Let the mean of the data be 5. If and are respectively the mean deviation about the mean and the variance of the data, then is equal to ________ . [2023]

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Solution

Q.
Let the mean of the data

$x$ 1 3 5 7 9

Frequency $(f)$ 4 24 28 $α$ 8

be 5. If $m$ and $σ^{2}$ are respectively the mean deviation about the mean and the variance of the data, then $\frac{3 α}{m + σ^{2}}$ is equal to ________ . [2023]

Ans.
(8)

Given, $\bar{x} = 5$

$\Rightarrow \frac{\sum x_{i} f_{i}}{\sum f_{i}} = \frac{4 + 72 + 140 + 7 α + 72}{64 + α} = 5$

$\Rightarrow 288 + 7 α = 320 + 5 α \Rightarrow 2 α = 32 \Rightarrow α = 16$

Now, $M.D. (\bar{x}) = \frac{\sum f_{i} | x_{i} - \bar{x} |}{\sum f_{i}}$

$= \frac{4 \times 4 + 24 \times 2 + 28 \times 0 + 16 \times 2 + 8 \times 4}{80} = \frac{128}{80} = \frac{8}{5} \Rightarrow m = \frac{8}{5}$

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

$= \frac{4 \times 16 + 24 \times 4 + 28 \times 0 + 16 \times 4 + 8 \times 16}{80} = \frac{352}{80} = \frac{22}{5}$

$∴$ $\frac{3 α}{m + σ^{2}} = \frac{3 \times 16}{\frac{8}{5} + \frac{22}{5}} = \frac{48}{6} = 8$

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