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

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.