Measures of Central Tendency and DispersionMCQPYQ Jan 21Question 3131 of 473
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It is given that the mean (X\displaystyle X) is 10 and standard deviation (s.d.) is 3.2. If the observations are increased by 4, then the new mean and standard deviations are:

Options

AX=10\displaystyle X=10, s.d. =7.2\displaystyle =7.2
BX=14\displaystyle X=14, s.d. =3.2\displaystyle =3.2
CX=14\displaystyle X=14, s.d. =7.2\displaystyle =7.2
DX=10\displaystyle X=10, s.d. =3.2\displaystyle =3.2
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Correct Answer

✅ Option b — X=14\displaystyle X=14, s.d. =3.2\displaystyle =3.2

All Options:

  • AX=10\displaystyle X=10, s.d. =7.2\displaystyle =7.2
  • BX=14\displaystyle X=14, s.d. =3.2\displaystyle =3.2
  • CX=14\displaystyle X=14, s.d. =7.2\displaystyle =7.2
  • DX=10\displaystyle X=10, s.d. =3.2\displaystyle =3.2

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Detailed Solution & Explanation

**Given:** Original Mean Xˉ=10\displaystyle \bar{X} = 10, Original SD = 3.2 Each observation is increased by 4. **Effect on Mean:** When a constant c\displaystyle c is added to every observation, the mean also increases by c\displaystyle c: Xˉnew=Xˉ+c=10+4=14\bar{X}_{new} = \bar{X} + c = 10 + 4 = 14 **Effect on Standard Deviation:** Standard Deviation measures the spread of data around the mean. Adding a constant to all observations shifts the entire dataset uniformly — it does NOT change the spread. Therefore: SDnew=SDold=3.2SD_{new} = SD_{old} = 3.2 **Verification:** If xi\displaystyle x_i are the original observations and xi′=xi+4\displaystyle x_i' = x_i + 4: - New mean: xˉ′=xˉ+4=14\displaystyle \bar{x}' = \bar{x} + 4 = 14 - Deviation: xi′−xˉ′=(xi+4)−(xˉ+4)=xi−xˉ\displaystyle x_i' - \bar{x}' = (x_i + 4) - (\bar{x} + 4) = x_i - \bar{x} (unchanged) - Hence SD unchanged. **New Mean = 14, New SD = 3.2** Hence, **Option B** 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.

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