Measures of Central Tendency and DispersionMCQPYQ Sep 24Question 3147 of 473
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If each observation of a set is divided by 10, then the Standard Deviation of the new observation is:

Options

A110\displaystyle \frac{1}{10} of SD of original observation
B10th\displaystyle 10^{\text{th}} of SD of original observation
C100\displaystyle 100 of SD of original observation
D10\displaystyle 10 of SD of original observation
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Correct Answer

✅ Option a — 110\displaystyle \frac{1}{10} of SD of original observation

All Options:

  • A110\displaystyle \frac{1}{10} of SD of original observation
  • B10th\displaystyle 10^{\text{th}} of SD of original observation
  • C100\displaystyle 100 of SD of original observation
  • D10\displaystyle 10 of SD of original observation

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Detailed Solution & Explanation

**Concept:** When every observation is multiplied (or divided) by a constant k\displaystyle k, the Standard Deviation is also multiplied (or divided) by ∣k∣\displaystyle |k|. **Proof:** If xi′=xi10\displaystyle x_i' = \frac{x_i}{10}, then xˉ′=xˉ10\displaystyle \bar{x}' = \frac{\bar{x}}{10}. σ′=∑(xi10−xˉ10)2n=∑(xi−xˉ10)2n=110∑(xi−xˉ)2n=σ10\sigma' = \sqrt{\frac{\sum\left(\frac{x_i}{10} - \frac{\bar{x}}{10}\right)^2}{n}} = \sqrt{\frac{\sum\left(\frac{x_i - \bar{x}}{10}\right)^2}{n}} = \frac{1}{10}\sqrt{\frac{\sum(x_i - \bar{x})^2}{n}} = \frac{\sigma}{10} Therefore, the new SD = 110\displaystyle \frac{1}{10} of the original SD. 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.

Standard Deviation

A measure of the dispersion or spread of values in a data set around the mean. It is the square root of Variance. A lower standard deviation indicates data points are clustered close to the mean; a higher value indicates greater spread.

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