Chi-Square Test — Previous Year Questions

📊 Topic Overview

Metric	Value
Papers Appeared	22 of 27 (81.5%)
Priority	🔴 VERY HIGH
Section Distribution	Sec A: ~8
Most Repeated	"Compute Chi-square for the following data"
Probability	Extremely High — nearly certain as Section B computation

🎯 Must-Know Concepts

1. Chi-square (χ²) is a non-parametric test for categorical data
2. Two uses: goodness of fit and test of independence (contingency)
3. Formula: χ² = Σ[(O − E)² / E]
4. Expected frequency calculation for contingency tables
5. Yates' correction for 2×2 tables
6. Assumptions and limitations of chi-square

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🔥 Core Computation Procedure

Chi-Square Test of Independence (Contingency Table)

Question type: "Compute chi-square for the following data. Is the difference significant?"

Step 1: Set up the contingency table Arrange observed frequencies (O) in rows and columns.

Step 2: Calculate Expected Frequencies (E) $E_{ij} = \frac{Row\, Total_i \times Column\, Total_j}{Grand\, Total}$

Step 3: Compute Chi-Square $\chi^2 = \sum \frac{(O - E)^2}{E}$

Calculate (O − E), then (O − E)², then (O − E)²/E for each cell. Sum all values.

Step 4: Degrees of Freedom $df = (r - 1)(c - 1)$ where r = number of rows, c = number of columns.

Step 5: Chi-Square Table

Cell	O	E	O−E	(O−E)²	(O−E)²/E
Cell 1
Cell 2
...
Total					χ²

Step 6: Decision

• Find χ² critical from the chi-square distribution table at the computed df at α = 0.05
• If χ² obtained ≥ χ² critical: Reject H₀ → Significant association exists
• If χ² obtained < χ² critical: Retain H₀ → No significant association

Yates' Correction (for 2×2 tables with small expected frequencies)

When expected frequency in any cell < 5 in a 2×2 table: $\chi^2_{Yates} = \sum \frac{(|O - E| - 0.5)^2}{E}$

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📝 Conceptual Questions

Chi-square as Goodness of Fit (Section B — ~250 words)

The goodness of fit test examines whether observed frequencies differ significantly from expected (theoretical) frequencies.

Procedure:

1. State expected frequency for each category based on theoretical distribution
2. Compute χ² = Σ[(O − E)² / E]
3. df = k − 1 (k = number of categories)
4. Compare χ² obtained with χ² critical

Example: Testing whether a die is fair — expected frequency for each face = N/6.

Assumptions of Chi-Square:

1. Data must be in frequencies (counts), not percentages or proportions
2. Observations must be independent
3. Expected frequency in each cell should be at least 5 (use Yates' correction if not)
4. Data measured on nominal or ordinal scale

Advantages: Simple to compute; applicable to nominal data; no normality assumption.

Limitations: Cannot be used with continuous data directly; sensitive to small expected frequencies; only tests association, not causation.

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🧠 Memory Aid

Chi-Square Steps (SEDC):

• Set up observed table
• Expected frequencies (row × col / grand total)
• Difference: (O−E)²/E for each cell, sum = χ²
• Compare with critical value at df = (r−1)(c−1)

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📚 Related Content

• Chi-Square Test: Goodness of Fit and Independence
• Non-parametric Tests: Chi-square, Mann-Whitney

✅ Self-Assessment

1. What is the formula for expected frequency in a contingency table?
2. When should Yates' correction be applied?
3. For a 3×2 contingency table, what are the degrees of freedom?
4. What is the difference between goodness of fit and test of independence?
5. What is the minimum expected frequency rule for chi-square?

💡 Exam Tips

• Always draw the full table with O, E, (O−E), (O−E)², (O−E)²/E columns.
• Mention Yates' correction even if not applied — it shows awareness.
• State H₀: "There is no significant association between the two variables."
• Chi-square is non-parametric — mention this in your answer.
• For goodness of fit questions, clearly state what the expected frequencies represent.