Question

If the sum of mode and mean of a certain data is $129$ and its median is $63$. What is its mode?
${\text{(A) 69}}$
${\text{(B) 63}}$
${\text{(C) 60}}$
${\text{(D) 65}}$

Answer 1

First, we find the mean value by using the given data and solving that using the formula. An empirical relationship between the mean, median and mode of a distribution, then we find the mode value by using the solving data. Finally we get the required answer.

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

Complete step-by-step solution:
It is given that the question stated as the sum of the mode and mean of the certain data is $129$.
It can be written as mathematically:
$mode + mean = 129 \to (1)$
Also, the median of the data is $63$
Here we use the formula and we get,
${\text{mode = 3median - 2mean}}$
By putting the value of Median and Mode we get:
$\Rightarrow$ ${\text{mode = 3}} \times {\text{63 - 2}} \times {\text{mean}}$
On simplifying, we get:
$\Rightarrow$ ${\text{mode = 189 - 2}} \times {\text{mean}}$
Now we will take similar terms across the sign.
On taking mean across the sign it becomes positive and transferring across makes it negative therefore, it can be written as:
$\Rightarrow$ ${\text{mode}} + 2{\text{mean}} = 189 \to (2)$
Now on doing $(2) - (1)$ we get:
Mean $= 60$.
Now since we know the mean and median of the distribution, then we will find the mode:
$\Rightarrow$ ${\text{mode = 3median - 2mean}}$
On substituting the values, we get:
$\Rightarrow$ ${\text{mode = 3}} \times {\text{63 - 2}} \times {\text{60}}$
On multiplying the terms, we get:
$\Rightarrow$ ${\text{mode = 189 - 120}}$
On simplifying we get:
$\Rightarrow$ ${\text{mode = 69}}$,

Therefore, the correct option is $(A)$.

Note: A distribution, in which we can say the median, mean and mode are all the same is known as symmetrical distribution.
Also, a distribution which doesn’t have the mean, median and mode the same is called an asymmetrical distribution or a skewed distribution.
Now there exists a relationship between these three central tendencies which we 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:
$Mean - Median = \dfrac{{Mode - Mean}}{3}$
Let us cross multiply the term and we can write the equation as,
${\text{mode = 3median - 2mean}}$
Knowing any $2$ values, the third value can be calculated using this formula.