Central Tendency & DispersionMTP June 2023 Series IIQuestion 3062 of 473
All Questions

Which of the following is the correct relation between mean, median and mode

Options

AMedian = mode +23\displaystyle + \frac{2}{3} (mean −\displaystyle - mode)
B2\displaystyle 2Mean = Mode −3\displaystyle - 3Median
C2\displaystyle 2Mean = Mode +3\displaystyle + 3Median
DMode = 3\displaystyle 3Median +2\displaystyle + 2Mean
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Correct Answer

✅ Option a — Median = mode +23\displaystyle + \frac{2}{3} (mean −\displaystyle - mode)

All Options:

  • AMedian = mode +23\displaystyle + \frac{2}{3} (mean −\displaystyle - mode)
  • B2\displaystyle 2Mean = Mode −3\displaystyle - 3Median
  • C2\displaystyle 2Mean = Mode +3\displaystyle + 3Median
  • DMode = 3\displaystyle 3Median +2\displaystyle + 2Mean

Detailed Solution & Explanation

**Step 1: Recall the empirical formula.** Mode=3Median−2Mean\text{Mode} = 3\text{Median} - 2\text{Mean} **Step 2: Rearrange.** 2Mean=3Median−Mode2\text{Mean} = 3\text{Median} - \text{Mode} 2Mean+Mode=3Median2\text{Mean} + \text{Mode} = 3\text{Median} This can also be written as: 2Mean=3Median−Mode2\text{Mean} = 3\text{Median} - \text{Mode} Let's check Option C: 2Mean=Mode+3Median\displaystyle 2\text{Mean} = \text{Mode} + 3\text{Median}? From the formula: 2Mean=3Median−Mode\displaystyle 2\text{Mean} = 3\text{Median} - \text{Mode}, so Option C would require Mode + 3Median = 3Median - Mode → 2Mode = 0 → Mode = 0. This is incorrect. **Option A:** Median = Mode + (2/3)(Mean - Mode) = Mode + (2/3)Mean - (2/3)Mode = (1/3)Mode + (2/3)Mean = (Mode + 2Mean)/3 From the formula: 3Median = Mode + 2Mean → Median = (Mode + 2Mean)/3 ✓ This matches **Option A**. Hence, **Option A** is the correct answer.

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