Quick Reference - Module 12: Chi-Square Tests

Essential Formulas

Goodness of Fit

E = n × p

df = k - 1

Independence

E = (RT×CT)/GT

df = (r-1)(c-1)

Homogeneity

E = (RT×CT)/GT

df = (r-1)(c-1)

Test Statistic (ALL TESTS):

χ² = Σ[(O - E)² / E]

Sum over all categories or cells

Which Test to Use?

Test	Variables	Samples	Question	Example
Goodness of Fit	1 categorical	1 sample	Fits expected distribution?	Is die fair?
Independence	2 categorical	1 sample	Are variables related?	Gender vs party?
Homogeneity	1 categorical	2+ samples	Same distribution?	3 cities' views same?

Decision Flowchart

How many categorical variables?

↓

ONE variable

TWO variables

↓

How many samples?

1 sample
→ Goodness of Fit

Multiple samples
→ Homogeneity

1 sample
→ Independence

Conditions Checklist (ALL TESTS)

Random sampling or random assignment
Independent observations
All expected frequencies ≥ 5 (CRITICAL!)
Categorical data (use counts, not percentages)

7-Step Procedure

Step	Action
1. Hypotheses	H₀ vs Hₐ (state in context)
2. Conditions	Check all 4 conditions (especially E ≥ 5)
3. Expected	Calculate expected frequencies
4. Test Statistic	Calculate χ² = Σ[(O-E)²/E]
5. df	Find degrees of freedom
6. p-value/CV	Use table or technology
7. Conclusion	Decision + statement in context

Common Mistakes

Using proportions instead of counts
Not checking E ≥ 5 condition
Wrong df formula for test type
Claiming causation from association
Forgetting to sum over ALL cells
Using chi-square for quantitative data

Key Reminders

Chi-square distribution:

Always positive (χ² ≥ 0)
Right-skewed
Shape depends on df

Tests are always:

Right-tailed
Based on categorical data
About frequencies, not means

Hypotheses Templates

Test	H₀	Hₐ
Goodness of Fit	Data follows specified distribution	Data does NOT follow distribution
Independence	Variables are independent	Variables are associated
Homogeneity	Distributions are the same	At least one distribution differs

Decision Rules

Using Critical Value:

If χ² > CV → Reject H₀

If χ² ≤ CV → Fail to reject H₀

Using p-value:

If p-value < α → Reject H₀

If p-value ≥ α → Fail to reject H₀

Safaa Dabagh

Chi-Square Tests Quick Reference