Question

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

• A. 8
• B. 5
• C. 1
• D. 3

Answer 1

Answer: (D)

Given observations are 4, 7, 2, 8, 6, a and mean is 7.
We know
Mean = $\frac{4 + 7 + 2 + 8 + 6 + a }{6}$
$\Rightarrow \, 7 = \frac{4 + 7 + 2 + 8 + 6 + a}{6} \Rightarrow \, a = 15$
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} \,\text{observation} + \left(\frac{6}{2} +1\right)\text{th} \,\text{observation}}{2}$
$ = \frac{3\text{rd}\, \text{observation} + 4\text{th} \,\text{observation}}{2} $
$= \frac{6+7}{2} = \frac{13}{2}$
Now, Mean deviation $ = \frac{\sum\limits^{6}_{i=1} \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 $