Measures of Central Tendency and DispersionMCQPYQ Sep 24 Series IQuestion 2940 of 473
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The algebraic sum of the deviations of set of values from their arithmetic mean is:

Options

A>0\displaystyle >0
B<0\displaystyle <0
C0\displaystyle 0
DNone of these
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Correct Answer

✅ Option c — 0\displaystyle 0

All Options:

  • A>0\displaystyle >0
  • B<0\displaystyle <0
  • C0\displaystyle 0
  • DNone of these

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Detailed Solution & Explanation

Let x1,x2,…,xn\displaystyle x_1, x_2, \dots, x_n be a set of n\displaystyle n observations with arithmetic mean xˉ=1n∑i=1nxi\displaystyle \bar{x} = \frac{1}{n} \sum_{i=1}^n x_i. The algebraic sum of the deviations of these observations from their arithmetic mean is given by: ∑i=1n(xi−xˉ)=∑i=1nxi−∑i=1nxˉ\sum_{i=1}^n (x_i - \bar{x}) = \sum_{i=1}^n x_i - \sum_{i=1}^n \bar{x} Since xˉ\displaystyle \bar{x} is a constant: ∑i=1n(xi−xˉ)=∑i=1nxi−nxˉ\sum_{i=1}^n (x_i - \bar{x}) = \sum_{i=1}^n x_i - n\bar{x} Substituting ∑i=1nxi=nxˉ\displaystyle \sum_{i=1}^n x_i = n\bar{x}: ∑i=1n(xi−xˉ)=nxˉ−nxˉ=0\sum_{i=1}^n (x_i - \bar{x}) = n\bar{x} - n\bar{x} = 0 Thus, the algebraic sum of deviations from the arithmetic mean is always equal to 0\displaystyle 0. 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

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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