Let , R.

Question

Let $\alpha ,\beta \in R$. 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 :

Accepted Answer

$$\begin{aligned}
& \text { Mean }=\frac{-3+4+7+(-6)+\alpha+\beta}{6}=2 \
& \Rightarrow \alpha+\beta=10 \
& \text { Variance }=\frac{\sum x_i^2}{n}-\left(\frac{\bar{x}}{n}\right)^2=23 \
& \Rightarrow \sum x_i^2=27 \times 6 \
& \Rightarrow 9+16+49+36+\alpha^2+\beta^2=162 \
& \Rightarrow \alpha^2+\beta^2=52
\end{aligned}$$

We get $$\alpha$$ and $$\beta$$ as 4 and 6

So, mean deviation about mean

$$\begin{aligned}
& =\frac{|-3-2|+|4-2|+|7-2|+|-6-2|+|4-2|+|6-2|}{6} \
& =\frac{5+2+5+8+2+4}{6} \
& =\frac{13}{3}
\end{aligned}$$

Solutions · 1