If the variance of the frequency distribution x 2 3 4 5 6 7 8 Frequency f i 3 6 16 9 5 6 is 3, then is equal to _______. [2023]

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Solution

Q.
If the variance of the frequency distribution

$x_{𝑖}$ 2 3 4 5 6 7 8

Frequency $f_{i}$ 3 6 16 $α$ 9 5 6

is 3, then $α$ is equal to _______. [2023]

Ans.
(5)

$x_{i}$ $f_{i}$ $d_{i} = x_{i} - 5$ $d_{i}^{2}$ $f_{i} d_{i}^{2}$ $f_{i} d_{i}$

2 3 - 3 9 27 - 9

3 6 - 2 4 24 - 12

4 16 - 1 1 16 - 16

5 $α$ 0 0 0 0

6 9 1 1 9 9

7 5 2 4 20 10

8 6 3 9 54 18

Total $45 + α$ 150 0

$σ^{2} = \frac{\sum f_{i} d_{i}^{2}}{\sum f_{i}} - {(\frac{\sum f_{i} d_{i}}{\sum f_{i}})}^{2} = \frac{150}{45 + α} - {(\frac{0}{45 + α})}^{2}$

$\Rightarrow 3 = \frac{150}{45 + α} (∵ variance = 3)$

$\Rightarrow 135 + 3 α = 150 \Rightarrow 3 α = 15 \Rightarrow α = 5$

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