Choose a statistical test by the question it answers, the study design and the assumptions you can defend—not by a normality check alone. Parametric methods such as t tests and ANOVA model data through parameters; nonparametric methods often use ranks or signs. Neither label guarantees that a method fits your data, and the two approaches may target different effects.
What “parametric” and “nonparametric” mean
A parametric method makes inferences using a model described by parameters, such as a population mean. A t test and ANOVA are familiar examples. Their assumptions depend on the particular model and design; they are not one universal checklist.
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Nonparametric methods often use ranks, signs or other procedures that make fewer assumptions about a particular distribution. They can be useful with ordinal outcomes, ranked observations, skewed data or when a conventional model is unsuitable. “Nonparametric” does not mean assumption-free: independence, symmetry or other conditions may still matter. Penn State’s STAT 500 lesson on nonparametric tests and bootstrap introduces these methods and examples such as sign and Wilcoxon procedures.
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Start with the effect you want to estimate or test
Methods that look like alternatives may answer different questions. A t test commonly tests a difference in means. A rank-based procedure may instead assess rank distributions or relative ordering. A median interpretation for a rank test requires additional distributional conditions; the test name alone does not establish that interpretation. This distinction can explain why two valid methods produce different p-values: they need not be testing the same target. Jim Frost discusses this difference in “Nonparametric Tests vs. Parametric Tests”.
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Before selecting a procedure, state the target in plain language: for example, “the difference in average outcome,” “whether one group tends to have higher values,” or “the strength of a monotonic association.” If the statistical target does not match the practical question, choosing the right label will not fix the mismatch.
Match the procedure to the design
These are common starting points, not interchangeable recipes. Check the exact hypotheses and assumptions for your design before treating one method as a substitute for another.
| Data setup | Parametric example | Nonparametric example | Interpretation to check |
|---|---|---|---|
| One sample or paired measurements | One-sample or paired t test | Sign test or Wilcoxon signed-rank | The signed-rank test has method-specific conditions; it is not assumption-free. |
| Two independent groups | Two-sample t test | Mann–Whitney U / Wilcoxon rank-sum | Do not automatically describe the rank test as a test of medians. |
| More than two groups | One-way ANOVA | Kruskal–Wallis or Mood’s median test | Clarify the target effect and assumptions; these methods are not exact one-to-one equivalents. |
| Repeated measures or blocked comparisons | Factorial-design methods, when appropriate | Friedman test | Confirm that the procedure matches the repeated or blocked design and the hypothesis. |
| Ordinal data or monotonic association | Pearson correlation in suitable settings | Spearman correlation | Spearman addresses monotonic association; it is not a test for every nonlinear relationship. |
Penn State’s STAT 800 lesson gives an applied Mann–Whitney example and discusses alternatives including Fisher’s exact test, Kruskal–Wallis and one-sample Wilcoxon procedures.
Use this decision process
- Specify the target. Decide whether you need a mean difference, median-related quantity, rank tendency, probability of superiority or association. Do not assume these are the same effect.
- Describe the design. Identify whether observations are independent, paired, repeated or blocked, and whether the outcome is categorical, quantitative or ordinal.
- Check the measurement scale. Rank methods can be appropriate for ordinal outcomes, but ordinal measurement alone does not select a unique test.
- Review the assumptions of the candidate method. Check independence and any relevant distributional, symmetry, variance or shape conditions. A distribution-light method still has conditions.
- Inspect the data in context. Look at distributions, outliers and the study design. Do not choose a method solely because raw observations fail a normality check: parametric analyses can be robust to some nonnormality in suitable settings.
- Consider power and interpretation. A nonparametric method can have lower power in some comparable settings, but there is no universal penalty. Ask what effect the design can detect and what a result would mean for the intended question.
Why “nonnormal means nonparametric” is a poor rule
A normality test does not determine whether a method answers the right question. It also cannot, by itself, account for pairing, independence, sample context, outliers or the method’s other assumptions. A departure from normality may matter differently across designs and procedures; some parametric analyses tolerate some departures under suitable conditions.
Instead, compare candidates on the target effect, outcome scale, design, distributional and shape assumptions, sensitivity to outliers, and power for the alternative you care about. Those features—not a binary normal/non-normal label—explain which method is defensible.
Nonparametric tests still have assumptions
For a concrete example, Penn State’s STAT 415 lesson on Wilcoxon tests states that the Wilcoxon signed-rank procedure assumes a continuous random variable and a symmetric population probability distribution. Those conditions are specific to that procedure and setting; another rank or sign test has its own requirements.
Likewise, Mann–Whitney U (also called Wilcoxon rank-sum) should not automatically be reported as a median comparison. Its interpretation depends on distributional conditions. When groups differ in spread or shape, a simple median explanation may be misleading; describe the rank-based hypothesis actually tested and the evidence supporting any stronger interpretation.
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Name the procedure, the design and the effect or hypothesis it addresses. Report an effect estimate and uncertainty where appropriate, not only a p-value. Explain relevant assumptions and how you assessed them. If you considered a parametric and a nonparametric analysis, state whether they tested the same target; differing results are not automatically evidence that one analysis is wrong.
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