If the arithmetic mean of the following distribution is $8.2$ , then $a$ is equal to
$x_{i}$ : 1 3 5 9 11 13
$f_{i}$ : 3 2 7 a 4 8

5

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6

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9

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none of these

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Solution

The correct option is B $6$
We have
mean $= \frac{1 \times 3 + 3 \times 2 + 5 \times 7 + 9 a + 11 \times 4 + 13 \times 8}{3 + 2 + 7 + a + 4 + 8} = 8.2$ (given)
$\Rightarrow \frac{3 + 6 + 35 + 9 a + 44 + 104}{24 + a} = 8.2$
$\Rightarrow 9 a + 192 = 24 \times 8.2 + 8.2 a \Rightarrow 0.8 a = 4.8 \Rightarrow a = 6$

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