Central Tendency & DispersionPYQ Sep 24Question 3046 of 473
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For a moderately-skewed distribution, which of the following relationship holds?

Options

AMean - Mode = 3 (Mean - Median)
BMean - Mode = 3 (Mean - Median)
CMean - Median = 3 (Mean - Mode)
DMean - Median = 3 (Median - Mode)
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Correct Answer

✅ Option a — Mean - Mode = 3 (Mean - Median)

All Options:

  • AMean - Mode = 3 (Mean - Median)
  • BMean - Mode = 3 (Mean - Median)
  • CMean - Median = 3 (Mean - Mode)
  • DMean - Median = 3 (Median - Mode)

Detailed Solution & Explanation

**Step 1: Recall the empirical relationship (Karl Pearson's formula).** For a moderately skewed distribution: Mode=3×Median−2×Mean\text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean} **Step 2: Rearrange.** Mean−Mode=2Mean+Mode−3Median+Mean−Mode\text{Mean} - \text{Mode} = 2\text{Mean} + \text{Mode} - 3\text{Median} + \text{Mean} - \text{Mode} More directly: Mean−Mode=2Mean−(3Median−2Mean)+Mean−Mean\text{Mean} - \text{Mode} = 2\text{Mean} - (3\text{Median} - 2\text{Mean}) + \text{Mean} - \text{Mean} Simply: Mode=3Median−2Mean\text{Mode} = 3\text{Median} - 2\text{Mean} Mean−Mode=Mean−3Median+2Mean=3Mean−3Median=3(Mean−Median)\text{Mean} - \text{Mode} = \text{Mean} - 3\text{Median} + 2\text{Mean} = 3\text{Mean} - 3\text{Median} = 3(\text{Mean} - \text{Median}) **Step 3: The correct formula.** Mean - Mode = 3(Mean - Median). Both Options A and B appear identical. The standard empirical formula is: Mean−Mode=3(Mean−Median)\text{Mean} - \text{Mode} = 3(\text{Mean} - \text{Median}) Hence, **Option A** is the correct answer.

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