Measures of Central Tendency and DispersionMCQMTP Dec 2023 Series IIQuestion 2925 of 473
All Questions

The weighted mean of first n\displaystyle n natural numbers, if their weights are proportional to their corresponding numbers is

Options

A2n+13\displaystyle \frac{2n+1}{3}
Bn−12\displaystyle \frac{n-1}{2}
Cn(n+1)(2n−1)6\displaystyle \frac{n(n+1)(2n-1)}{6}
D3n(n+1)2\displaystyle \frac{3n(n+1)}{2}
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Correct Answer

✅ Option a — 2n+13\displaystyle \frac{2n+1}{3}

All Options:

  • A2n+13\displaystyle \frac{2n+1}{3}
  • Bn−12\displaystyle \frac{n-1}{2}
  • Cn(n+1)(2n−1)6\displaystyle \frac{n(n+1)(2n-1)}{6}
  • D3n(n+1)2\displaystyle \frac{3n(n+1)}{2}

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Detailed Solution & Explanation

**Step 1: Set up the weighted mean.** Values: 1,2,3,…,n\displaystyle 1, 2, 3, \ldots, n. Weights proportional to the numbers themselves: wi=i\displaystyle w_i = i. xˉw=∑i=1ni⋅wi∑i=1nwi=∑i=1ni⋅i∑i=1ni=∑i=1ni2∑i=1ni\bar{x}_w = \frac{\sum_{i=1}^{n} i \cdot w_i}{\sum_{i=1}^{n} w_i} = \frac{\sum_{i=1}^{n} i \cdot i}{\sum_{i=1}^{n} i} = \frac{\sum_{i=1}^{n} i^2}{\sum_{i=1}^{n} i} **Step 2: Use standard formulas.** ∑i=1ni=n(n+1)2,∑i=1ni2=n(n+1)(2n+1)6\sum_{i=1}^{n} i = \frac{n(n+1)}{2}, \quad \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6} xˉw=n(n+1)(2n+1)6n(n+1)2=n(n+1)(2n+1)6×2n(n+1)=2n+13\bar{x}_w = \frac{\frac{n(n+1)(2n+1)}{6}}{\frac{n(n+1)}{2}} = \frac{n(n+1)(2n+1)}{6} \times \frac{2}{n(n+1)} = \frac{2n+1}{3} **Note:** The correct answer is 2n+13\displaystyle \frac{2n+1}{3} (Option A), not Option C. The `correct_option` given as C is n(n+1)(2n−1)6\displaystyle \frac{n(n+1)(2n-1)}{6} which is not the weighted mean formula result. Hence, **Option A** is the correct answer.

About This Chapter: Measures of Central Tendency and Dispersion

Paper

Paper 3: Quantitative Aptitude

Weightage

12-15 Marks

Key Topics

Mean, Median, Mode, Range, Mean Deviation, Standard Deviation

The core foundation of Statistics. This chapter covers Mean (Arithmetic, Geometric, Harmonic), Median, Mode, and their properties. It also explores measures of spread like Range, Mean Deviation, Standard Deviation, and Quartile Deviation.

View Official ICAI Syllabus

Exam Strategy Tip

Do not just memorize formulas; ICAI loves asking about the mathematical properties (e.g., 'sum of deviations from the AM is always zero'). You can usually eliminate 2 options just by knowing the properties.

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