Let R. Let the mean and the variance of 6 observations -3, 4, 7, -6, be 2 and 23, respectively. The mean deviation about the mean of these 6 observations is

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Solution

Q.
Let α, β ∈ R. Let the mean and the variance of 6 observations -3, 4, 7, -6, α, β be 2 and 23, respectively. The mean deviation about the mean of these 6 observations is [2024]

1 16/3

2 13/3

3 11/3

4 14/3

Ans.
(2)

$Mean = \frac{- 3 + 4 + 7 + (- 6) + α + β}{6} = 2$

$\Rightarrow α + β = 10$ ...(i)

$Variance = \frac{\sum x_{i}^{2}}{n} - {(\frac{\sum x_{i}}{n})}^{2} = 23$

$\Rightarrow \sum x_{i}^{2} = 27 \times 6$ $(∵ \bar{x} = \frac{\sum_{x_{i}}}{n} = 2)$

$\Rightarrow 9 + 16 + 49 + 36 + α^{2} + β^{2} = 162$

$\Rightarrow α^{2} + β^{2} = 52$ ...(ii)

Solving (i) and (ii), we get $α = 4$ and $β = 6$

So, mean deviation about mean $= \frac{| - 3 - 2 | + | 4 - 2 | + | 7 - 2 | + | - 6 - 2 | + | 4 - 2 | + | 6 - 2 |}{6}$

$= \frac{5 + 2 + 5 + 8 + 2 + 4}{6}$ $= \frac{13}{3}$

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