Chi-Square Tests

What are Chi-Square Tests?

Chi-square tests are used to analyze categorical data—data that falls into distinct categories rather than continuous measurements. Unlike t-tests or ANOVA (which work with means), chi-square tests work with frequencies and counts.

Common applications include:

• Testing if observed data matches expected proportions
• Determining if two categorical variables are independent
• Analyzing survey responses
• Goodness-of-fit for probability distributions

Types of Chi-Square Tests

1. Chi-Square Goodness of Fit Test

Tests whether observed frequencies match expected frequencies for a single categorical variable.

Example: Do die rolls show equal frequencies for each number?

2. Chi-Square Test of Independence

Tests whether two categorical variables are independent or related.

Example: Is there a relationship between gender and political preference?

The Chi-Square Statistic

Both tests use the same basic formula:

Where:

• $O$ = Observed frequency (actual count)
• $E$ = Expected frequency (theoretical count)
• $\sum$ = Sum over all categories or cells

How It Works:

1. Calculate expected frequencies under the null hypothesis
2. Compare observed to expected for each category/cell
3. Larger differences → larger chi-square statistic
4. Larger chi-square → more evidence against null hypothesis

Chi-Square Goodness of Fit Test

Purpose

Test whether a single categorical variable follows a specific distribution.

Hypotheses

• $H_0$: The variable follows the specified distribution
• $H_A$: The variable does NOT follow the specified distribution

Steps

1. State hypotheses

2. Calculate expected frequencies

• $E_i = n \times p_i$
• Where $n$ = total sample size, $p_i$ = expected proportion for category $i$

3. Compute chi-square statistic $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$

4. Find degrees of freedom $df = k - 1$ Where $k$ = number of categories

5. Compare to critical value or find p-value

Chi-Square Test of Independence

Purpose

Test whether two categorical variables are independent or associated.

Hypotheses

• $H_0$: The two variables are independent (no relationship)
• $H_A$: The two variables are dependent (relationship exists)

Contingency Table

Data is organized in a contingency table (also called a cross-tabulation):

	Variable B: Category 1	Category 2	Category 3	Row Total
Variable A: Category 1	$O_{11}$	$O_{12}$	$O_{13}$	$R_1$
Category 2	$O_{21}$	$O_{22}$	$O_{23}$	$R_2$
Column Total	$C_1$	$C_2$	$C_3$	$n$

Expected Frequencies

Under independence:

Or more formally:

$E_{ij} = \frac{R_i \times C_j}{n}$

Degrees of Freedom

Where:

• $r$ = number of rows
• $c$ = number of columns

Assumptions and Conditions

For valid chi-square tests:

1. Independence

• Observations must be independent
• Each individual contributes to only one cell
• Random sampling is ideal

2. Expected Frequencies

• Rule of thumb: All expected frequencies should be ≥ 5
• If violated, consider:
  • Combining categories
  • Using Fisher’s exact test (for 2×2 tables)
  • Collecting more data

3. Sample Size

• Larger samples are better
• Small samples may produce unreliable results

Interpreting Results

Statistical Significance

• Small p-value (< 0.05): Reject null hypothesis
• Large p-value (≥ 0.05): Fail to reject null hypothesis

Effect Size: Cramér’s V

For test of independence, measure effect size with Cramér’s V:

Interpretation:

• 0.00 - 0.10: Negligible association
• 0.10 - 0.30: Weak association
• 0.30 - 0.50: Moderate association
• 0.50+: Strong association

Examining Residuals

To understand which cells contribute most to chi-square:

• Values > 2 or < -2 indicate cells that deviate substantially from expectation
• Help identify patterns in the data

Goodness of Fit: Chi-Square Distribution

The chi-square test statistic follows a chi-square distribution with appropriate degrees of freedom.

Properties:

• Always non-negative (≥ 0)
• Skewed right
• Approaches normal distribution as df increases
• Mean = df
• Variance = 2×df

Common Mistakes

1. Using Percentages Instead of Counts

• Chi-square requires actual counts, not percentages
• If you only have percentages, convert back to counts

2. Ignoring Expected Frequency Requirement

• Don’t use chi-square if expected frequencies are too small

3. Confusing Independence with Causation

• Rejecting independence means variables are associated
• Doesn’t prove one causes the other

4. Using Wrong Degrees of Freedom

• Goodness of fit: $df = k - 1$
• Independence: $df = (r-1)(c-1)$

Practical Applications

Market Research

• Customer preferences across demographics
• Brand loyalty studies

Medicine

• Disease rates across groups
• Treatment outcomes by category

Social Sciences

• Survey response patterns
• Voting behavior analysis

Quality Control

• Defect rates across production lines
• Product category distributions

Chi-Square vs. Other Tests

Comparison	Chi-Square	Alternative
Categorical vs. Continuous	Categorical data	t-test or ANOVA for continuous
Independence	Non-parametric	May need correlation for continuous
Small samples	Fisher’s exact test	When expected frequencies < 5
Ordinal data	Can use, but…	Consider Mann-Whitney or Kruskal-Wallis

Summary

In this lesson, you learned:

• Chi-square tests analyze categorical data using frequencies
• Goodness of fit tests if data matches a theoretical distribution
• Test of independence checks relationships between two categorical variables
• Formula: $\chi^2 = \sum (O - E)^2 / E$
• Expected frequencies should be ≥ 5 in all cells
• Degrees of freedom: $k-1$ for goodness of fit, $(r-1)(c-1)$ for independence
• Cramér’s V measures effect size for test of independence

Next Steps

Continue learning about categorical data analysis:

• Chi-Square Calculator - Compute tests automatically
• Odds Ratios - Another way to analyze categorical data
• Logistic Regression - Model categorical outcomes