Measures of Central Tendency and DispersionMCQPYQ Dec 22Question 3138 of 473
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If the sum of square of the values equals to 3390, Number of observations are 30 and Standard deviation is 7, what is the mean value of the above observations?

Options

A14\displaystyle 14
B11\displaystyle 11
C8\displaystyle 8
D5\displaystyle 5
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Correct Answer

✅ Option c — 8\displaystyle 8

All Options:

  • A14\displaystyle 14
  • B11\displaystyle 11
  • C8\displaystyle 8
  • D5\displaystyle 5

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Detailed Solution & Explanation

**Given:** ∑xi2=3390\displaystyle \sum x_i^2 = 3390, n=30\displaystyle n = 30, σ=7\displaystyle \sigma = 7 **Step 1: Use the variance formula.** σ2=∑xi2n−xˉ2\sigma^2 = \frac{\sum x_i^2}{n} - \bar{x}^2 72=339030−xˉ27^2 = \frac{3390}{30} - \bar{x}^2 49=113−xˉ249 = 113 - \bar{x}^2 **Step 2: Solve for xˉ2\displaystyle \bar{x}^2.** xˉ2=113−49=64\bar{x}^2 = 113 - 49 = 64 **Step 3: Solve for xˉ\displaystyle \bar{x}.** xˉ=64=8\bar{x} = \sqrt{64} = 8 Wait — the answer is 8, but the given `correct_option` is 'b' which is 11\displaystyle 11. Let me recheck: 339030=113\displaystyle \frac{3390}{30} = 113; 113−49=64\displaystyle 113 - 49 = 64; 64=8\displaystyle \sqrt{64} = 8. So the mean = **8**, which is Option C. Let me verify Option B (xˉ=11\displaystyle \bar{x} = 11): σ2=113−121=−8\displaystyle \sigma^2 = 113 - 121 = -8 — impossible (negative variance). Let me verify Option A (xˉ=14\displaystyle \bar{x} = 14): σ2=113−196=−83\displaystyle \sigma^2 = 113 - 196 = -83 — impossible. Let me verify Option D (xˉ=5\displaystyle \bar{x} = 5): σ2=113−25=88\displaystyle \sigma^2 = 113 - 25 = 88; σ=88≈9.38≠7\displaystyle \sigma = \sqrt{88} \approx 9.38 \neq 7. **The correct mathematical answer is xˉ=8\displaystyle \bar{x} = 8 (Option C).** 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

Standard Deviation

A measure of the dispersion or spread of values in a data set around the mean. It is the square root of Variance. A lower standard deviation indicates data points are clustered close to the mean; a higher value indicates greater spread.

Related Comparison Tables

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