Chi-Square (χ²): Analyzing Categorical Data

1. Parametric vs. Non-Parametric

When data violates normal distribution assumptions, or when dealing with nominal/ordinal categorical data, we use Non-Parametric (distribution-free) tests. The Chi-Square test is the most prominent.

2. Chi-Square Test for Independence

Evaluates whether two categorical variables are associated or completely independent.

• H₀: Variables A and B are independent (knowing A tells you nothing about B).
• H₁: Variables A and B are dependent.
• Data Structure: Contingency Table (Cross-tabulation).

Step-by-Step Calculation:

1. Calculate Expected Frequencies (E) for each cell:

E = (Row Total * Column Total) / Grand Total

2. Apply the χ² Formula:

χ² = Σ [ (Observed - Expected)² / Expected ]

3. Determine Degrees of Freedom:

df = (Rows - 1) * (Columns - 1)

4. Compare against χ² Critical Value table.

3. Chi-Square Goodness of Fit Test

Evaluates how well an observed distribution matches an expected theoretical distribution.

• Example: You have a random number generator in Python. You generate 600 numbers between 1 and 6.
• Expected: Each number should appear exactly 100 times.
• Observed: [1: 95, 2: 110, 3: 105, 4: 90, 5: 102, 6: 98]
• You run the Goodness of Fit test. If χ² is very high (p < 0.05), your algorithm is biased and not truly random. df = k - 1 (where k = number of categories).

4. Yates' Continuity Correction

The Chi-Square distribution is continuous, but categorical counts are discrete. For a 2x2 contingency table, this discrepancy can inflate the χ² value, increasing Type I errors. Yates' Correction adjusts the formula by subtracting 0.5 from the absolute difference before squaring: χ²_yates = Σ [ (|O - E| - 0.5)² / E ]

5. Core Assumptions

1. Data must be raw frequencies (counts), not percentages or ratios.
2. Categories are mutually exclusive.
3. No expected frequency E should be < 1, and no more than 20% of expected frequencies should be < 5. (If violated, use Fisher's Exact Test).

6. Complete Worked Example: Test for Independence

Problem: 200 students surveyed about their preferred programming language (Python/Java) and their background (Science/Arts). Test at α = 0.05 whether preference is independent of background.

Observed Frequencies (O):

Step 1: Expected Frequencies (E):

• E(Science, Python) = (100 × 120) / 200 = 60
• E(Science, Java) = (100 × 80) / 200 = 40
• E(Arts, Python) = (100 × 120) / 200 = 60
• E(Arts, Java) = (100 × 80) / 200 = 40

Step 2: Chi-Square Statistic: χ² = (70-60)²/60 + (30-40)²/40 + (50-60)²/60 + (50-40)²/40 χ² = 100/60 + 100/40 + 100/60 + 100/40 χ² = 1.67 + 2.50 + 1.67 + 2.50 = 8.33

Step 3: Degrees of Freedom: df = (2-1) × (2-1) = 1

Step 4: Critical Value: χ²₀.₀₅,₁ = 3.841

Step 5: Conclusion: χ² = 8.33 > 3.841 → Reject H₀ Conclusion: There is a significant association between student background and programming language preference.

7. Quick Reference: Chi-Square Critical Values (α = 0.05)