Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- A chi-square test compares observed counts in a two-way table with the expected counts you would see if H₀ were true.
- The chi-square statistic measures the distance between observed and expected counts, relative to the expected counts. It is never negative; larger values are stronger evidence against H₀.
- Chi-square distributions take only positive values and are skewed right. As the degrees of freedom increase, the skew becomes less pronounced and the peak moves right.
- The p-value of a chi-square test is always a right-tail area.
Key relationships
| Homogeneity | Independence | |
|---|---|---|
| Design | Independent random samples from 2+ populations, or a randomized experiment with 2+ treatments | One random sample from one population |
| Variables | One categorical variable | Two categorical variables per individual |
| H₀ | No difference in the distribution of [variable] across [populations/treatments] | No association between [variable 1] and [variable 2] in [population] |
| Hₐ | There is a difference in the distribution of [variable] across [populations/treatments] | There is an association between [variable 1] and [variable 2] in [population] |
| Randomization | Independent random samples or random assignment | One random sample |
| 10% condition | Each n ≤ 10% of its N (not needed for an experiment) | n ≤ 10% of N |
| Expected counts | All greater than 5 | All greater than 5 |
Assumptions and conventions
- Use counts, not percentages, in the table.
- Expected counts come from technology or from (row total × column total) ÷ table total (calculated in Topic 3.15).
- If the expected counts condition fails, do not carry out the test as it stands: take a larger sample, or combine categories if that still answers a sensible question.
Mistakes to avoid
- Choosing the test from the table’s shape instead of the design.
- Checking observed counts instead of expected counts.
- Hypotheses about the sample, or with no context.
- Hₐ: “all distributions are different” instead of “there is a difference”.
- Reading an association as cause in an observational study.
- Checking the 10% condition for an experiment.
Quick self-check
- Is a chi-square distribution with 2 degrees of freedom more or less skewed than one with 12? (More skewed. Skew weakens as df increases.)
- A shop asks one random sample of 300 customers their age group and preferred payment method. Which test? (Independence: one sample, two variables.)
- In a two-way table with 150 individuals, a row total is 60 and a column total is 45. What is that cell’s expected count, and does it meet the condition? (60 × 45 ÷ 150 = 18, which is greater than 5.)
Next: practice questions.