Let $\alpha, \beta \in \mathbf{R}$. Let the mean and the variance of 6 observations $-3,4,7,-6, \alpha, \beta$ be 2 and 23, respectively. The mean deviation about the mean of these 6 observations is :
Solution
<p>$$\begin{aligned}
& \text { Mean }=\frac{-3+4+7+(-6)+\alpha+\beta}{6}=2 \\
& \Rightarrow \alpha+\beta=10 \\
& \text { Variance }=\frac{\sum x_i^2}{n}-\left(\frac{\bar{x}}{n}\right)^2=23 \\
& \Rightarrow \sum x_i^2=27 \times 6 \\
& \Rightarrow 9+16+49+36+\alpha^2+\beta^2=162 \\
& \Rightarrow \alpha^2+\beta^2=52
\end{aligned}$$</p>
<p>We get $\alpha$ and $\beta$ as 4 and 6</p>
<p>So, mean deviation about mean</p>
<p>$$\begin{aligned}
& =\frac{|-3-2|+|4-2|+|7-2|+|-6-2|+|4-2|+|6-2|}{6} \\
& =\frac{5+2+5+8+2+4}{6} \\
& =\frac{13}{3}
\end{aligned}$$</p>
About this question
Subject: Mathematics · Chapter: Statistics · Topic: Measures of Central Tendency
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