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Question: If median and mode \( = 2.5\). Find the approximate value of the mean. \({\text{(A) 2.5}}\) \({\...

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

Explanation

Solution

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: mode = 3median - 2mean{\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 2.52.5
Here we can be written as mathematically we get:
Median = 2.5{\text{Median = 2}}{\text{.5}} and Mode = 2.5{\text{Mode = 2}}{\text{.5}}
Now we use the formula,
mode = 3median - 2mean{\text{mode = 3median - 2mean}}
On substituting the value of Median and Mode we get:
2.5=3(2.5)−2(Mean)2.5 = 3(2.5) - 2(Mean)
On multiplying the bracket term, we get:
2.5=7.5−2Mean2.5 = 7.5 - 2Mean
Now we will take like terms across the == sign.
On taking mean across the ==sign it becomes positive and transferring 2.52.5 across makes it negative therefore, it can be written as:
2Mean=7.5−2.52Mean = 7.5 - 2.5
On subtracting the RHS we get:
2Mean=52Mean = 5
On taking 22 across the == sign, it gets written in the denominator, it can be written as:
Mean=52Mean = \dfrac{5}{2}
On dividing the terms we get:
Mean=2.5Mean = 2.5

Therefore, the correct option is (A){\text{(A)}} which is 2.52.5.

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=Mode−Mean3Mean - Median = \dfrac{{Mode - Mean}}{3}
On simplification of this equation we get the empirical formula which is:
mode = 3median - 2mean{\text{mode = 3median - 2mean}}
Knowing any 22 values, the third value can be calculated using this formula.