If all the observations are increased by $6$, then the variance of the series will be

The correct answer is Option c: Unchanged. **Concept:** Variance is independent of change of origin. Adding any constant to all observations does NOT change variance. **Proof:** If $x_i' = x_i + 6$, then $\bar{x}' = \bar{x} + 6$. $$x_i' - \bar{x}' = x_i - \bar{x}$$ $$\sigma'^2 = \frac{\sum(x_i' - \bar{x}')^2}{n} = \frac{\sum(x_i - \bar{x})^2}{n} = \sigma^2$$ **Variance is Unchanged.** Hence, **Option C** is the correct answer.

Measures of Central Tendency and DispersionMCQMTP Sep 24 Series IIQuestion 3215 of 473

All Questions

If all the observations are increased by $\displaystyle 6$ , then the variance of the series will be

Options

AIncreased

BDecreased

CUnchanged

DNone of these

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Correct Answer

✅ Option c — Unchanged

All Options:

AIncreased
BDecreased
CUnchanged
DNone of these

Detailed Solution & Explanation

**Concept:** Variance is independent of change of origin. Adding any constant to all observations does NOT change variance. **Proof:** If

\displaystyle x_i' = x_i + 6

, then

\displaystyle \bar{x}' = \bar{x} + 6

x_i' - \bar{x}' = x_i - \bar{x}

\sigma'^2 = \frac{\sum(x_i' - \bar{x}')^2}{n} = \frac{\sum(x_i - \bar{x})^2}{n} = \sigma^2

**Variance is Unchanged.** 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.

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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If all the observations are increased by 6\displaystyle 66, then the variance of the series will be