Question

For a given set of data, if the difference between mean and mode is 36, what is the difference between mean and median?

• A. 6
• B. 12
• C. 24
• D. 36

Hint:

Memorize Pearson's empirical formula: \( \text{Mean} - \text{Mode} = 3(\text{Mean} - \text{Median}) \). This relationship is a common question on statistical exams.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
For moderately skewed, unimodal distributions, there is a consistent empirical relationship between the three main measures of central tendency: the mean, median, and mode.
Key Formula or Approach:
The empirical relationship established by Karl Pearson is expressed as:
\[ \text{Mean} - \text{Mode} \approx 3 \times (\text{Mean} - \text{Median}) \]
Using this formula, if we know the difference between the mean and the mode, we can calculate the difference between the mean and the median:
\[ \text{Mean} - \text{Median} = \frac{\text{Mean} - \text{Mode}}{3} \]

Step 2: Detailed Explanation:
Let us apply the given values to the formula:
The difference between the mean and the mode is given as 36:
\[ \text{Mean} - \text{Mode} = 36 \]
Substitute this value into the empirical formula:
\[ 36 = 3 \times (\text{Mean} - \text{Median}) \]
To find the difference between the mean and the median, divide both sides of the equation by 3:
\[ \text{Mean} - \text{Median} = \frac{36}{3} \]
\[ \text{Mean} - \text{Median} = 12 \]
Therefore, the difference between the mean and the median for this dataset is 12.

Step 3: Final Answer:
The difference between the mean and the median is 12.
Thus, the correct choice is (B).