Chi-Square Test Explained With an Example: A Beginner's Guide

A chi-square test checks whether two categorical variables are related — for example, whether gender is associated with preferring coffee or tea — by comparing the counts you actually observed against the counts you'd expect if there were no relationship at all. If you're staring at a table of counts (yes/no, male/female, pass/fail) and wondering whether the pattern is real or just chance, this is almost certainly the test you need. The confusion usually starts because there are two different chi-square tests, and picking the wrong one wastes hours.

Key Takeaways

• The chi-square test of independence tests whether two categorical variables are related (e.g., treatment group vs. recovery outcome).
• The chi-square goodness-of-fit test tests whether one categorical variable matches an expected distribution (e.g., are dice rolls evenly spread?).
• Chi-square works on counts, not averages — use it when your data are frequencies in categories, not measurements like height or score.
• A result is statistically significant when p < .05, meaning the association you see is unlikely to be due to chance.
• Report effect size with Cramér's V or phi (φ), because a significant p-value alone doesn't tell you how strong the relationship is.

What is a chi-square test in plain English?

A chi-square test asks one simple question: are the counts in my categories different from what I'd expect by chance? It works entirely with frequencies — how many people fall into each box of a table. If you surveyed 200 people and recorded whether they smoke and whether they exercise, a chi-square test tells you whether smoking and exercising are linked, or whether the split you observed is just random noise.

The name comes from the test statistic, written χ² (the Greek letter "chi", pronounced "kye"). The bigger χ² is, the further your observed counts are from the "no relationship" scenario — and the more likely the relationship is real.

When should I use a chi-square test?

Use a chi-square test when both of your variables are categorical — nominal groups like yes/no, male/female/other, or brand A/B/C — and you want to know if they're related. It is not for continuous data. If you're comparing average test scores between two groups, you want a t-test; if you're comparing more than two group means, you want ANOVA.

There are two versions, and knowing which you need is half the battle:

One practical rule: chi-square assumes each expected count is at least 5. If your table has tiny cells (many expected counts below 5), use Fisher's exact test instead — StatRyx flags this automatically and switches tests for you.

Chi-square test explained with a worked example

Let's walk through a chi-square test of independence with real numbers. Suppose you're a psychology student studying whether a new study-skills workshop affects exam outcomes. You track 120 students: some attended the workshop, some didn't, and you record whether each passed or failed.

Here's your observed contingency table:

Step 1: Calculate the expected counts

The expected count for each cell (assuming no relationship) is:

(row total × column total) ÷ grand total

For "Attended + Passed": (60 × 75) ÷ 120 = 37.5. Doing this for all four cells:

Step 2: Compute the chi-square statistic

For each cell, calculate (observed − expected)² ÷ expected, then add them all up:

• (45 − 37.5)² ÷ 37.5 = 1.50
• (15 − 22.5)² ÷ 22.5 = 2.50
• (30 − 37.5)² ÷ 37.5 = 1.50
• (30 − 22.5)² ÷ 22.5 = 2.50

Sum: χ² = 1.50 + 2.50 + 1.50 + 2.50 = 8.00

Step 3: Find the degrees of freedom and p-value

Degrees of freedom = (rows − 1) × (columns − 1) = (2 − 1) × (2 − 1) = 1.

With χ²(1) = 8.00, the p-value is .005 — well below .05. Because p is small, you reject the idea that workshop attendance and passing are unrelated. Attendance and exam outcome are associated.

Step 4: Measure the effect size

A significant result tells you the relationship is real, not how big it is. For a 2×2 table, use phi (φ):

φ = √(χ² ÷ N) = √(8.00 ÷ 120) = √0.0667 = 0.26

A φ of about 0.26 is a small-to-moderate effect. For larger tables, use Cramér's V instead, interpreted the same way (0.1 = small, 0.3 = medium, 0.5 = large).

What counts as a significant result?

A chi-square result is statistically significant when the p-value is below your threshold — almost always .05 in psychology and the social sciences. A p of .005, as in our example, means there's only a 0.5% chance you'd see a difference this large if attendance and passing were genuinely unrelated. Crucially, "significant" does not mean "large" or "important" — that's what the effect size (φ or Cramér's V) is for. Always report both.

How do I report a chi-square test in APA 7 format?

APA 7 wants the test statistic, degrees of freedom, sample size, exact p-value, and an effect size. Italicise χ² and p, and drop the leading zero on p. For our example:

A chi-square test of independence showed a significant association between workshop attendance and exam outcome, χ²(1, N = 120) = 8.00, p = .005, φ = .26.

That single sentence is publication-ready. The (1, N = 120) gives the degrees of freedom and total sample size; φ = .26 communicates a small-to-moderate effect.

Getting the notation exactly right — italics, no leading zero, effect size included — is where a lot of manuscripts get flagged. StatRyx runs the chi-square test, checks your assumptions, and hands you the APA 7 sentence already formatted, so you can paste it straight into your results section.

Chi-square vs. Fisher's exact vs. the t-test

Choosing the right test trips up most non-statisticians. Here's the quick decision guide:

If you're deciding between comparing counts and comparing means, see our guide on choosing the right statistical test — it maps your variable types to the correct analysis in under a minute.

Stop calculating this by hand — run it free in StatRyx → Try StatRyx

Frequently Asked Questions

What is the difference between a chi-square test and a t-test?