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: Understanding the Question:
The topic is Statistics, specifically the relationship between the three measures of central tendency: Mean, Median, and Mode.
We are given the values of the Mean and Mode of a dataset and need to calculate its Median.

Step 2: Key Formula or Approach:
There is an empirical relationship that connects the three measures of central tendency for a moderately asymmetrical distribution. This is known as the Empirical Formula:
\[ \text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean} \]
We can rearrange this formula to solve directly for the Median:
\[ 3 \times \text{Median} = \text{Mode} + 2 \times \text{Mean} \]
\[ \text{Median} = \frac{\text{Mode} + 2 \times \text{Mean}}{3} \]

Step 3: Detailed Explanation:

• Identify the given values from the problem statement:
$\text{Mean} = 12$
$\text{Mode} = 21$

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

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

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

• Transpose $-24$ to the left-hand side of the equation to isolate the term containing the Median:
\[ 21 + 24 = 3\text{ Median} \]
\[ 45 = 3\text{ Median} \]

• Divide both sides by 3 to find the Median:
\[ \text{Median} = \frac{45}{3} \]
\[ \text{Median} = 15 \]


Step 4: Final Answer:
The median of the given data is 15, which corresponds to option (C).