Central Tendency & DispersionPYQ 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

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.

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