Question

For the values of \[{x_1},\;{x_2},{\text{ }} \ldots \ldots \ldots .{x_{101}}\] of a distribution ${x_1} < {x_2} < {x_3} \ldots \ldots . < {x_{100}} < {x_{101}}$. The mean deviation of this distribution with respect to a number $k$ will be minimum when $k$ is equal to –
a) ${x_1}$
b) ${x_{51}}$
c) ${x_{50}}$
d) $\dfrac{{{x_1} + {x_2} + ....... + {x_{101}}}}{{101}}$

Answer 1

Hint: In this question, they give one distribution. From that distribution we have to choose which term is equal to k when the mean deviation of this distribution with respect to k will be minimum. We solve this problem by taking k as the median of the given observation.

Complete step-by-step answer:
As we know mean deviation is minimum when median has been taken.
Therefore, here $k$ is taken as the median of the given observations.
Total number of observations in the question is $101$.
Now we have to apply the formula of median which is $\dfrac{{n + 1}}{2}$
Since $k$ is the median and $n = 101$ in this question therefore we can write-
$ \Rightarrow k = \dfrac{{n + 1}}{2}$
By substituting the total number of observations,
$ \Rightarrow k = \dfrac{{101 + 1}}{2}$
Adding the terms we get,
$ \Rightarrow k = \dfrac{{102}}{2}$
Dividing the terms we get,
$ \Rightarrow k = 51$
Hence, $k = 51^{th}$ observation
$\therefore $ Thus, $k = {x_{51}}$

So, the correct answer is “Option b”.

Note: Mean, median and mode are the three kinds of averages which have wide application in statistics.
Median is generally used to find the middle value or centre of a set of data or information.
It can also be used for an open-end distribution and it is more useful than mean.
The main formula which is used for finding out the value of median for an odd discrete series is $\dfrac{{n + 1}}{2}$ where n represents the total number of data or numbers available in a distribution and the formula for even discrete series is $\dfrac{1}{2}[\dfrac{n}{2} + (\dfrac{n}{2} + 1)]$
It is only applicable in quantitative data but not qualitative.
Value of the median is not dependent on all the values of the data in a data set and it does not depend on the individual value of a particular data of a set.