Central Tendency & DispersionPYQ Sep 24 Series IQuestion 2940 of 473
All Questions

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

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.

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