Measures of Central Tendency and DispersionMCQMTP Nov 19Question 2883 of 473
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The mean of the values of 1,2,3,…,n\displaystyle 1, 2, 3, \dots, n with respective frequencies x,2x,3x,…,nx\displaystyle x, 2x, 3x, \dots, nx is.

Options

An+12\displaystyle \frac{n+1}{2}
Bn2\displaystyle \frac{n}{2}
C2n+13\displaystyle \frac{2n+1}{3}
D2n+16\displaystyle \frac{2n+1}{6}
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Correct Answer

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

All Options:

  • An+12\displaystyle \frac{n+1}{2}
  • Bn2\displaystyle \frac{n}{2}
  • C2n+13\displaystyle \frac{2n+1}{3}
  • D2n+16\displaystyle \frac{2n+1}{6}

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Detailed Solution & Explanation

**Step 1: Set up the weighted mean formula.** Values: 1,2,3,…,n\displaystyle 1, 2, 3, \ldots, n. Frequencies: x,2x,3x,…,nx\displaystyle x, 2x, 3x, \ldots, nx (i.e., frequency of value r\displaystyle r is rx\displaystyle rx). xˉ=∑r=1nr⋅(rx)∑r=1nrx=x∑r=1nr2x∑r=1nr\bar{x} = \frac{\sum_{r=1}^{n} r \cdot (rx)}{\sum_{r=1}^{n} rx} = \frac{x \sum_{r=1}^{n} r^2}{x \sum_{r=1}^{n} r} **Step 2: Use standard summation formulas.** ∑r=1nr=n(n+1)2,∑r=1nr2=n(n+1)(2n+1)6\sum_{r=1}^{n} r = \frac{n(n+1)}{2}, \qquad \sum_{r=1}^{n} r^2 = \frac{n(n+1)(2n+1)}{6} **Step 3: Compute the ratio.** xˉ=n(n+1)(2n+1)6n(n+1)2=n(n+1)(2n+1)6×2n(n+1)=2(2n+1)6=2n+13\bar{x} = \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{2(2n+1)}{6} = \frac{2n+1}{3} Hence, **Option C** 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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