Measures of Central Tendency and DispersionMCQMTP June 24 Series IIQuestion 3210 of 473
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For a set of $100$ observations, taking assumed mean as $4$, the sum of the deviations is $-11$ cm, and the sum of the squares of these deviations is $257$ cm2\displaystyle ^2. The coefficient of variation is:

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Correct Answer

✅ Option a — $41.13\%$

All Options:

  • A$41.13\%$
  • BNone of these
  • C$40.13\%$
  • D$40.13\%$

Detailed Solution & Explanation

**Given:** n=100\displaystyle n = 100, Assumed Mean A=4\displaystyle A = 4, ∑di=−11\displaystyle ∑\displaystyle \sum d_i = -11, ∑di2=257\displaystyle ∑\displaystyle \sum d_i^2 = 257 (where di=xi−A\displaystyle d_i = x_i - A) **Step 1: Calculate actual mean.** xˉ=A+∑din=4+−11100=4−0.11=3.89\displaystyle \bar{x} = A + \frac{∑\displaystyle \sum d_i}{n} = 4 + −11100=4−0\displaystyle \frac{-11}{100} = 4 - 0.11 = 3.89 **Step 2: Calculate Variance.** σ2=∑di2n−(∑din)2=257100−(−11100)2\displaystyle \sigma^2 = \frac{∑\displaystyle \sum d_i^2}{n} - \left(∑din\displaystyle \frac{\sum d_i}{n}\right)^2 = \displaystyle \frac{257}{100} - \left(−11100\displaystyle \frac{-11}{100}\right)^2 =2.57−0.0121=2.5579= 2.57 - 0.0121 = 2.5579 **Step 3: Calculate SD.** σ=2.5579≈1.5993\displaystyle \sigma = \sqrt{2.5579} ≈\displaystyle \approx 1.5993 **Step 4: Calculate CV.** CV=σxˉ×100=1.59933.89×100≈41.11%CV = σxˉ\displaystyle \frac{\sigma}{\bar{x}} ×\displaystyle \times 100 = 1.59933.89\displaystyle \frac{1.5993}{3.89} ×\displaystyle \times 100 ≈\displaystyle \approx 41.11\% This is approximately $41.13\%$ (Option A). 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.

Key Concepts to Understand

Set

A well-defined collection of distinct objects (called elements or members). Sets are denoted by capital letters and elements are listed within curly braces. Key operations include Union (∪), Intersection (∩), and Complement.

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