Theoretical DistributionsMCQMTP Dec 2022 Series IIQuestion 3539 of 230
All Questions

If for a normal distribution Q1=54.52\displaystyle Q_1 = 54.52 and Q3=78.86\displaystyle Q_3 = 78.86, then the median of the distribution is

Options

A12.17\displaystyle 12.17
B39.43\displaystyle 39.43
C66.69\displaystyle 66.69
DNone of these
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Correct Answer

✅ Option c — 66.69\displaystyle 66.69

All Options:

  • A12.17\displaystyle 12.17
  • B39.43\displaystyle 39.43
  • C66.69\displaystyle 66.69
  • DNone of these

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Detailed Solution & Explanation

**Median of a Normal Distribution from Quartiles** **Given:** - First quartile (Q1\displaystyle Q_1) = 54.52\displaystyle 54.52 - Third quartile (Q3\displaystyle Q_3) = 78.86\displaystyle 78.86 **Step 1: Apply properties of the Normal Distribution** A normal distribution is perfectly symmetrical. In any symmetric distribution: 1. The mean, median, and mode are all equal. 2. The first quartile (Q1\displaystyle Q_1) and third quartile (Q3\displaystyle Q_3) are equidistant from the median (second quartile Q2\displaystyle Q_2). Therefore, the median is the simple arithmetic mean of Q1\displaystyle Q_1 and Q3\displaystyle Q_3: textMedian=fracQ1+Q32\\text{Median} = \\frac{Q_1 + Q_3}{2} **Step 2: Substitute the values** textMedian=frac54.52+78.862\\text{Median} = \\frac{54.52 + 78.86}{2} textMedian=frac133.382=66.69\\text{Median} = \\frac{133.38}{2} = 66.69 **Discrepancy Note:** The mathematically derived median is 66.69\displaystyle 66.69, which corresponds to **Option C**. The textbook answer key incorrectly lists **Option B** (39.43\displaystyle 39.43) as the correct answer (which is exactly fracQ32\displaystyle \\frac{Q_3}{2}, a mathematically incorrect formulation). We have presented the correct mathematical derivation leading to Option C. Hence, **Option C** is the correct answer.

About This Chapter: Theoretical Distributions

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Binomial, Poisson, Normal Distribution

This chapter covers Binomial, Poisson, Normal Distribution and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 4-6 Marks weightage. Focus on understanding core concepts rather than memorizing.

Key Concepts to Understand

Median

The middle value of a data set when arranged in ascending or descending order. It divides the distribution into two equal halves. For even n: Median = average of (n/2)th and (n/2 + 1)th values. It is NOT affected by extreme values.

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For a Poisson distribution,

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