Measures of Central Tendency and DispersionMCQPYQ Jun 24Question 3145 of 473
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If Arithmetic mean and Coefficient of variations of x\displaystyle x are 5 and 20 respectively, the variance of 12−3x\displaystyle 12-3x is

Options

A9\displaystyle 9
B81\displaystyle 81
C3\displaystyle 3
D100\displaystyle 100
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Correct Answer

✅ Option a — 9\displaystyle 9

All Options:

  • A9\displaystyle 9
  • B81\displaystyle 81
  • C3\displaystyle 3
  • D100\displaystyle 100

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Detailed Solution & Explanation

**Given:** xˉ=5\displaystyle \bar{x} = 5, CV of x\displaystyle x = 20 **Step 1: Find SD of x\displaystyle x.** CV=σxxˉ×100CV = \frac{\sigma_x}{\bar{x}} \times 100 20=σx5×10020 = \frac{\sigma_x}{5} \times 100 σx=20×5100=1\sigma_x = \frac{20 \times 5}{100} = 1 **Step 2: Find Var(x\displaystyle x).** Var(x)=σx2=12=1\text{Var}(x) = \sigma_x^2 = 1^2 = 1 **Step 3: Find Var(12−3x\displaystyle 12 - 3x).** Var(a+bx)=b2⋅Var(x)\text{Var}(a + bx) = b^2 \cdot \text{Var}(x) Var(12−3x)=(−3)2⋅Var(x)=9×1=9\text{Var}(12 - 3x) = (-3)^2 \cdot \text{Var}(x) = 9 \times 1 = 9 Wait — the answer is 9, but the given `correct_option` is 'b' (81). Let me check if I made an error. σx=1\displaystyle \sigma_x = 1, Var(x)=1\displaystyle \text{Var}(x) = 1. Var(12−3x)=9×1=9\displaystyle \text{Var}(12-3x) = 9 \times 1 = 9. So Option A (9) is the mathematically correct answer. Option B (81) would require Var(x)=9\displaystyle \text{Var}(x) = 9, i.e., σx=3\displaystyle \sigma_x = 3. That would happen if CV = 60, not 20. 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

Variance

A measure of dispersion equal to the average of the squared deviations of each data point from the mean. Formula: σ² = Σ(xᵢ − x̄)² / n. Standard Deviation = √Variance.

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