Question

Relation among mean, median and mode is:
A. Mode \[ = \]3Median \[ + \]2Mean
B. Mode \[ = \]3Median \[ - \]2Mean
C. Mode \[ = \]3Mean \[ + \]2Median
D. Mode \[ = \]3Mean\[ - \]2Median

Answer 1

Hint: Use the general empirical relationship between the mean, median and mode of a skewed distribution to calculate the relation between mean, median and mode in terms of mode.
* For a skewed distribution we have the empirical relationship between mean, median and mode as
Mean \[ - \] Mode \[ = \]3(Mean \[ - \] Median)

Complete step-by-step solution:
We are given the empirical relationship between mean, median and mode as
Mean \[ - \] Mode \[ = \]3(Mean \[ - \] Median)
If we multiply the constant value in right hand side of the equation to each term inside the bracket, we have
\[ \Rightarrow \]Mean \[ - \] Mode \[ = \]3Mean \[ - \] 3Median
Bring all terms except Mode to right hand side of the equation
\[ \Rightarrow \]\[ - \] Mode \[ = \]3Mean \[ - \] 3Median\[ - \]Mean
Pair the same terms together in a bracket
\[ \Rightarrow \]\[ - \] Mode \[ = \](3Mean \[ - \] Mean)\[ - \]3Median
Calculate the sum or difference of terms in bracket in right hand side of the equation
\[ \Rightarrow \]\[ - \] Mode \[ = \]2Mean \[ - \] 3Median
Multiply both sides of the equation by -1
\[ \Rightarrow \]\[ - 1 \times - \] Mode \[ = - 1 \times \](2Mean \[ - \] 3Median)
Multiply terms outside the bracket to terms inside the bracket in right hand side of the equation
\[ \Rightarrow \]\[ - 1 \times - \] Mode \[ = - 1 \times 2\]Mean \[ - 1 \times - 3\]Median
Use the concept that multiplication of two negative signs gives a positive sign as their product.
\[ \Rightarrow \]Mode \[ = \] 3Median \[ - \] 2Mean
\[\therefore \]Relation between mean, median and mode is Mode \[ = \] 3Median \[ - \] 2Mean

\[\therefore \]Correct option is B.

Note: Mean is nothing but the sum of observations divided by the total number of observations. Median is the middle most element of a sorted sequence. Mode is the number which is repeated the most number of times in a given sequence.