Question

If the ratio of mode and median of a certain data is $6:5$, then find the ratio of its mean and median :

• A. 3 : 5
• B. 3 : 10
• C. 9 : 10
• D. 10 : 9

Hint:

This problem relies on knowing the empirical relationship between the measures of central tendency: Mode = 3 Median - 2 Mean. This formula is approximate but widely used in statistics for moderately skewed distributions.

Answer 1

The Correct Option is C

Step 1: Recall the empirical relationship between Mean, Median, and Mode.
For a moderately skewed distribution, there is an empirical relationship (often called the Emprical Mean-Median-Mode Formula or Karl Pearson's empirical relation):
Mode = 3 Median - 2 Mean Step 2: Write down the given ratio.
Given: Ratio of Mode to Median = 6 : 5
This means $\frac{\text{Mode}}{\text{Median}} = \frac{6}{5}$.
So, Mode = $\frac{6}{5}$ Median.
Step 3: Substitute the given ratio into the empirical formula.
Substitute Mode = $\frac{6}{5}$ Median into the formula:
$\frac{6}{5}$ Median = 3 Median - 2 Mean Step 4: Rearrange the equation to find the relationship between Mean and Median.
Move the terms involving Median to one side and Mean to the other side:
2 Mean = 3 Median - $\frac{6}{5}$ Median
2 Mean = $(\frac{15}{5} - \frac{6}{5})$ Median
2 Mean = $\frac{9}{5}$ Median
Step 5: Find the ratio of Mean and Median.
We need to find $\frac{\text{Mean}}{\text{Median}}$.
From the equation 2 Mean = $\frac{9}{5}$ Median, divide both sides by 2 and by Median:
$\frac{\text{Mean}}{\text{Median}} = \frac{9}{5 \times 2}$
$\frac{\text{Mean}}{\text{Median}} = \frac{9}{10}$
So, the ratio of mean and median is 9 : 10. Step 6: Compare with the given options.
The calculated ratio is 9 : 10, which matches option (3). (3) 9 : 10