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

The six observations are $$4,\,7,\,2,\,8,\,6$$ and $$a$$, whose mean is given to be $$7$$.

Mean formula: $$\text{Mean}=\dfrac{\text{Sum of observations}}{\text{Number of observations}}$$

Hence $$\dfrac{4+7+2+8+6+a}{6}=7 \; \Rightarrow \; 27+a=42 \; \Rightarrow \; a=15$$.

Including $$a$$, the data set becomes $$2,\,4,\,6,\,7,\,8,\,15$$ (arranged in ascending order).

For an even number $$n=6$$ of observations,

$$\text{Median}= \dfrac{\text{3rd term}+\text{4th term}}{2}= \dfrac{6+7}{2}=6.5$$.

Compute the absolute deviations from the median:

$$|2-6.5|=4.5,\; |4-6.5|=2.5,\; |6-6.5|=0.5,\; |7-6.5|=0.5,\; |8-6.5|=1.5,\; |15-6.5|=8.5$$

Sum of these deviations $$=4.5+2.5+0.5+0.5+1.5+8.5=18$$.

Mean deviation from the median is therefore $$\dfrac{18}{6}=3$$.

Option D which is: $$3$$