Question

If the mean of 26, 19, 15, 24, x is x, then find the median of the data?
(a) 23
(b) 22
(c) 20
(d) 21

Answer 1

Hint: We start solving the problem by recalling the definition of mean of ‘n’ numbers \[{{a}_{1}}\], \[{{a}_{2}}\],……, \[{{a}_{n}}\] as $\dfrac{{{a}_{1}}+{{a}_{2}}+......+{{a}_{n}}}{n}$. We then use this definition for the given numbers in the problem and make necessary calculations to get the value of ‘x’. We then rewrite the data in ascending order by substituting the value of x. We then check whether the total number of terms present is odd or even and we then use the definition of median for the ascending order to get the required answer.

Complete step by step answer:
According to the problem, we are given that the mean of 26, 19, 15, 24, x is x. We need to find the median of that data.
Let us first find the value of x.
We know that the mean of ‘n’ numbers \[{{a}_{1}}\], \[{{a}_{2}}\],……, \[{{a}_{n}}\] is $\dfrac{{{a}_{1}}+{{a}_{2}}+......+{{a}_{n}}}{n}$.
So, we have $\dfrac{26+19+15+24+x}{5}=x$.
$\Rightarrow 84+x=5x$.
$\Rightarrow 84=4x$.
$\therefore x=21$.
So, we have found the value of x as 21.
Now, let us write the data again. The data is 26, 19, 15, 24, 21. Let us rewrite these numbers in ascending order.
So, we get the data in ascending order as 15, 19, 21, 24, 26.
We can see that there are 5 numbers in total. We know that the median of n numbers which are written in ascending order (where n is odd) is defined as $\left( \dfrac{n+1}{2} \right)$th number.
So, the median is $\left( \dfrac{5+1}{2} \right)={{3}^{rd}}$ number in the data written in ascending order.
We can say that the ${{3}^{rd}}$ number in 15, 19, 21, 24, 26 is 21.
So, the median of the given data is 21.
The correct option for the given problem is (d).

Note:
We should not directly tell the median for the data which is not in ascending or descending order. We should know that in order to find the median of any given data, we first need to write the given terms in ascending or descending order which is a very important property in this problem. We can find the mode of the given data using the formula $\operatorname{mode}=3median-2mean$.