Mathematics - Statistics Question with Solution | TestHub

Given observations are 4, 7, 2, 8, 6, $a$ and mean is 7. We know $$\begin{array}{rl}& \text{ Mean }=\frac{4+7+2+8+6+a}{6}\\ & \Rightarrow 7=\frac{4+7+2+8+6+a}{6}\Rightarrow a=15\end{array}$$ Now, given observations can be written in ascending order which is $2,4,6,7,8,15$ Since, No. of observation is even $\therefore$ Median $=\frac{\left(\frac{6}{2}\right)\text{ th observation }+\left(\frac{6}{2}+1\right)\text{ th observation }}{2}$ $=\frac{\text{ 3rd observation }+4\text{ th observation }}{2}$ $=\frac{6+7}{2}=\frac{13}{2}$ $$\begin{array}{rl}& \text{ Now, Mean deviation }=\frac{{\sum }_{i=1}^6\left|x_i-\frac{13}{2}\right|}{6}\\ & =\frac{\left|4-\frac{13}{2}\right|+\left|7-\frac{13}{2}\right|+\left|2-\frac{13}{2}\right|+\left|8-\frac{13}{2}\right|+\left|6-\frac{13}{2}\right|+\left|15-\frac{13}{2}\right|}{6}\\ & =\frac{\frac{5}{2}+\frac{1}{2}+\frac{9}{2}+\frac{3}{2}+\frac{1}{2}+\frac{17}{2}}{6}=\frac{18}{6}=3\\ & \end{array}$$