Question

The empirical relationship between mean, median and mode is:
A.$$\text{Mean}>\text{Median}>\text{Mode}$$
B.$$\text{Mean}=\text{Median}=\text{Mode}$$
C.$$\text{Mode}-\text{Mean}=3\left(\text{Median}-\text{Mean}\right)$$
D.$$\text{Mean}-\text{Mode}=3\left(\text{Mean}-\text{Median}\right)$$

Answer 1

Hint: First we will use the formula of the relationship between mean, median and mode is $$\text{Mode}=3\text{Median}-2\text{Mean}$$. Then simplify the given relationship by adding and subtracting Mean on both sides to find the required value.

Complete step-by-step answer:
We know that the formula of the relationship between mean, median and mode is $$\text{Mode}=3\text{Median}-2\text{Mean}$$.

Subtracting and adding the value of mean in the above formula of relationship to find the empirical relationship, we get
$$\begin{gathered} \Rightarrow \text{Mode}-\text{Mean}=3\text{Median}-2\text{Mean}-\text{Mean} \\ \Rightarrow \text{Mode}-\text{Mean}=3\text{Median}-3\text{Mean} \end{gathered}$$
Taking the 3 common from the right hand side of the above equation, we get
$$\Rightarrow \text{Mode}-\text{Mean}=3\left(\text{Median}-\text{Mean}\right)$$
Therefore, the required value is $$\text{Mode}-\text{Mean}=3\left(\text{Median}-\text{Mean}\right)$$.

Hence, option C is correct.

Note: We know that the mean or an average of a data set is found by adding all numbers in the data set and then dividing by the number of values in the set. The median is the middle value when a data set is ordered from least to greatest. The mode is the number that occurs most often in a data set.