Question

How do you find the median of a set of values when there is an even number of values?

Answer 1

Hint: We will use the concepts of measurement of central tendency to solve this problem related to statistics. We will define all the three methods of measuring central tendency with a few examples and through them, we will try to solve this problem.

Complete step by step solution:
In mathematics and mainly in statistics, there are three ways of measuring central tendency of a given set of values (also called as observations). Those are:
(1) Mean (2) Median (3) Mode
Consider a set of \[n\] observations which are \[{a_1},{a_2},{a_3},{a_4},{a_5},........,{a_n}\]
For these observations, let us find the central tendency value.
Mean:
It is the average value of all observations and is equal to the sum of all observations divided by number of observations. It is denoted by \[\overline x \].
So, mathematically, we write it as,
\[\overline x = \dfrac{{{\text{sum of observations}}}}{{{\text{no}}{\text{.of observations}}}} = \dfrac{{{a_1} + {a_2} + {a_3} + {a_4} + {a_5} + ........ + {a_n}}}{n}\]
Median:
It is defined as the middle value of a given set of observations.
To find the median, first we need to arrange the set of observations in increasing order.
Let us assume that, the given set \[{a_1},{a_2},{a_3},{a_4},{a_5},........,{a_n}\] is in increasing order.
So, median is equal to
(i) \[{\left( {\dfrac{{n + 1}}{2}} \right)^{th}}\] term, if \[n\] is an odd number.
(ii) Average of \[{\left( {\dfrac{n}{2}} \right)^{th}}\] term and \[{\left( {\dfrac{n}{2} + 1} \right)^{th}}\] term, if \[n\] is an even number.
Mode:
It is defined as the most repeated observation in the given set of observations.
If in the set \[{a_1},{a_2},{a_3},{a_4},{a_5},........,{a_n}\], suppose that \[{a_2}\] value is repeating for 4 times and no other term is not repeating, then mode is equal to \[{a_2}\].
Let us take another example and solve it.
Consider the set of observations \[8,2,5,5,9,2,5,4\]
Here, mean is equal to \[\overline x = \dfrac{{8 + 2 + 5 + 5 + 9 + 2 + 5 + 4}}{8} = \dfrac{{40}}{8} = 5\]
In the given set, 5 is repeated three times. So, mode is equal to 5.
There are eight observations. After arranging them in increasing order, we get, \[2,2,4,5,5,5,8,9\].
As there are even number of observations, median is equal to Average of \[{\left( {\dfrac{n}{2}} \right)^{th}}\] term and \[{\left( {\dfrac{n}{2} + 1} \right)^{th}}\] term.
\[ \Rightarrow {\text{median }} = {\text{ avg}}{\text{. of }}{\left( {\dfrac{8}{2}} \right)^{th}}{\text{term and }}{\left( {\dfrac{8}{2} + 1} \right)^{th}}{\text{term}}\]
\[ \Rightarrow {\text{median }} = {\text{ avg}}{\text{. of }}{{\text{4}}^{th}}{\text{ term and }}{{\text{5}}^{th}}{\text{ term}}\]
So, we can write it as,
\[ \Rightarrow {\text{median }} = {\text{ }}\dfrac{{{4^{th}}term + {5^{th}}term}}{2}\]
\[ \Rightarrow {\text{median }} = {\text{ }}\dfrac{{5 + 5}}{2} = \dfrac{{10}}{2} = 5\]
So, Median is equal to 5.

Note:
A set of observations is also called ‘data’. They can be any numbers such as natural numbers, whole numbers, integers and rational numbers too. There is another method for finding the mean of a data, which is the deviation method. Here we assume a mean from the given data, and find deviations and then add the assumed mean and mean of deviations, to get the mean.