TL;DR — Quick Answer
The t-test, ANOVA, and chi-square test are the three most commonly used statistical tests in research, and each answers a different kind of question. A t-test compares the means of two groups on a continuous variable (e.g., do male and female employees differ in average job satisfaction?). ANOVA extends this to three or more groups (e.g., does satisfaction differ across four departments?). A chi-square test works with categorical data, testing whether two categorical variables are associated (e.g., is gender related to preferred work mode?). Choose based on two questions: what type of data do you have, and how many groups are you comparing?
Most researchers meet these three tests in their very first quantitative study, and most confusion about them traces to a single misunderstanding: the tests are not interchangeable tools of varying sophistication — they are answers to different questions about different kinds of data. This guide explains what each test does, when to use it, what its results mean, and the assumptions and alternatives you need to know to defend your choice. It works as a companion to our broader guide on how to choose the right statistical test, going deeper on the three tests you are most likely to actually run.
The Decision in One Sentence Each
- T-test: two groups, continuous outcome — do the two means differ?
- ANOVA: three or more groups, continuous outcome — do any of the means differ?
- Chi-square: two categorical variables — are the categories associated?
Everything else in this guide is elaboration of those three sentences.
The T-Test: Comparing Two Means
The t-test (formally Student’s t-test, developed by William Sealy Gosset writing as “Student” in 1908) evaluates whether the difference between two means is larger than would plausibly arise from sampling variation alone. It comes in three forms, and selecting the right one matters:
- Independent-samples t-test — two separate groups of people or cases. Example: comparing mean exam anxiety between students taught online and students taught in person.
- Paired-samples t-test — the same group measured twice. Example: employee stress scores before and after a wellbeing intervention. Pairing removes between-person variation, making this design considerably more powerful when it applies.
- One-sample t-test — one group compared against a known or hypothesised value. Example: testing whether your sample’s mean weekly working hours differ from the national figure of 48.
The test produces a t statistic and a p-value. The t statistic expresses the group difference in units of its own variability; the p-value converts that into the probability of observing such a difference if no true difference exists. If the p-value falls below your significance level (conventionally 0.05), the difference is declared statistically significant — a concept unpacked fully in our guide on statistical significance and the p-value.
Key assumptions: the outcome variable is continuous (interval or ratio, or a composite scale score treated as such); observations are independent (for the independent-samples version); the outcome is approximately normally distributed within each group — though the test is robust to moderate violations at reasonable sample sizes; and the two groups have roughly equal variances. When variances are clearly unequal, report the Welch-corrected version, which statistical software provides alongside the standard result. When normality fails badly in small samples, the non-parametric alternative is the Mann–Whitney U test (independent groups) or the Wilcoxon signed-rank test (paired data).
ANOVA: Comparing Three or More Means
Analysis of Variance (ANOVA), developed by Ronald Fisher, answers the natural next question: what if there are more than two groups? Comparing four departments with repeated t-tests (A vs B, A vs C, and so on) would require six tests, and each additional test inflates the chance of a false positive somewhere among them. ANOVA solves this by asking one omnibus question — is there any difference among these means? — in a single test.
Despite its name, ANOVA compares means by analysing variance: it partitions the total variability in the data into variability between groups and variability within groups. The resulting F statistic is the ratio of the two. If groups genuinely differ, between-group variability outweighs within-group noise and F grows large; the p-value again judges whether the observed F is beyond what chance would produce.
The common designs:
- One-way ANOVA — one grouping factor with three or more levels. Example: job satisfaction across four departments.
- Two-way ANOVA — two grouping factors simultaneously, which additionally tests their interaction. Example: satisfaction by department and employment type, including whether the departmental pattern differs between permanent and contract staff.
- Repeated-measures ANOVA — the same cases measured at three or more time points; the multi-group extension of the paired t-test.
The step most often missed: a significant ANOVA result says only that at least one group differs from at least one other — it does not say which. Identifying the specific differences requires post-hoc tests (Tukey’s HSD is the standard choice for equal-variance designs), which perform the pairwise comparisons with the false-positive inflation properly controlled. An ANOVA reported without post-hoc follow-up is an unfinished analysis, and examiners flag it.
Assumptions mirror the t-test’s: continuous outcome, independent observations, approximate normality within groups, and equality of variances across groups (assessed with Levene’s test in most software). The non-parametric alternative is the Kruskal–Wallis test, with its own post-hoc procedures. Incidentally, a t-test is simply ANOVA with two groups — run both on two groups and the p-values are identical, with F equal to t squared.
Chi-Square: Testing Association Between Categorical Variables
The first two tests need a continuous outcome. But much research data is purely categorical: gender, department, yes/no responses, preferred work mode, adoption vs non-adoption of a technology. The chi-square test of independence (χ²), developed by Karl Pearson, tests whether two categorical variables are associated — whether the distribution of one differs across the levels of the other.
The logic is intuitive. Cross-tabulate the two variables into a contingency table — say, gender (2 rows) by preferred work mode (3 columns). If the variables are unrelated, the proportions choosing each work mode should be roughly the same for both genders; the test computes the cell counts expected under that independence and measures how far the observed counts deviate. Large total deviation yields a large χ² statistic and a small p-value, indicating association.
A second use of the same statistic, the chi-square goodness-of-fit test, compares one categorical variable’s distribution against a hypothesised pattern — for example, whether complaints are evenly distributed across weekdays.
Assumptions and cautions: observations must be independent (each respondent appears in exactly one cell — chi-square is not for repeated measures); the data are counts, never percentages; and expected counts must be adequate — the standard rule is that no more than 20% of cells have expected counts below 5, and none below 1. For small 2×2 tables that violate this, report Fisher’s exact test instead. Finally, a significant chi-square establishes association, not strength: report an effect size such as Cramér’s V alongside it, because with large samples even trivial associations reach significance.
Choosing Between Them: A Worked Set of Examples
Consider a study of technology adoption in 40 companies — the kind of survey design discussed in our guide on questionnaire design. The same dataset can call for all three tests, depending on the question:
- Do adopting and non-adopting companies differ in mean employee count? Two groups, continuous outcome → independent-samples t-test.
- Does mean adoption-readiness score differ across small, medium, and large companies? Three groups, continuous outcome → one-way ANOVA, with Tukey post-hoc tests if significant.
- Is company size category associated with adoption status (yes/no)? Two categorical variables → chi-square test of independence.
- Did readiness scores change from before to after training in the same companies? Same cases, two time points → paired-samples t-test.
Notice that the choice never depends on which test is “stronger” — it is fully determined by the structure of the question and the measurement level of the variables, concepts covered in our guides on variables in research and descriptive vs inferential statistics.
Reporting the Results
Standard reporting conventions (APA style) for each, with illustrative numbers:
- T-test: “Adopting companies (M = 142.3, SD = 38.1) employed significantly more staff than non-adopting companies (M = 98.7, SD = 41.5), t(38) = 3.42, p = .002, d = 1.09.”
- ANOVA: “Readiness differed significantly across company sizes, F(2, 37) = 6.85, p = .003, η² = .27. Tukey post-hoc comparisons showed large companies scored significantly higher than small companies (p = .002), with no other pairwise differences.”
- Chi-square: “Company size was significantly associated with adoption status, χ²(2, N = 40) = 8.11, p = .017, Cramér’s V = .45.”
Note that each report includes an effect size (d, η², Cramér’s V) — increasingly a requirement of journals and examiners, because significance alone says nothing about practical magnitude. All three tests, along with their assumption checks and post-hoc procedures, are available through menus in SPSS; see our beginner’s guide to SPSS for the software side.
Frequently Asked Questions
Can I use a t-test for three groups by running it three times?
No — repeated pairwise t-tests inflate the overall false-positive rate. Use ANOVA followed by post-hoc tests, which control that inflation properly.
Which test do I use for Likert-scale data?
For composite multi-item scale scores, t-tests and ANOVA are standard practice. For a single Likert item treated strictly as ordinal, Mann–Whitney or Kruskal–Wallis are the more defensible choices — and if you have grouped the responses into categories, chi-square applies.
What sample size do these tests need?
There is no universal minimum, but as working guidance: t-tests and ANOVA behave well from roughly 30 cases per group; chi-square is governed by the expected-count rule above rather than total N. Formal power analysis is the rigorous route — see our guide on calculating sample size.
What if my data fail the normality assumption?
With moderate samples, mild non-normality is rarely fatal — these tests are robust. With small samples or severe skew, switch to the non-parametric counterpart: Mann–Whitney (independent t-test), Wilcoxon (paired t-test), Kruskal–Wallis (one-way ANOVA).
Final Thoughts
The t-test, ANOVA, and chi-square cover a remarkable share of the questions quantitative research actually asks — which is precisely why they are the tests examiners and reviewers scrutinise most closely. The discipline is always the same: let the question and the measurement level of your variables dictate the test, verify the assumptions, follow a significant omnibus result to its post-hoc conclusions, and report effect sizes alongside p-values. Master these three and their non-parametric alternatives, and you can analyse — and defend — the majority of survey-based research designs with confidence.