For a set of 100 observations, taking assumed mean as 4 , the sum of the deviations is -11 cm, and the sum of the squares of these deviations is 257 cm 2 . The coefficient of variation is:

Option a: 41.13%. Solution: Given: n = 100 , Assumed Mean A = 4 , sum di = -11 , sum di2 = 257 (where di = xi - A ) Step 1: Calculate actual mean. x = A + sum di/n = 4 + -11/100 = 4 - 0.11 = 3.89 Step 2: Calculate Variance. sigma2 = sum di2/n - (sum di/n)2 = 257/100 - (-11/100)2 = 2.57 - 0.0121 = 2.5579 Step 3: Calculate SD. sigma = sqrt(2.5579) ≈ 1.5993 Step 4: Calculate CV. CV = sigma/x × 100 = 1.5993/3.89 × 100 ≈ 41.11% This is approximately 41.13% (Option A). Hence, Option A is the correct answer.

Measures of Central Tendency and DispersionMCQMTP June 24 Series IIQuestion 3210 of 473

All Questions

For a set of $\displaystyle 100$ observations, taking assumed mean as $\displaystyle 4$ , the sum of the deviations is $\displaystyle -11$ cm, and the sum of the squares of these deviations is $\displaystyle 257$ cm $\displaystyle ^2$ . The coefficient of variation is:

Options

\displaystyle 41.13\%

BNone of these

\displaystyle 40.13\%

\displaystyle 40.13\%

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Correct Answer

✅ Option a — $\displaystyle 41.13\%$

All Options:

A $\displaystyle 41.13\%$
BNone of these
C $\displaystyle 40.13\%$
D $\displaystyle 40.13\%$

Detailed Solution & Explanation

**Given:**

\displaystyle n = 100

, Assumed Mean

\displaystyle A = 4

\displaystyle \sum d_i = -11

\displaystyle \sum d_i^2 = 257

(where

\displaystyle d_i = x_i - A

) **Step 1: Calculate actual mean.**

\bar{x} = A + \frac{\sum d_i}{n} = 4 + \frac{-11}{100} = 4 - 0.11 = 3.89

**Step 2: Calculate Variance.**

\sigma^2 = \frac{\sum d_i^2}{n} - \left(\frac{\sum d_i}{n}\right)^2 = \frac{257}{100} - \left(\frac{-11}{100}\right)^2

= 2.57 - 0.0121 = 2.5579

**Step 3: Calculate SD.**

\sigma = \sqrt{2.5579} \approx 1.5993

**Step 4: Calculate CV.**

CV = \frac{\sigma}{\bar{x}} \times 100 = \frac{1.5993}{3.89} \times 100 \approx 41.11\%

This is approximately

\displaystyle 41.13\%

(Option A). Hence, **Option A** 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

Set

A well-defined collection of distinct objects (called elements or members). Sets are denoted by capital letters and elements are listed within curly braces. Key operations include Union (∪), Intersection (∩), and Complement.

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For a set of 100\displaystyle 100100 observations, taking assumed mean as 4\displaystyle 44, the sum of the deviations is −11\displaystyle -11−11 cm, and the sum of the squares of these deviations is 257\displaystyle 257257 cm2\displaystyle ^22. The coefficient of variation is: