Question

The coefficient of mean deviation from the median of observations 40, 62, 54, 90, 68, 76 is-
A. 2.16
B. 0.2
C. 5
D. None of the above

Answer 1

Hint: The mean deviation about the median is defined as the ratio of the difference between the median and the observations in the entry and the median. Mathematically, it can be defined as-
$MD = \mathop \sum \limits_{i = 1}^n \dfrac{{\left| {M - {a_i}} \right|}}{M}$
Here M is the median and ai’s are the observations.

Complete step-by-step solution-
First we will arrange the data in ascending order, which can be written as-
40, 54, 62, 68, 76, 90
The number of entries are 6(even), so the median is the average of the ${\left( {\dfrac{{\text{n}}}{2}} \right)^{th}}\;and\;{\left( {\dfrac{{\text{n}}}{2} + 1} \right)^{th}}$ terms.
So the median is the average of the 3rd and the 4th terms.
${\text{M}} = \dfrac{{62 + 68}}{2} = 65$
Now, we will find the mean deviation using the given formula as-
$\begin{gathered}
  MD = \dfrac{{\left| {40 - 65} \right| + \left| {54 - 65} \right| + \left| {62 - 65} \right| + \left| {68 - 65} \right| + \left| {76 - 65} \right| + \left| {90 - 65} \right|}}{{65}} \\
  MD = \dfrac{{25 + 11 + 3 + 3 + 11 + 25}}{{65}} = \dfrac{{78}}{{65}} = 1.2 \\
\end{gathered} $
This is the required answer. The correct option is D. None of the above.

Note: The most common mistake is the application of the formula of mean deviation in the question. Students forget to use the modulus in the formula and solve the summation incorrectly, this leads to a wrong answer.