Parametric vs Non-Parametric: The Reasoning Behind the Choice, Not Just the Flowchart

Two tests, two different sentences

Stevens' classic distinction between scales of measurement explains why the sentence changes with the test.¹ A t-test treats your outcome as an interval scale: the numbers carry real distances, so "10 units apart" and "20 units apart" mean what they say, and averaging is a meaningful operation. Its output can therefore say how large a difference is, in the outcome's own units, with a confidence interval around that number.

Mann-Whitney U doesn't touch the raw values at all. It ranks every observation across both groups and compares the rank sums.² That is a genuinely different question: not "how far apart are the means," but "if you picked one observation from each group at random, which one would tend to be larger." A significant Mann-Whitney result supports the second claim. It does not support the first, no matter how tempting it is to write "the groups differed by X units" once you already have a significant P value.

The same trial, two different questions

Both results below come from the Mayo Clinic's primary biliary cirrhosis (PBC) trial, the same 312 randomised patients.³ One compares patient age between those with and without spider angiomata; the other compares serum bilirubin between those with and without hepatomegaly. Each was routed to its test the same way: check normality first, then let the result decide.

Example data: Mayo Clinic primary biliary cirrhosis (PBC) trial (Fleming and Harrington, 1991); full worked example in The Independent t-Test. Computed with scipy, not generated.

Here every group cleared 30 observations and passed Shapiro-Wilk, so the t-test is licensed to say exactly how many years apart the two group means are, and how confident that estimate is. The P value is not significant, but the sentence it supports is still a specific, interval-scale claim: a mean difference, with a confidence interval.

Bilirubin is strongly right-skewed in both groups, so Shapiro-Wilk rejects normality and the comparison routes to Mann-Whitney U instead: U = 17635.5, P < .001, r = 0.39 (medium).⁴ That result licenses "bilirubin tends to run higher in patients with hepatomegaly." It does not license "bilirubin is 1.55 mg/dL higher," even though 2.55 minus 1.00 is right there in the table. The medians describe the two groups; the test itself only supports the weaker, rank-based claim.

Why the sentence matters more than the button

Choosing the test is really choosing the sentence you're entitled to write in your results section. Overclaim it, report a mean-style difference from a rank-based test, and a statistical reviewer has grounds to send the manuscript back. Underclaim it, hedge a genuine interval-scale result into vague rank language when the t-test's assumptions were actually met, and you've thrown away information the data supported. Neither direction is free.

It is never a statistical error to run the more conservative, rank-based test where a parametric test would also have been valid; it costs some power, not validity. Running a t-test when its assumptions are violated is the error that changes conclusions, which is why the normality check has to come first and actually be checked, not assumed.

Writing it correctly either way

For a t-test result: report both group means with SDs, name which version of the t-test was used (Student's or Welch's) and why, the statistic with degrees of freedom, the exact P value, and Cohen's d with its confidence interval. For a Mann-Whitney result: report both group medians with IQRs, the U statistic, the exact P value, and effect size r. Neither sentence substitutes for the other, and a reviewer checking your methods against your results table should find them consistent with the test you named.

Frequently asked questions

What does an independent t-test let you claim that Mann-Whitney U doesn't?

A t-test licenses a claim about means on a scale where distance is meaningful: group A averages a specific number of units more than group B, with a confidence interval around that difference. Mann-Whitney U licenses a weaker, rank-based claim: values in one group tend to run higher than in the other, without a defensible numeric gap.

Is it ever wrong to use Mann-Whitney U when a t-test's assumptions are actually met?

No. Using the rank-based test where a parametric test would also have been valid is conservative, not an error; it costs some statistical power but not validity. The reverse, running a t-test when normality is violated, is the error reviewers and methodologists flag, because it can produce an untrustworthy P value.

Does Mann-Whitney U only test whether medians differ?

No, and this is a common misreading. Mann-Whitney U tests whether one distribution is stochastically larger than the other, which can be driven by a difference in spread or shape, not only a difference in the median. Two groups with identical medians can still return a significant result if their distributions differ in shape.

References

1. Stevens SS. On the theory of scales of measurement. Science. 1946;103(2684):677-680. https://doi.org/10.1126/science.103.2684.677
2. Hart A. Mann-Whitney test is not just a test of medians: differences in spread can be important. BMJ. 2001;323(7309):391-393. https://doi.org/10.1136/bmj.323.7309.391
3. Fleming TR, Harrington DP. Counting Processes and Survival Analysis. New York: John Wiley and Sons; 1991.
4. Nahm FS. Nonparametric statistical tests for the continuous data: the basic concept and the practical use. Korean Journal of Anesthesiology. 2016;69(1):8-14. https://doi.org/10.4097/kjae.2016.69.1.8