Correlation and RegressionPYQ Sept 25Question 4490 of 188
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For a group of students, the sum of squares of differences in ranks for Maths and Physics marks are found to be 60, which is 120 times the value of rank correlation coefficient. How many students are there in the group?

Options

A8
B10
C9
D12
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Correct Answer

✅ Option c — 9

All Options:

  • A8
  • B10
  • C9
  • D12

Detailed Solution & Explanation

Let rk\displaystyle r_k be Spearman's rank correlation coefficient, and n\displaystyle n be the number of students in the group.
1. **Given Data**:
- Sum of squares of differences in ranks: ∑d2=60\displaystyle \sum d^2 = 60
- The sum of squares is 120\displaystyle 120 times the value of the correlation coefficient:
∑d2=120×rk\sum d^2 = 120 \times r_k
60=120×rk  ⟹  rk=60120=0.560 = 120 \times r_k \implies r_k = \frac{60}{120} = 0.5
2. **Formula for Spearman's Rank Correlation Coefficient**:
rk=1−6∑d2n(n2−1)r_k = 1 - \frac{6 \sum d^2}{n(n^2 - 1)}
3. **Calculation**:
Substitute rk=0.5\displaystyle r_k = 0.5 and ∑d2=60\displaystyle \sum d^2 = 60 into the formula:
0.5=1−6(60)n(n2−1)0.5 = 1 - \frac{6(60)}{n(n^2 - 1)}
0.5=1−360n(n2−1)0.5 = 1 - \frac{360}{n(n^2 - 1)}
360n(n2−1)=1−0.5=0.5\frac{360}{n(n^2 - 1)} = 1 - 0.5 = 0.5
n(n2−1)=3600.5=720n(n^2 - 1) = \frac{360}{0.5} = 720
n(n−1)(n+1)=720n(n-1)(n+1) = 720
We need to find three consecutive integers whose product is 720\displaystyle 720. Let us factor 720\displaystyle 720:
720=8×9×10720 = 8 \times 9 \times 10
Comparing the terms, we get n=9\displaystyle n = 9. Thus, there are 9 students in the group.
Hence, **Option C** is the correct answer.

About This Chapter: Correlation and Regression

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Correlation Coefficient, Regression Equations

This chapter covers Correlation Coefficient, Regression Equations 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

Correlation

A statistical measure that expresses the extent to which two variables are linearly related.

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