Parametric tests assume your data follow a specific distribution (usually a normal, bell-shaped curve) and compare means, while nonparametric tests make no such assumption and typically compare medians or ranks — so you choose parametric when your data are roughly normal and nonparametric when they are skewed, ordinal, or small. If you've ever stared at a normality plot wondering whether your dissertation data "count" as normal enough for a t-test, you're asking exactly the right question — and getting it wrong can quietly invalidate your results.
Key Takeaways
- Parametric tests (t-test, ANOVA, Pearson correlation) assume normally distributed data and compare means; they are more powerful when their assumptions hold.
- Nonparametric tests (Mann-Whitney U, Wilcoxon, Kruskal-Wallis, Spearman) make no distribution assumption and work on ranks or medians, making them safer for skewed or ordinal data.
- Use parametric tests when your outcome is continuous, roughly normal, and your sample is reasonably large (often n ≥ 30 per group as a rough rule of thumb).
- Use nonparametric tests when data are ordinal (e.g. Likert scales), heavily skewed, have outliers, or your sample is small.
- Each parametric test has a nonparametric "cousin" — for example, the independent t-test maps to the Mann-Whitney U test.
What Is the Difference Between Parametric and Nonparametric Tests?
The difference comes down to assumptions. A parametric test assumes your data come from a known distribution — almost always the normal distribution — and it uses the actual values (means, variances) to draw conclusions. A nonparametric test throws that assumption away, converts your data into ranks or works with medians, and asks whether groups differ without needing a bell curve.
Think of it like measuring a race. A parametric approach uses the exact finishing times of every runner. A nonparametric approach only cares about the order they crossed the line — 1st, 2nd, 3rd. If a couple of runners had wildly unusual times (outliers), the ranked version isn't thrown off, because 1st is still just 1st.
Parametric tests are more powerful — meaning they're better at detecting a real effect — when their assumptions are met. Nonparametric tests are more robust — they keep working when assumptions break. That trade-off is the whole decision.
When Should I Use a Parametric Test?
Use a parametric test when your outcome variable is continuous, approximately normally distributed, and you have a reasonably large sample. These conditions let the test's mathematics behave, giving you accurate p-values and more statistical power.
Reach for a parametric test when:
- Your dependent variable is measured on an interval or ratio scale (reaction time, blood pressure, test scores).
- The data are roughly symmetric with no extreme outliers — check with a histogram, Q-Q plot, or Shapiro-Wilk test.
- For comparing groups, variances are similar (homogeneity of variance).
- Your sample is large enough that the Central Limit Theorem helps — often cited as n ≥ 30 per group, though this is a guideline, not a law.
Common parametric tests include the independent-samples t-test, paired t-test, one-way ANOVA, and Pearson's r correlation.
When Should I Use a Nonparametric Test?
Use a nonparametric test when your data are ordinal, clearly non-normal, contain outliers you can't justify removing, or when your sample is too small to check assumptions confidently. In these situations a nonparametric test protects you from false conclusions that a parametric test would happily produce.
Choose a nonparametric test when:
- Your outcome is ordinal — Likert-scale agreement ratings, pain scores from 1–10, tumour stages.
- A normality test fails or your histogram is visibly skewed.
- You have influential outliers that distort the mean.
- Your groups are small (e.g. n = 8 per condition), making normality impossible to verify.
If you're specifically weighing the two-group case, our guide on Mann-Whitney U vs the t-test walks through that exact fork.
Which Nonparametric Test Replaces Which Parametric Test?
Almost every parametric test has a direct nonparametric counterpart that answers the same research question with fewer assumptions. This is the single most useful table to keep beside your analysis plan.
| Research question | Parametric test | Nonparametric equivalent |
|---|---|---|
| Compare 2 independent groups | Independent-samples t-test | Mann-Whitney U test |
| Compare 2 related/paired scores | Paired-samples t-test | Wilcoxon signed-rank test |
| Compare 3+ independent groups | One-way ANOVA | Kruskal-Wallis H test |
| Compare 3+ related conditions | Repeated-measures ANOVA | Friedman test |
| Relationship between 2 variables | Pearson's r | Spearman's rho (ρ) |
When you're unsure which row you're on, StatRyx checks your variable types and assumptions automatically and points you to the correct test — so you don't have to memorise this grid.
A Worked Example: Same Data, Two Tests
Suppose you're comparing exam anxiety scores between two teaching methods, with 12 students per group.
Method A anxiety scores: 22, 25, 27, 24, 26, 23, 28, 25, 24, 26, 27, 95
Method B anxiety scores: 30, 32, 29, 31, 33, 30, 34, 31, 32, 30, 29, 33
Notice that Method A has an obvious outlier (95) — perhaps a data-entry error or one extremely anxious student.
The parametric result
Run an independent-samples t-test and that single outlier inflates Method A's mean and variance dramatically. You might get something like t(22) = −0.71, p = .485 — a non-significant result suggesting the methods don't differ. The outlier has masked a real pattern.
The nonparametric result
Run a Mann-Whitney U test, which ranks all 24 scores. The outlier becomes "the highest rank" and nothing more — it can't distort the comparison. You'd find something like U = 18.5, z = −3.24, p = .001, showing Method B genuinely produces higher anxiety scores.
The lesson: the same data gave opposite conclusions. With a skewing outlier, the nonparametric test told the truer story. This is exactly the kind of trap StatRyx flags before you report the wrong number.
How Do I Report These Tests in APA 7 Format?
APA 7 wants the test statistic, degrees of freedom (where applicable), the exact p-value, and an effect size.
- Independent t-test: t(22) = 2.14, p = .043, d = 0.62
- Mann-Whitney U: U = 18.50, z = −3.24, p = .001, r = .66
- One-way ANOVA: F(2, 42) = 4.31, p = .019, η² = .17
- Kruskal-Wallis: H(2) = 9.88, p = .007
Note the conventions: test statistics and p are italicised, and p-values drop the leading zero (.043, not 0.043). Always pair the statistic with an effect size — reviewers increasingly require it.
What Happens If I Pick the Wrong One?
Choosing the wrong test doesn't just cost points in review — it can flip your conclusion, as the worked example above shows. Running a parametric test on badly skewed data can produce **inflated Type I error