Measures of Central Tendency and DispersionMCQPYQ Dec. 21Question 2867 of 473
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For a data having odd number of values, the difference between the first and the middle value is equal to the difference between the last and the middle value; similarly, the difference between the second and middle values is equal to that of second last and middle value so on. Therefore, the middle value is equal to

Options

AHalf of the range
BHalf of standard deviation
CMode
DMean
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Correct Answer

✅ Option d — Mean

All Options:

  • AHalf of the range
  • BHalf of standard deviation
  • CMode
  • DMean

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Detailed Solution & Explanation

**Step 1: Analyze the symmetry condition.** For a symmetrically placed dataset with an odd number of values, the values are equally spaced around the middle value. This means every xi\displaystyle x_i and xn+1−i\displaystyle x_{n+1-i} are equidistant from the middle value. **Step 2: Algebraic reasoning.** If the middle value is m\displaystyle m, then for each pair: xi−m=−(xn+1−i−m)x_i - m = -(x_{n+1-i} - m) This means the deviations cancel in pairs, so ∑(xi−m)=0\displaystyle \sum(x_i - m) = 0. But this is the defining property of the arithmetic mean: ∑(xi−xˉ)=0\displaystyle \sum(x_i - \bar{x}) = 0. Therefore, m=xˉ\displaystyle m = \bar{x}, i.e., the middle value equals the **mean**. Hence, **Option D** 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.

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