Kendall's Tau Correlation — Previous Year Questions

📊 Topic Overview

Metric	Value
Papers Appeared	12 of 27 (44.4%)
Priority	🟠 HIGH
Section Distribution	Sec A: ~6
Most Repeated	"When do we use Kendall's Tau? Find out Tau value for the following data."

🎯 Must-Know Concepts

1. Kendall's tau (τ) is a non-parametric rank correlation coefficient
2. Based on concordant and discordant pairs
3. Formula: τ = (P − Q) / [n(n−1)/2]
4. When to use: ordinal data, or when comparing with Spearman's rho
5. Tau is generally lower in magnitude than Spearman's rho for the same data

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🔥 Core Computation Procedure

Step-by-Step Kendall's Tau

Step 1: Rank both variables Assign ranks to both X and Y.

Step 2: Arrange by rank of X Sort all pairs in ascending order of Rank X.

Step 3: Count Concordant (P) and Discordant (Q) pairs

For each pair in the sorted order, examine all pairs below it:

• Concordant: A pair below has a higher rank on Y → adds to P
• Discordant: A pair below has a lower rank on Y → adds to Q

Sum all concordant pairs → P
Sum all discordant pairs → Q

Step 4: Apply the formula $\tau = \frac{P - Q}{\frac{n(n-1)}{2}}$

Where n(n−1)/2 = total number of pairs compared.

Step 5: Interpret

• τ = +1: Perfect positive concordance
• τ = −1: Perfect negative concordance (all discordant)
• τ = 0: No relationship

Step 6: Significance test (for larger samples, convert to z-score)

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📝 Conceptual: Kendall's Tau vs Spearman's Rho (Section B — ~250 words)

Aspect	Kendall's Tau (τ)	Spearman's Rho (ρ)
Basis	Concordant and discordant pairs	Rank differences (D²)
Formula	(P−Q) / [n(n−1)/2]	1 − 6ΣD²/n(n²−1)
Range	−1 to +1	−1 to +1
Value	Generally lower than rho for same data	Generally higher than tau
Interpretation	Probability that a randomly chosen pair is concordant minus discordant	Degree of monotonic rank relationship
Preferred	Smaller samples, ties present	Standard rank correlation

When to use Kendall's Tau:

1. When data consists of natural ranks or ordinal measurements
2. When sample is small
3. When you want a probability-based interpretation (the difference in probability of concordant vs discordant pairs)
4. When there are many ties in the data

Both tau and rho are equivalent to Pearson's r when the data are bivariate normally distributed and the relationship is linear.

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🧠 Memory Aid

Kendall's Tau Steps (RACP):

• Rank and sort by rank of X
• All pairs below: count Concordant (P) and Discordant (Q)
• Compute τ = (P−Q) / [n(n−1)/2]
• Positive = more concordant pairs

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📚 Related Content

• Spearman Rho and Kendall Tau
• Kendall Tau Rank Order Correlation

✅ Self-Assessment

1. What is the difference between a concordant and a discordant pair?
2. Write the formula for Kendall's tau.
3. If P = 18 and Q = 2 with n = 6, what is tau?
4. Why is Kendall's tau usually smaller than Spearman's rho?
5. When would you use Kendall's tau instead of Pearson's r?

💡 Exam Tips

• Show the full pair-comparison table — this is where marks are earned.
• The formula n(n−1)/2 = total pairs; double-check your count.
• "Compare Pearson's r with Kendall's Tau" is a frequent pairing — prepare the comparison table.
• Verification: P + Q = n(n−1)/2 (if no ties).