Central Tendency & DispersionPYQ May 18Question 3025 of 473
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Relation between mean, median and mode is

Options

Amean - mode = 3\displaystyle 3 (mean - median)
Bmean - median = 3\displaystyle 3 (mean - mode)
Cmean - median = 2\displaystyle 2 (mean - mode)
Dmean - mode = 3\displaystyle 3 (mean - median)
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Correct Answer

✅ Option a — mean - mode = 3\displaystyle 3 (mean - median)

All Options:

  • Amean - mode = 3\displaystyle 3 (mean - median)
  • Bmean - median = 3\displaystyle 3 (mean - mode)
  • Cmean - median = 2\displaystyle 2 (mean - mode)
  • Dmean - mode = 3\displaystyle 3 (mean - median)

Detailed Solution & Explanation

**Step 1: Recall the empirical relationship between mean, median, and mode.** For a moderately skewed distribution, Karl Pearson's empirical formula states: Mode=3×Median−2×Mean\text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean} **Step 2: Rearrange to find the relation.** Rearranging: 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}) So the standard relation is: Mean−Mode=3(Mean−Median)\text{Mean} - \text{Mode} = 3(\text{Mean} - \text{Median}) **Step 3: Check options.** - Option A and D are identical and state: Mean - Mode = 3(Mean - Median) — this is the correct standard formula. - Option C states: Mean - Median = 2(Mean - Mode) — let's verify: From Mean - Mode = 3(Mean - Median): Mean - Median = (Mean - Mode)/3 ≠ 2(Mean - Mode) **Correct formula: Mean - Mode = 3(Mean - Median).** This matches **Options A and D** (they appear to be identical in text). Hence, **Option A** is the correct answer.

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