Let the mean and the standard deviation of the observation 2, 3, 3, 4, 5, 7, a, b be 4 and respectively. Then the mean deviation about the mode of these observations is : [2025]

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Solution

Q.
Let the mean and the standard deviation of the observation 2, 3, 3, 4, 5, 7, a, b be 4 and $\sqrt{2}$ respectively. Then the mean deviation about the mode of these observations is : [2025]

1 1

2 $\frac{3}{4}$

3 2

4 $\frac{1}{2}$

Ans.
(1)

Given observations are 2, 3, 3, 4, 5, 7, a, b.

Also, mean = 4 and S.D. = $\sqrt{2}$

$\Rightarrow \frac{24 + a + b}{8} = 4 \Rightarrow a + b = 8$

${(\sqrt{2})}^{2} = \frac{2^{2} + 3^{2} + 3^{2} + 4^{2} + 5^{2} + 7^{2} + a^{2} + b^{2}}{8} - 16$

$\Rightarrow 112 + a^{2} + b^{2} = 18 \times 8$

$\Rightarrow a^{2} + b^{2} = 32 \Rightarrow a = b = 4$

New numbers are 2, 3, 3, 4, 5, 7, 4, 4

Arrange the numbers in a ascending order

2, 3, 3, 4, 4, 4, 5, 7

So, mode = 4.

$∴$ Mean deviation about mode

$= \frac{2 + 1 + 1 + 0 + 1 + 3 + 0 + 0}{8} = 1$ .

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