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MannWhitneyTest
✖
MannWhitneyTest

✖
MannWhitneyTest[{data1,data2}]

tests whether the medians of data1 and data2 are equal.

✖
MannWhitneyTest[dspec,μ0]

tests the median difference against μ0.

✖
MannWhitneyTest[dspec,μ0,"property"]

returns the value of "property".

Details and Options

  • MannWhitneyTest performs a hypothesis test on data1 and data2 with null hypothesis that the true median difference against that .
  • By default, a probability value or -value is returned.
  • A small -value suggests that it is unlikely that is true.
  • The data in dspec can be univariate {x1,x2,…} or multivariate {{x1,y1,…},{x2,y2,…},…}.
  • The argument μ0 can be a real number or a real vector with length equal to the dimension of the data.
  • MannWhitneyTest assumes that the data is elliptically symmetric about a common spatial median in the multivariate case.
  • MannWhitneyTest[dspec,μ0,"HypothesisTestData"] returns a HypothesisTestData object htd that can be used to extract additional test results and properties using the form htd["property"].
  • MannWhitneyTest[dspec,μ0,"property"] can be used to directly give the value of "property".
  • Properties related to the reporting of test results include:
  • "DegreesOfFreedom"the degrees of freedom used in a test
    "PValue"list of -values
    "PValueTable"formatted table of -values
    "ShortTestConclusion"a short description of the conclusion of a test
    "TestConclusion"a description of the conclusion of a test
    "TestData"list of pairs of test statistics and -values
    "TestDataTable"formatted table of -values and test statistics
    "TestStatistic"list of test statistics
    "TestStatisticTable"formatted table of test statistics
  • The following options can be used:
  • AlternativeHypothesis "Unequal"the inequality for the alternative hypothesis
    MaxIterations Automaticmax iterations for multivariate median tests
    Method Automaticthe method to use for computing -values
    SignificanceLevel 0.05cutoff for diagnostics and reporting
  • For univariate samples, MannWhitneyTest performs the Mann–Whitney -test for median differences of independent samples. The following methods are available:
  • Automaticchooses depending on the size of the sample
    "Asymptotic"large sample statistic is asymptotically normally distributed
    "Exact"small sample values can be computed exactly
    "Permutation"simulation based
  • For the "Asymptotic" and "Permutation" methods, a correction for ties is applied; the test statistic is corrected for continuity, and it is assumed to follow a NormalDistribution.
  • For multivariate samples, MannWhitneyTest performs an extension of the Mann–Whitney -test using spatial ranks. The test statistic is assumed to asymptotically follow a ChiSquareDistribution[dim] where dim is the dimension of dspec.
  • For the MannWhitneyTest, a cutoff is chosen such that is rejected only if . The value of used for the "TestConclusion" and "ShortTestConclusion" properties is controlled by the SignificanceLevel option. By default, is set to 0.05.

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

Test whether the medians of two independent populations differ:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-lq5ldd

The median difference :

In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-soqc9
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-bolh4v
Out[3]=3

At the 0.05 level, the medians are significantly different:

In[4]:=4
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-csendg
Out[4]=4

Compare the locations of multivariate populations:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-2lnxj

The median difference vector :

In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-nj57pj
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-b6etdn
Out[3]=3

At the 0.05 level, is not significantly different from {1,2}:

In[4]:=4
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-j55r4w
Out[4]=4

Scope  (9)Survey of the scope of standard use cases

Testing  (6)

Test versus :

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-ekpnsf

The -values are generally small when the locations are not equal:

In[4]:=4
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-bud71n
Out[4]=4

The -values are generally large when the locations are equal:

In[5]:=5
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-nnksx4
Out[5]=5

Test versus :

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-exhx0q

The order of the datasets affects the test results:

In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-f3e5kf
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-jqamzo
Out[3]=3

Test whether the median difference vector of two multivariate populations is the zero vector:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-cbbn53
In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-fnaxqn
In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-f9ns4d
Out[3]=3

Alternatively, test against {1,0,-1,0}:

In[4]:=4
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-hkcfuq
Out[4]=4

Create a HypothesisTestData object for repeated property extraction:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-0x5m5
In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-cc8eh1

The properties available for extraction:

In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-frvg20
Out[3]=3

Extract some properties from a HypothesisTestData object:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-c03go
In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-bpn9dr

The -value and test statistic:

In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-365dq
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-bn5rjv
Out[4]=4

Extract any number of properties simultaneously:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-bycagv
In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-dmu6hk

The -value and test statistic from a Mann–Whitney test:

In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-i6fwj7
Out[3]=3

Reporting  (3)

Tabulate the test results:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-ba6zb1
In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-hb1mu5
In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-hh3kq
Out[3]=3

Retrieve the entries from a test table for customized reporting:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-cg07e5
In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-98x7a
In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-fforak
Out[3]=3

Tabulate -values or test statistics:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-fr3ezf
In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-blo8x
In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-g8i1dt
Out[3]=3

The -value from the table:

In[4]:=4
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-o0wuj
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-dw7vzl
Out[5]=5

The test statistic from the table:

In[6]:=6
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-bitsqd
Out[6]=6

Generalizations & Extensions  (2)Generalized and extended use cases

The Mann–Whitney test ignores the time stamps when the input is a TimeSeries:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-7py5sj
In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-0467fg
In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-57bf57
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-vvd88
Out[4]=4

The Mann–Whitney test recognizes the path structure of a TemporalData with exactly two paths:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-qry3ls
In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-b1l4g0
Out[2]=2

Use the values directly:

In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-jdvl55
Out[3]=3

Options  (12)Common values & functionality for each option

AlternativeHypothesis  (4)

A two-sided test is performed by default:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-bgoxnx

Test versus :

In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-he0w0s
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-jqt2u8
Out[3]=3

Perform a two-sided test or a one-sided alternative:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-kwmm8d

Test versus :

In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-dy0fuc
Out[2]=2

Test versus :

In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-g8h639
Out[3]=3

Test versus :

In[4]:=4
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-f19ykz
Out[4]=4

Perform tests with one-sided alternatives when μ0 is given:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-cay5xk
In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-v01gr
Out[2]=2

Test versus :

In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-ci67wa
Out[3]=3

Test versus :

In[4]:=4
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-m3sbc
Out[4]=4

The two-sided test is symmetric under the exchange of the datasets:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-e2hwfp
In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-k0o4ez
Out[2]=2

The one-sided tests are not symmetric under the exchange of the datasets:

In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-n8xq3
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-p54su9
Out[4]=4

However, for the permutation method even the (default) two-sided test is not symmetric:

In[5]:=5
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-gezxwu
Out[5]=5

MaxIterations  (2)

Set the maximum number of iterations to use for multivariate tests:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-wsdxy

By default, 250 iterations are allowed:

In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-l6esri
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-eb62z0
Out[4]=4

Setting the maximum number of iterations may result in lack of convergence:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-z0x96

The -values are not equivalent:

In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-hu3jt7
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-xar3i
Out[3]=3

Method  (5)

For small samples, -values are computed using an exact formula by default:

In[5]:=5
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-n5553
In[6]:=6
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-798j7
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-d8n542
Out[7]=7

For large samples, -values are computed using asymptotic test statistic distributions by default:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-v21sy
In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-fifdab
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-b9g9ag
Out[3]=3

Permutation methods can be used:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-hy6ohx
In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-f69ts
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-drrw70
Out[3]=3

Set the number of permutations to use:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-cv172p
In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-co5msl
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-dicab0
Out[3]=3

By default, random permutations are used:

In[4]:=4
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-ktxmmb
Out[4]=4

Set the seed used for generating random permutations:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-cjrea
In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-0du0
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-f7dru0
Out[3]=3

SignificanceLevel  (1)

The significance level is used for "TestConclusion" and "ShortTestConclusion":

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-bhkod7
In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-lasldz
In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-hykroc
In[4]:=4
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-bvt7nt
In[5]:=5
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-hpqqgh
In[6]:=6
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-flavjg
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-m2oyg2
Out[7]=7

Applications  (3)Sample problems that can be solved with this function

Test whether the medians of some populations are equal:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-4kn67
In[4]:=4
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-elrcz
Out[4]=4

The medians of the first two populations are similar:

In[5]:=5
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-iwddm4
Out[5]=5

The median of the third population is different from the first:

In[6]:=6
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-bwrr6i
Out[6]=6

It has been observed that the duration of Old Faithful geyser eruptions is proportional to the time elapsed since the previous eruption:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-f5fyyr
In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-doox3i
In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-sv0lc
Out[3]=3

Assuming one hour is a long wait for an eruption, test the statement that long waits lead to long eruption durations:

In[4]:=4
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-po4rt
In[5]:=5
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-50pb8
Out[5]=5

Two hundred Australian crabs were collected, and five morphological measures were taken for each crab. The data is organized by type and gender:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-izhcxv
In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-tfoq2
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-cwtxjk

Determine if there is a difference in the first four morphological measures for the two varieties:

In[4]:=4
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-b3cj4o
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-hfs4ix
Out[5]=5

Compare the morphological measures across the genders:

In[6]:=6
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-gxu7d3
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-nuygkj
Out[7]=7

Properties & Relations  (4)Properties of the function, and connections to other functions

For univariate data, the test statistic follows a NormalDistribution[0,1] under :

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-b9u4tx
In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-mn9zg
In[4]:=4
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-be6r5c

A large sample approximation to the NormalDistribution:

In[5]:=5
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-i6xczx
In[6]:=6
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-evb0jl
In[7]:=7
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-dwvfqv
Out[7]=7
In[8]:=8
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-eb3599
Out[8]=8

For multivariate data, the test statistic follows a ChiSquareDistribution[dim] under :

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-fcw44i
In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-mktqe4
In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-kx8eo0
In[4]:=4
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-2v5dw
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-bfkrj9
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-pdfbx5
Out[6]=6

The test statistic is computed by pooling and ranking the data:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-czw9ty
In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-liumv

In the absence of ties, Ordering can compute the ranks:

In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-z1gqk
In[4]:=4
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-easm4
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-gbsp9h
Out[5]=5

The exact -value matches the frequencies of the test statistic when considering every possible permutation for a given size of the datasets:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-cjs4w1
In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-p701ns
Out[2]=2

Show few permutations:

In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-gd3kz7
Out[3]=3

Compute the statistics and -values for all the permutation pairs:

In[4]:=4
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-eerfer

Show few results:

In[5]:=5
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-c7w8lu
Out[5]=5

Compute frequencies by normalizing the counts of the statistic:

In[6]:=6
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-jhfe4u

The statistic is an integer between and the product of the lengths of the data inputs—here :

In[7]:=7
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-eh8wbb
Out[7]=7

Since the -value is the cumulative mass function (CDF) of the test distribution, it needs to be compared to the cumulative frequencies:

In[8]:=8
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-dv312t
Out[8]=8

By convention, the -values start at :

In[9]:=9
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-3wkst
Out[9]=9

The exact -value is computed using the following recursion relation:

In[10]:=10
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-1g2hk
In[11]:=11
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-h1yxsm
In[12]:=12
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-p7ie3
Out[12]=12

Possible Issues  (1)Common pitfalls and unexpected behavior

For the permutation method, the two-sided test is not symmetric under the exchange of the datasets:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-nwe11v
In[2]:=2
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-hhzqx3
Out[2]=2

Neat Examples  (1)Surprising or curious use cases

Compute the statistic when the null hypothesis is true:

In[1]:=1
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-2qqg3c
In[3]:=3
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-ywy3ty

The test statistic given a particular alternative:

In[4]:=4
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-612aqt
In[5]:=5
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-c5cy2n

Compare the distributions of the test statistics:

In[6]:=6
✖
https://wolfram.com/xid/0b7ekgesiu28r68m-87eb6q
Out[6]=6
Wolfram Research (2010), MannWhitneyTest, Wolfram Language function, https://reference.wolfram.com/language/ref/MannWhitneyTest.html.
✖
Wolfram Research (2010), MannWhitneyTest, Wolfram Language function, https://reference.wolfram.com/language/ref/MannWhitneyTest.html.

Text

Wolfram Research (2010), MannWhitneyTest, Wolfram Language function, https://reference.wolfram.com/language/ref/MannWhitneyTest.html.

✖
Wolfram Research (2010), MannWhitneyTest, Wolfram Language function, https://reference.wolfram.com/language/ref/MannWhitneyTest.html.

CMS

Wolfram Language. 2010. "MannWhitneyTest." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MannWhitneyTest.html.

✖
Wolfram Language. 2010. "MannWhitneyTest." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MannWhitneyTest.html.

APA

Wolfram Language. (2010). MannWhitneyTest. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MannWhitneyTest.html

✖
Wolfram Language. (2010). MannWhitneyTest. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MannWhitneyTest.html

BibTeX

@misc{reference.wolfram_2025_mannwhitneytest, author="Wolfram Research", title="{MannWhitneyTest}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/MannWhitneyTest.html}", note=[Accessed: 29-March-2025 ]}

✖
@misc{reference.wolfram_2025_mannwhitneytest, author="Wolfram Research", title="{MannWhitneyTest}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/MannWhitneyTest.html}", note=[Accessed: 29-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_mannwhitneytest, organization={Wolfram Research}, title={MannWhitneyTest}, year={2010}, url={https://reference.wolfram.com/language/ref/MannWhitneyTest.html}, note=[Accessed: 29-March-2025 ]}

✖
@online{reference.wolfram_2025_mannwhitneytest, organization={Wolfram Research}, title={MannWhitneyTest}, year={2010}, url={https://reference.wolfram.com/language/ref/MannWhitneyTest.html}, note=[Accessed: 29-March-2025 ]}