Question

If the mean and mode of a data are 12 and 21 respectively, then its median is :

• A. 6
• B. 13.5
• C. 15
• D. 14

Hint:

To memorize the empirical formula easily, associate the words with their lengths:
"3 Median - 2 Mean = 1 Mode"
Since "Median" has more letters (6) than "Mean" (4), it gets the larger multiplier (3), and "Mean" gets the smaller multiplier (2).
This simple word-length association prevents confusing the multipliers.

Answer 1

The Correct Option is C

Step 1: Use the "Mean minus Mode" form of the empirical relation instead of the "Mode equals" form.
The same well known empirical relation between mean, median and mode can be written in a different but equivalent shape, and starting from that shape changes the order of the arithmetic.

Step 2: Write down this form of the relation.
\[ \text{Mean} - \text{Mode} = 3(\text{Mean} - \text{Median}) \]
This says the gap between the mean and the mode is always three times the gap between the mean and the median, for a moderately skewed distribution.

Step 3: Compute the left-hand side using the given values.
We are given $\text{Mean} = 12$ and $\text{Mode} = 21$.
\[ \text{Mean} - \text{Mode} = 12 - 21 = -9 \]

Step 4: Solve for the gap between mean and median.
\[ -9 = 3(\text{Mean} - \text{Median}) \]
Divide both sides by 3:
\[ \text{Mean} - \text{Median} = -3 \]

Step 5: Solve for the median.
\[ \text{Median} = \text{Mean} - (-3) = \text{Mean} + 3 = 12 + 3 = 15 \]

Step 6: Final answer.
The median of the data is 15, which is option (C).
\[ \boxed{\text{Median} = 15} \]