Question

The empirical relationship among the three measures of central tendency (median, mode, and mean) is:

• A. $ 2 \cdot \text{Median} = 3 \cdot \text{Mode} + \text{Mean} $
• B. $ \text{Median} = \text{Mode} + 2 \cdot \text{Mean} $
• C. $ 2 \cdot \text{Mode} = 3 \cdot \text{Median} + \text{Mean} $
• D. $ 2 \cdot \text{Mean} = 3 \cdot \text{Median} + 2 \cdot \text{Mode} $

Hint:

The empirical relationship between the mean, median, and mode is useful for understanding the skewness of a distribution. Remember that for a moderately skewed distribution, the formula is $ \text{Mode} \approx 3 \cdot \text{Median} - 2 \cdot \text{Mean} $. Rearranging this formula gives the correct answer.

Answer 1

The Correct Option is D

Step 1: Understand the Empirical Relationship. In statistics, there is an empirical relationship between the mean, median, and mode for a moderately skewed distribution. This relationship is given by: \[ \text{Mode} \approx 3 \cdot \text{Median} - 2 \cdot \text{Mean} \] Rearranging this formula gives: \[ 2 \cdot \text{Mean} = 3 \cdot \text{Median} + 2 \cdot \text{Mode} \] Step 2: Analyze Each Option. Let's evaluate each option based on the empirical relationship: 1. \( 2 \cdot \text{Median} = 3 \cdot \text{Mode} + \text{Mean} \):
Incorrect, as this does not match the standard empirical relationship. 2. \( \text{Median} = \text{Mode} + 2 \cdot \text{Mean} \):
Incorrect, as this does not match the standard empirical relationship. 3. \( 2 \cdot \text{Mode} = 3 \cdot \text{Median} + \text{Mean} \):
Incorrect, as this does not match the standard empirical relationship. 4. \( 2 \cdot \text{Mean} = 3 \cdot \text{Median} + 2 \cdot \text{Mode} \):
Correct, as this matches the standard empirical relationship. Step 3: Final Answer. \[ (4) 2 \cdot \text{Mean} = 3 \cdot \text{Median} + 2 \cdot \text{Mode} \]