You have your data, you know your research question, and now you face the fork in the road every methods textbook warns you about: parametric or nonparametric? Students tend to treat this choice as either a formality (run the t-test, nobody checks) or a source of paralysis (three days spent staring at a histogram). Neither approach serves you well. This guide explains what the distinction actually means, how to check the assumptions that drive the decision, and when a violated assumption matters enough to change your test.
What "Parametric" Actually Means
A parametric test makes specific claims about the population your sample came from. The word comes from "parameter," the fixed numbers that describe a population distribution. When you run an independent-samples t-test, you are implicitly saying: I believe the populations behind these two groups are roughly normal, and the t-test's math depends on estimating their means and variances. The test borrows statistical strength from that assumed shape. If the assumption holds, you get precise results from relatively small samples.
Nonparametric tests drop most of those claims. Instead of working with the raw values, they typically convert your data to ranks. The Mann-Whitney U test, for instance, lines up every observation from both groups, ranks them from smallest to largest, and asks whether one group's ranks sit systematically higher than the other's. Because ranks do not care whether your distribution is normal, skewed, or shaped like a ski jump, the test works under far weaker conditions.
The trade is straightforward. Parametric tests assume more and, when those assumptions are true, detect effects more efficiently. Nonparametric tests assume less and pay for that flexibility with somewhat lower power under ideal conditions. Everything else in this article is about deciding which side of that trade suits your data.
Parametric tests analyze raw values and assume a population distribution (usually normal). Nonparametric tests analyze ranks and make minimal distributional assumptions. The choice is a trade between efficiency and safety.
The Four Assumptions of Parametric Tests
Most parametric tests you will meet as a student (t-tests, ANOVA, Pearson correlation) rest on four assumptions. Two of them get all the attention; the other two cause the real disasters.
1. Normality
The classic assumption. Strictly speaking, the requirement applies to the residuals or to the sampling distribution of the mean, not to the raw data themselves, but in practice you assess it by examining the distribution of your outcome variable within each group. A reaction-time variable with a long right tail, an income variable where three participants earn ten times the median, a depression scale where 60% of your community sample scores near zero: these are the situations where normality breaks down. Our full guide to normality testing walks through the details.
2. Interval or Ratio Measurement
Parametric tests compute means, and means only make sense when the distance between adjacent values is constant. The gap between 20 and 21 degrees Celsius equals the gap between 30 and 31. Compare that to a satisfaction item scored 1 to 5: is the jump from "dissatisfied" to "neutral" really the same size as the jump from "satisfied" to "very satisfied"? For single Likert items the honest answer is no, which points toward rank-based methods. Multi-item scale scores (say, a 20-item questionnaire summed to a total) behave close enough to interval data that most researchers treat them parametrically.
3. Homogeneity of Variance
For tests comparing groups, the spread within each group should be similar. A treatment group with a standard deviation of 4 compared against a control group with a standard deviation of 15 violates this, and the violation distorts the t-test's error rate, especially when group sizes differ. Unequal variances paired with unequal group sizes is the combination that genuinely wrecks results: if the smaller group has the larger variance, your true Type I error rate can climb well above the nominal 5%.
4. Independence of Observations
Each data point should be unrelated to every other data point. Measure the same person twice and put both scores in an independent-samples test, and you have violated independence. Sample 30 students from the same three classrooms, and their scores are clustered rather than independent. No statistical fix rescues a broken independence assumption after the fact, and no nonparametric test fixes it either. Wilcoxon assumes independence between pairs just as the paired t-test does. If you are unsure whether your design produces paired or independent data, our guide on paired versus independent samples covers exactly this question.
Students obsess over normality, but independence is the assumption that cannot be repaired or dodged. Get your design right first; distribution shape is a second-order problem.
How to Check Each Assumption
Checking assumptions takes about ten minutes in SPSS, R, or jamovi. Here is the standard toolkit.
Shapiro-Wilk for Normality
The Shapiro-Wilk test evaluates the null hypothesis that your data come from a normal distribution. A p-value below .05 signals significant departure from normality. Run it within each group, not on the pooled sample, since two normal groups with different means can produce a pooled distribution that looks bimodal.
One warning before you take the result at face value. Shapiro-Wilk's sensitivity scales with sample size. With n = 12, it will miss serious skew. With n = 400, it flags trivial wobbles that no test would ever notice in practice. Treat it as one piece of evidence rather than a verdict.
Q-Q Plots for a Visual Check
A quantile-quantile plot graphs your observed values against the values a perfect normal distribution would produce. Normal data hug the diagonal reference line. Skewed data peel away at one end in a curve; heavy-tailed data form an S shape. With practice, a Q-Q plot tells you more than any test statistic because it shows you what kind of departure you have and how bad it is. A gentle wobble around the line is fine. A hockey stick is a problem.
Levene's Test for Homogeneity of Variance
Levene's test checks whether group variances differ significantly. SPSS prints it automatically alongside every independent-samples t-test. If Levene's p-value drops below .05, do not abandon ship: read the second row of the SPSS output instead, labeled "equal variances not assumed." That row reports Welch's t-test, which corrects for unequal variances and, in the view of many methodologists, should be the default anyway. For ANOVA, the parallel fix is Welch's ANOVA.
Independence: Check Your Design, Not Your Data
There is no button for this one. Ask yourself how the data were collected. Repeated measurements from the same participants, siblings in the same sample, patients from the same clinic, responses collected in intact groups: each of these creates dependence you must handle through your choice of test or model, not through an assumption check.
The Decision Table: Every Parametric Test Has a Rank-Based Twin
When assumptions fail badly enough to matter, each common parametric test has a nonparametric counterpart that answers a closely related question.
| Research situation | Parametric test | Nonparametric alternative |
|---|---|---|
| Compare 2 independent groups | Independent-samples t-test | Mann-Whitney U test |
| Compare 2 related measurements | Paired-samples t-test | Wilcoxon signed-rank test |
| Compare 3+ independent groups | One-way ANOVA | Kruskal-Wallis H test |
| Compare 3+ related measurements | Repeated-measures ANOVA | Friedman test |
| Relationship between 2 continuous variables | Pearson correlation | Spearman's rho |
Notice the phrase "closely related question" above. The swap is not perfectly clean. A t-test compares means; Mann-Whitney compares whole rank distributions, which under certain conditions amounts to comparing medians. Usually the distinction changes nothing about your conclusions, but your write-up should describe what the test actually examined. Report medians and interquartile ranges alongside a Mann-Whitney result, not means and standard deviations.
If you want a guided route through this table, our Statistical Test Selector asks you a short series of questions about your variables and design and points you to the appropriate test. The broader logic behind test selection is covered in how to choose a statistical test.
When Violations Actually Matter
Here is the part your textbook probably undersold. Parametric tests tolerate a surprising amount of assumption violation, and the reason is the Central Limit Theorem: as sample size grows, the sampling distribution of the mean approaches normality regardless of the shape of the underlying population. The t-test cares about that sampling distribution, not about your histogram.
Some practical thresholds, drawn from decades of simulation studies:
Below about 15 per group, distribution shape matters a lot. A skewed variable with n = 10 per group is a genuine problem, and rank-based tests earn their keep here. Between roughly 15 and 30 per group, moderate skew is usually tolerable, though strong skew or extreme outliers still distort results. Above 30 per group, the t-test handles most realistic distributions well, and by n = 50 or so per group even markedly skewed data rarely change the substantive conclusion. Simulation work by Fagerland (2012) and others suggests that with large samples, the bigger worry flips: Mann-Whitney can detect differences in shape or spread that have nothing to do with the medians you think you are comparing.
Two caveats keep this from becoming a free pass. First, the CLT protects the Type I error rate, not your power; heavy tails and outliers still drag power down, so a rank-based test can actually be the more sensitive choice for genuinely nasty distributions. Second, the CLT says nothing about the variance and independence assumptions. A sample of 500 does not rescue you from clustered observations.
If your data are non-normal and your sample is small, you also have options beyond switching tests, including transformations and bootstrapping. We compare those routes in dealing with non-normal data.
With 30+ participants per group, mild to moderate non-normality rarely justifies abandoning a t-test or ANOVA. Below 15 per group, take distribution shape seriously.
Two Mistakes Students Make Constantly
Defaulting to Nonparametric "Just to Be Safe"
This one sounds virtuous and costs you power. A student sees one Shapiro-Wilk p-value of .03 in a sample of 200, panics, and runs Kruskal-Wallis on everything. Two problems follow. The mild non-normality was irrelevant at that sample size, so the caution bought nothing. And the rank-based test threw away information in the raw values, making a real effect harder to detect and eliminating access to the standard effect sizes, confidence intervals, and post-hoc procedures that supervisors and reviewers expect. Nonparametric tests are a tool for specific situations, not a protective charm.
Ignoring Assumptions Entirely
The opposite failure is running a t-test on 9 participants per group with a floor-effect outcome where half the sample scored zero, then reporting p = .048 as a finding. At that sample size, with that distribution, the p-value is not trustworthy. Markers notice. A dissertation that reports "Shapiro-Wilk indicated significant non-normality in the control group, W = 0.81, p = .002, so the Mann-Whitney U test was used" demonstrates exactly the competence examiners look for, and it takes two sentences.
The honest middle path: check assumptions every time, report what you found, and let sample size inform how much weight the checks carry.
The Power Question
Power is the probability your test detects an effect that truly exists, and it is where the parametric-nonparametric trade shows up in numbers. When data really are normal, the Mann-Whitney U test operates at about 95% of the t-test's efficiency (the asymptotic relative efficiency is 3/π ≈ 0.955). In concrete terms, a study needing 100 participants for adequate power with a t-test would need roughly 105 with Mann-Whitney. That is a modest tax, which is why the "just to be safe" habit, while wasteful, is rarely catastrophic.
The comparison reverses under heavy tails. When outliers or extreme skew contaminate the data, rank-based tests can be substantially more powerful than their parametric twins, because a single wild value inflates a t-test's variance estimate while barely moving the ranks. So the power argument cuts both ways: with clean normal data the parametric test wins by a little, and with heavy-tailed data the nonparametric test can win by a lot.
Sample size planning should reflect your choice. If you expect skewed data and plan a Wilcoxon or Mann-Whitney analysis from the start, build the extra 5 to 15% into your recruitment target rather than discovering the shortfall after data collection.
A Working Procedure
Pulling it together into the sequence we use with students who bring us their thesis data:
Start with the design. Confirm independence and identify whether your samples are paired or independent, since this determines the test family before distributions enter the picture. Then check measurement level: single ordinal items point toward rank-based tests regardless of anything else. Next, examine distributions within each group using a Q-Q plot and Shapiro-Wilk, interpreting both in light of your sample size. Check Levene's test for group comparisons, remembering that Welch's correction handles variance problems without leaving the parametric family. Only then choose: parametric when assumptions hold or your samples are large enough for the CLT to cover mild violations, nonparametric when samples are small and distributions are clearly non-normal, or when your outcome is genuinely ordinal.
Whatever you choose, write down why. One sentence in your methods section explaining the choice does more for your credibility than the choice itself. And if you are staring at your own dataset unsure which branch applies, the Statistical Test Selector will get you to a defensible answer in under two minutes.