Question 1

What is the chi-square test used for?

Accepted Answer

The chi-square test has two main uses: (1) Goodness of fit — tests whether a single categorical variable follows a hypothesised distribution (e.g., "Are die rolls equally likely across all 6 faces?"). (2) Test of independence — tests whether two categorical variables are associated (e.g., "Is gender independent of product preference?"). Both use the same χ² formula but different degrees of freedom.

Question 2

How do I calculate expected frequencies?

Accepted Answer

For goodness of fit: Expected = total sample size × hypothesised proportion. Example: 120 customers, expected 25% each in 4 regions → E = 120 × 0.25 = 30 per region. For independence test in a contingency table: E = (row total × column total) / grand total. All expected frequencies should be ≥ 5 for the χ² approximation to be valid.

Question 3

What does the p-value mean in a chi-square test?

Accepted Answer

The p-value is the probability of observing a χ² statistic this large (or larger) if the null hypothesis were true. p < 0.05 means: the observed frequencies differ from expected more than would occur by chance 95% of the time → reject H₀. p ≥ 0.05 means: insufficient evidence to reject H₀ at α = 0.05.

Question 4

What are degrees of freedom in chi-square?

Accepted Answer

For goodness of fit: df = k − 1, where k is the number of categories. For a test of independence (r×c table): df = (rows − 1) × (columns − 1). Example: a 2×3 contingency table → df = (2−1)(3−1) = 2. Higher df requires a larger χ² value to reach significance.

Question 5

What are the assumptions of the chi-square test?

Accepted Answer

1) Random sampling: observations are independent. 2) Adequate expected counts: all expected frequencies should be ≥ 5. If any E < 5, combine categories or use Fisher's exact test. 3) Categorical data: chi-square is not appropriate for continuous variables. 4) Mutually exclusive categories: each observation falls into exactly one category.

Question 6

What is a chi-square critical value?

Accepted Answer

The critical value is the χ² threshold that p = α. For α = 0.05: df=1 → χ²_crit = 3.841; df=2 → 5.991; df=3 → 7.815; df=4 → 9.488; df=5 → 11.070. If your calculated χ² exceeds the critical value, reject H₀. Most modern practice reports exact p-values rather than comparing to critical values.

df	χ² critical (α = 0.05)	χ² critical (α = 0.01)	Typical use case
1	3.841	6.635	2-category goodness of fit
2	5.991	9.210	3-category or 2×2 contingency
3	7.815	11.345	4-category goodness of fit
4	9.488	13.277	5-category test
5	11.070	15.086	6-category test
9	16.919	21.666	2×5 contingency table

Condition	Chi-Square GoF	Fisher's Exact	McNemar
All expected counts ≥ 5	✓ Standard	Also works	—
Any expected count < 5	—	✓ Use this	—
Paired / before-after categorical data	—	—	✓
2×2 table, large sample	✓ (Yates correction)	✓	✓ if paired
3+ categories, large sample	✓	—	—
Need effect size alongside p-value	Add Cramér's V	Add odds ratio	—

Chi-Square Calculator

Formula

Chi-Square Critical Values (α = 0.05)

Common Mistakes

Chi-Square vs Other Categorical Tests — Which to Use?

Case Study: Candy Colour Preference Survey — Testing Equal Distribution

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Category	Observed (O)	Expected (E)