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:

An easy way to memorize this empirical relation is:
"3 Medians minus 2 Means equals 1 Mode."
Keep alphabetical order in mind: Median (longest word, multiplier 3) comes first, then Mean (shorter word, multiplier 2).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem provides the mean and the mode of a dataset and asks us to compute the median.
The values given are:
- $\text{Mean} = 12$
- $\text{Mode} = 21$

Step 2: Key Formula or Approach:
For any moderately skewed frequency distribution, there is an empirical relationship between the three measures of central tendency (mean, median, and mode):
\[ \text{Mode} = 3 \text{ Median} - 2 \text{ Mean} \]
We will substitute the given values of mean and mode into this formula to solve for the median.

Step 3: Detailed Explanation:

• State the empirical relation:
\[ \text{Mode} = 3 \text{ Median} - 2 \text{ Mean} \]

• Substitute the given parameters ($\text{Mode} = 21$ and $\text{Mean} = 12$) into the equation:
\[ 21 = 3 \text{ Median} - 2(12) \]

• Simplify the multiplication term on the right-hand side:
\[ 21 = 3 \text{ Median} - 24 \]

• Isolate the term containing the median by adding 24 to both sides:
\[ 21 + 24 = 3 \text{ Median} \]
\[ 45 = 3 \text{ Median} \]

• Solve for the median by dividing both sides by 3:
\[ \text{Median} = \frac{45}{3} \]
\[ \text{Median} = 15 \]


Step 4: Final Answer:
The median of the given data is 15.
Hence, option (C) is correct.