Question

If median and mode $ = 2.5$. Find the approximate value of the mean.
${\text{(A) 2.5}}$
${\text{(B) 4}}$
${\text{(B) 5}}$
${\text{(B) 6}}$

Answer 1

Hint: Here we have to find out the approximate value of the mean. Also, we have an empirical relationship between the mean, median and mode of a distribution, we will use that to find the missing value.

Formula used: ${\text{mode = 3median - 2mean}}$

Complete step-by-step solution:
It is given that the question stated as the median and the mode of the distribution is the same which is $2.5$
Here we can be written as mathematically we get:
${\text{Median = 2}}{\text{.5}}$ and ${\text{Mode = 2}}{\text{.5}}$
Now we use the formula,
${\text{mode = 3median - 2mean}}$
On substituting the value of Median and Mode we get:
$2.5 = 3(2.5) - 2(Mean)$
On multiplying the bracket term, we get:
$2.5 = 7.5 - 2Mean$
Now we will take like terms across the $ = $ sign.
On taking mean across the $ = $sign it becomes positive and transferring $2.5$ across makes it negative therefore, it can be written as:
$2Mean = 7.5 - 2.5$
On subtracting the RHS we get:
$2Mean = 5$
On taking $2$ across the $ = $ sign, it gets written in the denominator, it can be written as:
$Mean = \dfrac{5}{2}$
On dividing the terms we get:
$Mean = 2.5$

Therefore, the correct option is ${\text{(A)}}$ which is $2.5$.

Note: A distribution in which the mean, median and mode are the same is called a symmetrical distribution.
And, a distribution which doesn’t have the mean, median and mode the same is called an asymmetrical distribution or a skewed distribution.
There exists a relationship between all the three central tendencies which is called the empirical relation.
The relation is that the distance between the mean and median in a distribution is almost about one-third of the distance between the mean and the mode, this can be written mathematically as:
$Mean - Median = \dfrac{{Mode - Mean}}{3}$
On simplification of this equation we get the empirical formula which is:
${\text{mode = 3median - 2mean}}$
Knowing any $2$ values, the third value can be calculated using this formula.