Measures of Central Tendency and DispersionMCQMTP June 24 Series IQuestion 3208 of 473
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The sum of the squares of deviations of a set of observations has the smallest value, when the deviations are taken from their

Options

AA. M.
BH. M.
CG. M.
DNone of these
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Correct Answer

✅ Option a — A. M.

All Options:

  • AA. M.
  • BH. M.
  • CG. M.
  • DNone of these

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Detailed Solution & Explanation

**Theorem:** The sum of squared deviations ∑(xi−c)2\displaystyle \sum(x_i - c)^2 is minimized when c=xˉ\displaystyle c = \bar{x} (the Arithmetic Mean). **Proof:** Let f(c)=∑i=1n(xi−c)2\displaystyle f(c) = \sum_{i=1}^{n}(x_i - c)^2. f′(c)=−2∑i=1n(xi−c)=0f'(c) = -2\sum_{i=1}^{n}(x_i - c) = 0 ∑xi=nc\sum x_i = nc c=∑xin=xˉc = \frac{\sum x_i}{n} = \bar{x} f′′(c)=2n>0\displaystyle f''(c) = 2n > 0, confirming this is a minimum. Therefore, the sum of squared deviations is smallest when deviations are taken from the **Arithmetic Mean (A.M.)**. 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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