Mann–Whitney U test

The Mann–Whitney U test (also called the Mann–Whitney–Wilcoxon (MWW/MWU), Wilcoxon rank-sum test, or Wilcoxon–Mann–Whitney test) is a nonparametric statistical test of the null hypothesis that randomly selected values X and Y from two populations have the same distribution.

Version

v1 · 2026-09-28 · History

Domain-specific #

10548

Domain group

Formal Sciences

Origin domain

Experimental Design & Statistics

Subdomain

Nonparametric Statistics → Experimental Design & Statistics

Core Idea

Mann–Whitney U test is treated here as the recurring nonparametric statistics identity summarized by this source-grounded definition: The Mann–Whitney U test (also called the Mann–Whitney–Wilcoxon (MWW/MWU), Wilcoxon rank-sum test, or Wilcoxon–Mann–Whitney test) is a nonparametric statistical test of the null hypothesis that randomly selected values X and Y from two populations have the same distribution.

The Mann–Whitney U test (also called the Mann–Whitney–Wilcoxon (MWW/MWU), Wilcoxon rank-sum test, or Wilcoxon–Mann–Whitney test) is a nonparametric statistical test of the null hypothesis that randomly selected values X and Y from two populations have the same distribution. The value of U calculated by the test can be converted to a measure of effect size by dividing it by the maximum value of U, which is the product of the sizes of the two samples being compared. This measure is the probability that the value of a random observation from the higher group will be greater than that of a random observation from the lower group.

Nonparametric tests used on two dependent samples are the sign test and the Wilcoxon signed-rank test. A measure of the central tendencies of the two groups (means or medians; since the Mann–Whitney U test is an ordinal test, medians are usually recommended). The responses are at least ordinal (i.e., one can at least say, of any two observations, which is the greater),.

For Mann–Whitney U test, the abstraction is narrower than the article's general subject matter: a positive case must preserve The Mann–Whitney U test (also called the Mann–Whitney–Wilcoxon (MWW/MWU), Wilcoxon rank-sum test, or Wilcoxon–Mann–Whitney test) is a nonparametric statistical test of the null hypothesis that randomly selected values X and Y from two populations have the same distribution. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in nonparametric statistics, which is why this identity is domain-specific rather than prime.

Structural Signature

Sig role-phrases:

• Defining carrier — As a sample statistic, the common language effect size is computed by forming all possible pairs between the two groups, then finding the proportion of pairs that support a direction (say, that items from group 1 are larger than items from group 2).
• Constitutive relation — This probability can be simply calculated by dividing U by its maximum value, which is the product of the two sample sizes.
• Operating condition — Rank the animals by the time they take to complete the course, so give the first animal home rank 12, the second rank 11, and so forth.
• Recognition evidence — The sum of the ranks achieved by the hares is , leading to.
• Admissible variation — Treatment A decreased weight by HLΔ = 5 kg (0.95 CL [2, 9] kg, , ).".
• Characteristic consequence — If the number of ties is small (and especially if there are no large tie bands) ties can be ignored when doing calculations by hand.
• Failure boundary — Because of this equivalence, the common language effect size is also directly related to the Gini coefficient of a classifier (equivalent to Somers' D in this context) by the linear transformation G = 2f - 1.

What It Is Not

• Not the whole field of nonparametric statistics. The node requires the specific identity stated by The Mann–Whitney U test (also called the Mann–Whitney–Wilcoxon (MWW/MWU), Wilcoxon rank-sum test, or Wilcoxon–Mann–Whitney test) is a nonparametric statistical test of the null hypothesis that randomly selected values X and Y from two populations have the same distribution.
• Not an over-broad reading. However it would be rare to find such an extensive report in a document whose major topic was not statistical inference.
• Not an over-broad reading. Hence, except in special cases, the Mann–Whitney U test and the t-test do not test the same hypotheses and should be compared with this in mind.
• Not an over-broad reading. This comparison in efficiency, however, should be interpreted with caution, as Mann–Whitney and the t-test do not test the same quantities.
• Not automatically Nemenyi test. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application

Mann–Whitney U test applies literally inside nonparametric statistics wherever the source-defined carrier and relation can be established. Its documented habitats include:

• The sample sizes. In practice some of this information may already have been supplied and common sense should be used in deciding whether to repeat it.
• Calculations. The smaller value of U 1 and U 2 is the one used when consulting significance tables.
• ExamplesIllustration of calculation methods. Using the direct method, we take each tortoise in turn, and count the number of hares it beats, getting 6, 1, 1, 1, 1, 1, which means that.
• A more computationally-efficient form with factored out. Note that since , the mean used in the normal approximation is the mean of the two values of U.
• A more computationally-efficient form with factored out. Therefore, the absolute value of the z-statistic calculated will be same whichever value of U is used.
• Effect sizes. It is a widely recommended practice for scientists to report an effect size for an inferential test.

Outside nonparametric statistics, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Theory or should be marked as analogy.

Clarity

A clear use of Mann–Whitney U test names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is The Mann–Whitney U test (also called the Mann–Whitney–Wilcoxon (MWW/MWU), Wilcoxon rank-sum test, or Wilcoxon–Mann–Whitney test) is a nonparametric statistical test of the null hypothesis that randomly selected values X and Y from two populations have the same distribution. The strongest recognition evidence in the frozen account is: The sum of the ranks achieved by the hares is , leading to. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification However it would be rare to find such an extensive report in a document whose major topic was not statistical inference. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity

Mann–Whitney U test compresses multiple nonparametric statistics details into a stable diagnostic relation. The source shows both the central mechanism—this probability can be simply calculated by dividing U by its maximum value, which is the product of the two sample sizes.—and the practical consequence—if the number of ties is small (and especially if there are no large tie bands) ties can be ignored when doing calculations by hand. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning

1. Type the carrier. Identify the nonparametric statistics entities to which the claim applies.
2. State the relation. Use the source-grounded identity: The Mann–Whitney U test (also called the Mann–Whitney–Wilcoxon (MWW/MWU), Wilcoxon rank-sum test, or Wilcoxon–Mann–Whitney test) is a nonparametric statistical test of the null hypothesis that randomly selected values X and Y from two populations have the same distribution.
3. Check operation and conditions. Rank the animals by the time they take to complete the course, so give the first animal home rank 12, the second rank 11, and so forth.
4. Demand recognition evidence. The sum of the ranks achieved by the hares is , leading to.
5. Test variation. Change an implementation or setting while preserving treatment A decreased weight by HLΔ = 5 kg (0.95 CL [2, 9] kg, , ).".
6. Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
7. Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Theory.

Knowledge Transfer

Within the home domain. Knowledge about Mann–Whitney U test transfers literally when a new case preserves the same carrier type, relation, and recognition test. In practice some of this information may already have been supplied and common sense should be used in deciding whether to repeat it. The smaller value of U 1 and U 2 is the one used when consulting significance tables.

Beyond the home domain. No canonical parent is asserted for Mann–Whitney U test. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples

Canonical

AUC k,k will always be zero but, unlike in the two-class case, generally , which is why the M measure sums over all (k,ℓ) pairs, in effect using the average of AUC k,ℓ and AUC ℓ,k. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → The Mann–Whitney U test (also called the Mann–Whitney–Wilcoxon (MWW/MWU), Wilcoxon rank-sum test, or Wilcoxon–Mann–Whitney test) is a nonparametric statistical test of the null hypothesis that randomly selected values X and Y from two populations have the same distribution; recognition evidence → The sum of the ranks achieved by the hares is , leading to

Applied / In Practice

Some books tabulate statistics equivalent to U, such as the sum of ranks in one of the samples, rather than U itself. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Calculations; invariant → The Mann–Whitney U test (also called the Mann–Whitney–Wilcoxon (MWW/MWU), Wilcoxon rank-sum test, or Wilcoxon–Mann–Whitney test) is a nonparametric statistical test of the null hypothesis that randomly selected values X and Y from two populations have the same distribution; boundary → the case exits the class when however it would be rare to find such an extensive report in a document whose major topic was not statistical inference

Structural Tensions

T1 — Stable identity versus admissible variation. However it would be rare to find such an extensive report in a document whose major topic was not statistical inference. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. Hence, except in special cases, the Mann–Whitney U test and the t-test do not test the same hypotheses and should be compared with this in mind. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. This comparison in efficiency, however, should be interpreted with caution, as Mann–Whitney and the t-test do not test the same quantities. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. If one desires a simple shift interpretation, the Mann–Whitney U test should not be used when the distributions of the two samples are very different, as it can give erroneous interpretation of significant results. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. As a sample statistic, the common language effect size is computed by forming all possible pairs between the two groups, then finding the proportion of pairs that support a direction (say, that items from group 1 are larger than items from group 2). The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Mann–Whitney U test literally, co-instantiate Theory, or only resemble it?

T6 — Autonomy versus reduction. This probability can be simply calculated by dividing U by its maximum value, which is the product of the two sample sizes. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Mann–Whitney U test distinguish that the broader parent Theory leaves together?

Structural–Framed Character

Mann–Whitney U test is structural-leaning. Its structural side is the repeatable organization summarized by The Mann–Whitney U test (also called the Mann–Whitney–Wilcoxon (MWW/MWU), Wilcoxon rank-sum test, or Wilcoxon–Mann–Whitney test) is a nonparametric statistical test of the null hypothesis that randomly selected values X and Y from two populations have the same distribution. Its framed side is the nonparametric statistics vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: Rank the animals by the time they take to complete the course, so give the first animal home rank 12, the second rank 11, and so forth. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Theory. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent

What is skeletal. The Mann–Whitney U test (also called the Mann–Whitney–Wilcoxon (MWW/MWU), Wilcoxon rank-sum test, or Wilcoxon–Mann–Whitney test) is a nonparametric statistical test of the null hypothesis that randomly selected values X and Y from two populations have the same distribution. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: As a sample statistic, the common language effect size is computed by forming all possible pairs between the two groups, then finding the proportion of pairs that support a direction (say, that items from group 1 are larger than items from group 2). This probability can be simply calculated by dividing U by its maximum value, which is the product of the two sample sizes. It further constrains recognition and variation through: Rank the animals by the time they take to complete the course, so give the first animal home rank 12, the second rank 11, and so forth. The sum of the ranks achieved by the hares is , leading to.

What is domain-bound. nonparametric statistics supplies the operative entities, technical vocabulary, warrants, and exceptions that make Mann–Whitney U test literal. Its documented scope includes the condition that In practice some of this information may already have been supplied and common sense should be used in deciding whether to repeat it. Another bounded application condition is that The smaller value of U 1 and U 2 is the one used when consulting significance tables. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—Treatment A decreased weight by HLΔ = 5 kg (0.95 CL [2, 9] kg, , ).".—and future graph densification may discover a defensible relation only if it preserves that boundary.

Instantiates / Related Primes

• Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Mann–Whitney U test. The reviewed identity is: The Mann–Whitney U test (also called the Mann–Whitney–Wilcoxon (MWW/MWU), Wilcoxon rank-sum test, or Wilcoxon–Mann–Whitney test) is a nonparametric statistical test of the null hypothesis that randomly selected values X and Y from two populations have the same distribution. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
• Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Neighborhood in Abstraction Space

Mann–Whitney U test sits in a sparse region of the domain-specific corpus (62^nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Statistical Tests & Choice Measurement (7 abstractions)

Nearest neighbors

• Interacting Particle System — 0.85
• Entropy estimation — 0.85
• Scale parameter — 0.85
• Reproductive value (population genetics) — 0.84
• Score (statistics) — 0.84

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With

• Theory. The parent omits the specialist differentia. Tell: Can the case establish The Mann–Whitney U test (also called the Mann–Whitney–Wilcoxon (MWW/MWU), Wilcoxon rank-sum test, or Wilcoxon–Mann–Whitney test) is a nonparametric statistical test of the null hypothesis that randomly selected values X and Y from two populations have the same distribution?
• Nemenyi test. A rank-based post-hoc multiple-comparison procedure that identifies pairs of treatments whose average ranks differ beyond a familywise-error-controlled critical distance after repeated-block comparison. Tell: Which entry's carrier, operation, and failure condition are satisfied?
• Kruskal–Wallis Test. An omnibus nonparametric test that pools and ranks observations from independent groups, then measures whether their rank sums differ more than expected under an exchangeable common-distribution null. Tell: Which entry's carrier, operation, and failure condition are satisfied?
• Wald test. A hypothesis test comparing an unrestricted parameter estimate with a constrained null value using its estimated covariance as a precision weight. Tell: Which entry's carrier, operation, and failure condition are satisfied?
• A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Mann–Whitney U test remain present if the detector or downstream effect changed?
• A metaphorical analogue. A similar shape outside nonparametric statistics lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Theory?

References

• Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Mann%E2%80%93Whitney_U_test (revision 1369120478).
• Preserved source candidate: https://www.jstor.org/stable/2283092
• Preserved source candidate: https://journals.sagepub.com/doi/10.1177/1536867X1201200202
• Preserved source candidate: https://www.researchgate.net/publication/279580873
• Preserved source candidate: https://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_nonparametric/bs704_nonparametric4.html
• Preserved source candidate: https://www.math.ucdavis.edu/~saito/data/roc/fawcett-roc.pdf
• Preserved source candidate: http://doi.apa.org/getdoi.cfm?doi=10.1037/a0024338
• Preserved source candidate: https://onlinepubs.trb.org/onlinepubs/nchrp/cd-22/manual/v2chapter6.pdf
• Preserved source candidate: https://link.springer.com/10.1007/978-3-030-02914-2

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.