{2} + 6 {(10 - 8)}^{2} + 3 {(12 - 8)}^{2} + 3 {(17 - 8)}^{2}}{22}$

$= \frac{3 \times 64 + 2 \times 49 + 3 \times 9 + 2 \times 4 + 6 \times 4 + 3 \times 16 + 3 \times 81}{22}$

$= \frac{640}{22} = 29.09 \approx 29$

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