If the mean of the set of observations x1, x2, x3, ..., xn is x , then the mean of the observation xi + ki , where i = 1, 2, 3, ..., n

Option d: x + k/2(n+1). Solution: Step 1: Compute the new mean. New observations are yi = xi + ki for i = 1, 2, ..., n . y = =1n(xi + ki)n = xi + ksumi=1n in Step 2: Substitute known values. sum xi/n = x and sumi=1n i = n(n+1)/2 y = x + k × (n+1)/2n = x + k(n+1)/2 Note: The derived answer is x + k(n+1)/2 (Option D), not Option A. Option A says x + k(n+1) , which is twice the correct value. The correct derivation gives Option D. Hence, Option D is the correct answer.

Measures of Central Tendency and DispersionMCQMTP Nov 18Question 2887 of 473

All Questions

If the mean of the set of observations $\displaystyle x_1, x_2, x_3, \dots, x_n$ is $\displaystyle \bar{x}$ , then the mean of the observation $\displaystyle x_i + ki$ , where $\displaystyle i = 1, 2, 3, \dots, n$

Options

\displaystyle \bar{x} + k(n+1)

\displaystyle \bar{x} + kn

\displaystyle \bar{x} + \frac{k}{n}

\displaystyle \bar{x} + \frac{k}{2}(n+1)

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Correct Answer

✅ Option d — $\displaystyle \bar{x} + \frac{k}{2}(n+1)$

All Options:

A $\displaystyle \bar{x} + k(n+1)$
B $\displaystyle \bar{x} + kn$
C $\displaystyle \bar{x} + \frac{k}{n}$
D $\displaystyle \bar{x} + \frac{k}{2}(n+1)$

Detailed Solution & Explanation

**Step 1: Compute the new mean.** New observations are

\displaystyle y_i = x_i + ki

for

\displaystyle i = 1, 2, \ldots, n

\bar{y} = \frac{\sum_{i=1}^{n}(x_i + ki)}{n} = \frac{\sum x_i + k\sum_{i=1}^{n} i}{n}

**Step 2: Substitute known values.**

\frac{\sum x_i}{n} = \bar{x} \quad \text{and} \quad \sum_{i=1}^{n} i = \frac{n(n+1)}{2}

\bar{y} = \bar{x} + \frac{k \cdot \frac{n(n+1)}{2}}{n} = \bar{x} + \frac{k(n+1)}{2}

**Note:** The derived answer is

\displaystyle \bar{x} + \frac{k(n+1)}{2}

(Option D), not Option A. Option A says

\displaystyle \bar{x} + k(n+1)

, which is twice the correct value. The correct derivation gives **Option D**. 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.

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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If the mean of the set of observations x1,x2,x3,…,xn\displaystyle x_1, x_2, x_3, \dots, x_nx1​,x2​,x3​,…,xn​ is xˉ\displaystyle \bar{x}xˉ, then the mean of the observation xi+ki\displaystyle x_i + kixi​+ki, where i=1,2,3,…,n\displaystyle i = 1, 2, 3, \dots, ni=1,2,3,…,n