Measures of Central Tendency and DispersionMCQPYQ July 21Question 2863 of 473
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There are n\displaystyle n numbers. When 50 is subtracted from each of these number the sum of the numbers so obtained is −10\displaystyle -10. When 46 is subtracted from each of the original n\displaystyle n numbers, then the sum of numbers so obtained is 70. What is the mean of the original n\displaystyle n numbers?

Options

A56.8\displaystyle 56.8
B25.7\displaystyle 25.7
C49.5\displaystyle 49.5
D53.8\displaystyle 53.8
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Correct Answer

✅ Option c — 49.5\displaystyle 49.5

All Options:

  • A56.8\displaystyle 56.8
  • B25.7\displaystyle 25.7
  • C49.5\displaystyle 49.5
  • D53.8\displaystyle 53.8

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Detailed Solution & Explanation

**Step 1: Set up equations using the given conditions.** Let the n\displaystyle n original numbers have sum S\displaystyle S and mean xˉ\displaystyle \bar{x}, so S=nxˉ\displaystyle S = n\bar{x}. **Condition 1:** When 50 is subtracted from each number: ∑(xi−50)=−10  ⟹  S−50n=−10⋯(1)\sum(x_i - 50) = -10 \implies S - 50n = -10 \quad \cdots(1) **Condition 2:** When 46 is subtracted from each number: ∑(xi−46)=70  ⟹  S−46n=70⋯(2)\sum(x_i - 46) = 70 \implies S - 46n = 70 \quad \cdots(2) **Step 2: Subtract equation (1) from equation (2).** (S−46n)−(S−50n)=70−(−10)(S - 46n) - (S - 50n) = 70 - (-10) 4n=80  ⟹  n=204n = 80 \implies n = 20 **Step 3: Find S\displaystyle S using equation (1).** S−50(20)=−10  ⟹  S=1000−10=990S - 50(20) = -10 \implies S = 1000 - 10 = 990 **Step 4: Compute the mean.** xˉ=Sn=99020=49.5\bar{x} = \frac{S}{n} = \frac{990}{20} = 49.5 **Note:** The computed mean is 49.5, which matches Option C, not Option A (56.8). Let us recheck. **Recheck:** S=990\displaystyle S = 990, n=20\displaystyle n = 20, mean =49.5\displaystyle = 49.5. So the correct answer by computation is **49.5**. 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.

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