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
\[
   \Rightarrow {\text{Mode}} - {\text{Mean}} = 3{\text{Median}} - 2{\text{Mean}} - {\text{Mean}} \\
   \Rightarrow {\text{Mode}} - {\text{Mean}} = 3{\text{Median}} - 3{\text{Mean}} \\
 \]
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.