Measures of Central Tendency and DispersionMCQPYQ June 24Question 3141 of 473

In a given set if all data are of same value then variance would be:

Options

\displaystyle 0

\displaystyle 1

\displaystyle -1

\displaystyle 0.5

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Correct Answer

✅ Option a — $\displaystyle 0$

All Options:

A $\displaystyle 0$
B $\displaystyle 1$
C $\displaystyle -1$
D $\displaystyle 0.5$

Detailed Solution & Explanation

**Concept:** Variance measures the average squared deviation from the mean.

\text{Variance} = \frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n}

**If all data values are the same:** Let every

\displaystyle x_i = c

(a constant). **Step 1:** Calculate the mean.

\bar{x} = \frac{nc}{n} = c

**Step 2:** Calculate each deviation.

x_i - \bar{x} = c - c = 0 \quad \text{for all } i

**Step 3:** Calculate variance.

\text{Variance} = \frac{\sum_{i=1}^{n}(0)^2}{n} = \frac{0}{n} = 0

When all observations are identical, there is no spread/variability, so the variance = **0**. 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.

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.

Related Comparison Tables

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