If the mean of 4,7,2,8,6 and a is 7 , then the mean deviation from the median of these observations is

Given observations are 4, 7, 2, 8, 6, $a$a and mean is 7. We know $$\begin{aligned} & \text { Mean }=\frac{4+7+2+8+6+a}{6} \\ & \Rightarrow 7=\frac{4+7+2+8+6+a}{6} \Rightarrow a=15 \end{aligned}$$​ Mean =64+7+2+8+6+a​⇒7=64+7+2+8+6+a​⇒a=15​ Now, given observations can be written in ascending order which is $2,4,6,7,8,15$2,4,6,7,8,15 Since, No. of observation is even $\therefore$∴ Median $=\frac{\left(\frac{6}{2}\right) \text { th observation }+\left(\frac{6}{2}+1\right) \text { th observation }}{2}$=2(26​) th observation +(26​+1) th observation ​ $=\frac{\text { 3rd observation }+4 \text { th observation }}{2}$=2 3rd observation +4 th observation ​ $=\frac{6+7}{2}=\frac{13}{2}$=26+7​=213​ $$\begin{aligned} & \text { Now, Mean deviation }=\frac{\sum_{i=1}^6\left|x_i-\frac{13}{2}\right|}{6} \\ & =\frac{\left|4-\frac{13}{2}\right|+\left|7-\frac{13}{2}\right|+\left|2-\frac{13}{2}\right|+\left|8-\frac{13}{2}\right|+\left|6-\frac{13}{2}\right|+\left|15-\frac{13}{2}\right|}{6} \\ & =\frac{\frac{5}{2}+\frac{1}{2}+\frac{9}{2}+\frac{3}{2}+\frac{1}{2}+\frac{17}{2}}{6}=\frac{18}{6}=3 \\ & \end{aligned}$$​ Now, Mean deviation =6∑i=16​​xi​−213​​​=6​4−213​​+​7−213​​+​2−213​​+​8−213​​+​6−213​​+​15−213​​​=625​+21​+29​+23​+21​+217​​=618​=3​