Correlation and RegressionMCQPYQ June 19Question 3675 of 188
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Given the following series: X 10 13 12 15 8 15 Y 12 16 18 16 7 18 The rank correlation coefficient r = 1−frac6sumd2+sumfracmi(mi2−1)12n(n2−1)\displaystyle 1 - \\frac{6 \\sum d^2 + \\sum \\frac{m_i(m_i^2 - 1)}{12}}{n(n^2 - 1)}

Options

A1−frac6sumd2n(n2−1)\displaystyle 1 - \\frac{6 \\sum d^2}{n(n^2 - 1)}
B1−frac6sumd2+sumfracmi(mi2−1)12n(n2−1)\displaystyle 1 - \\frac{6 \\sum d^2 + \\sum \\frac{m_i(m_i^2 - 1)}{12}}{n(n^2 - 1)}
C1−6sumd2+fracsummi(mi2−1)12frac1n(n2−1)\displaystyle 1 - 6 \\sum d^2 + \\frac{\\sum m_i(m_i^2 - 1)}{12} \\frac{1}{n(n^2 - 1)}
D1−6sumd2+fracsummi(mi2−1)12frac1n(n2−1)\displaystyle 1 - 6 \\sum d^2 + \\frac{\\sum m_i(m_i^2 - 1)}{12} \\frac{1}{n(n^2 - 1)}
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Correct Answer

✅ Option b — 1−frac6sumd2+sumfracmi(mi2−1)12n(n2−1)\displaystyle 1 - \\frac{6 \\sum d^2 + \\sum \\frac{m_i(m_i^2 - 1)}{12}}{n(n^2 - 1)}

All Options:

  • A1−frac6sumd2n(n2−1)\displaystyle 1 - \\frac{6 \\sum d^2}{n(n^2 - 1)}
  • B1−frac6sumd2+sumfracmi(mi2−1)12n(n2−1)\displaystyle 1 - \\frac{6 \\sum d^2 + \\sum \\frac{m_i(m_i^2 - 1)}{12}}{n(n^2 - 1)}
  • C1−6sumd2+fracsummi(mi2−1)12frac1n(n2−1)\displaystyle 1 - 6 \\sum d^2 + \\frac{\\sum m_i(m_i^2 - 1)}{12} \\frac{1}{n(n^2 - 1)}
  • D1−6sumd2+fracsummi(mi2−1)12frac1n(n2−1)\displaystyle 1 - 6 \\sum d^2 + \\frac{\\sum m_i(m_i^2 - 1)}{12} \\frac{1}{n(n^2 - 1)}

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Detailed Solution & Explanation

**Spearman's Rank Correlation with Tied Ranks:** When observations have **tied ranks** (repeated values), we use the correction factor. The formula becomes: rs=1−frac6left[sumd2+sumfracmi(mi2−1)12right]n(n2−1)r_s = 1 - \\frac{6\\left[\\sum d^2 + \\sum \\frac{m_i(m_i^2-1)}{12}\\right]}{n(n^2-1)} where mi\displaystyle m_i = number of times the i\displaystyle i-th value is repeated (tied). **Why this formula:** When ranks are tied, each tied observation is assigned the average of the ranks they would have occupied. The correction term fracmi(mi2−1)12\displaystyle \\frac{m_i(m_i^2-1)}{12} is added to adjust for these ties. **Verification with the given data:** - X\displaystyle X: 15 appears twice (tied) Rightarrowm1=2\displaystyle \\Rightarrow m_1 = 2 - Y\displaystyle Y: 16 appears twice (tied), 18 appears twice (tied) Rightarrowm2=2,m3=2\displaystyle \\Rightarrow m_2 = 2, m_3 = 2 So the corrected formula (Option B) is appropriate here. The formula in **Option B** is the correct Spearman's formula for tied ranks. Hence, **Option B** 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.

Related Comparison Tables

Correlation vs Regression: Statistics for CA Foundation

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