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SignTest
✖
SignTest

✖
SignTest[data]

tests whether the median of data is zero.

✖
SignTest[{data1,data2}]

tests whether the median of data1– data2 is zero.

✖
SignTest[dspec,μ0]

tests a location measure against μ0.

✖
SignTest[dspec,μ0,"property"]

returns the value of "property".

Details and Options

  • SignTest tests the null hypothesis against the alternative hypothesis :
  • data
    {data1,data2}
  • where μ is the population median for data and μ12 is the median of the paired differences of the two datasets .
  • 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,…},…}.
  • If two samples are given, they must be of equal length.
  • The argument μ0 can be a real number or a real vector with length equal to the dimension of the data.
  • SignTest[dspec,μ0,"HypothesisTestData"] returns a HypothesisTestData object htd that can be used to extract additional test results and properties using the form htd["property"].
  • SignTest[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
  • For univariate samples, SignTest performs the sign test for medians of paired samples. The test statistic is assumed to follow a BinomialDistribution[n,1/2] where n is the number of elements in dspec not equal to μ0.
  • For multivariate samples, SignTest performs an affine invariant test for paired samples using spatial signs. The test statistic is assumed to follow a ChiSquareDistribution[dim] where dim is the dimension of dspec.
  • 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 the SignTest, 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  (4)Summary of the most common use cases

Test whether the median of a population is zero:

In[1]:=1
✖
https://wolfram.com/xid/0enwmqlye-bcn0x0
In[2]:=2
✖
https://wolfram.com/xid/0enwmqlye-brgvwg
Out[2]=2

Test whether the spatial median of a multivariate population is some value:

In[1]:=1
✖
https://wolfram.com/xid/0enwmqlye-etsac
In[2]:=2
✖
https://wolfram.com/xid/0enwmqlye-cgw1ht
Out[2]=2

Compute the test statistic:

In[3]:=3
✖
https://wolfram.com/xid/0enwmqlye-j03gr3
Out[3]=3

Compare the median difference for paired data to a particular value:

In[1]:=1
✖
https://wolfram.com/xid/0enwmqlye-dum1jr
In[2]:=2
✖
https://wolfram.com/xid/0enwmqlye-c3rena
Out[2]=2

Report the test results in a table:

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

Create a HypothesisTestData object for repeated property extraction:

In[1]:=1
✖
https://wolfram.com/xid/0enwmqlye-crwuz3
In[2]:=2
✖
https://wolfram.com/xid/0enwmqlye-jwsqlx
Out[2]=2

A list of available properties:

In[3]:=3
✖
https://wolfram.com/xid/0enwmqlye-hi2110
Out[3]=3

Extract a single property or a list of properties:

In[4]:=4
✖
https://wolfram.com/xid/0enwmqlye-cad9bv
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0enwmqlye-b6k7bw
Out[5]=5

Scope  (13)Survey of the scope of standard use cases

Testing  (10)

Test versus :

In[1]:=1
✖
https://wolfram.com/xid/0enwmqlye-mk1rwj

The -values are typically large when the median is close to μ0:

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

The -values are typically small when the location is far from μ0:

In[4]:=4
✖
https://wolfram.com/xid/0enwmqlye-b15esj
Out[4]=4

Using Automatic is equivalent to testing for a median of zero:

In[1]:=1
✖
https://wolfram.com/xid/0enwmqlye-bbn60w
In[2]:=2
✖
https://wolfram.com/xid/0enwmqlye-cv9w6t
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0enwmqlye-nloyf
Out[3]=3

Test versus :

In[1]:=1
✖
https://wolfram.com/xid/0enwmqlye-ccr6o1

The -values are typically large when the median is close to μ0:

In[2]:=2
✖
https://wolfram.com/xid/0enwmqlye-fk03hb
Out[2]=2

The -values are typically small when the location is far from μ0:

In[3]:=3
✖
https://wolfram.com/xid/0enwmqlye-e810zu
Out[3]=3

Test whether the median vector of a multivariate population is the zero vector:

In[1]:=1
✖
https://wolfram.com/xid/0enwmqlye-bqd543
In[2]:=2
✖
https://wolfram.com/xid/0enwmqlye-hr2vr
Out[2]=2

Alternatively, test against {0.1,0,-0.05,0}:

In[3]:=3
✖
https://wolfram.com/xid/0enwmqlye-cze22
Out[3]=3

Test versus :

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

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

In[2]:=2
✖
https://wolfram.com/xid/0enwmqlye-bud71n
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0enwmqlye-nnksx4
Out[3]=3

Test versus :

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

The order of the datasets affects the test results:

In[2]:=2
✖
https://wolfram.com/xid/0enwmqlye-f3e5kf
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0enwmqlye-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/0enwmqlye-cbbn53
In[2]:=2
✖
https://wolfram.com/xid/0enwmqlye-fnaxqn
In[3]:=3
✖
https://wolfram.com/xid/0enwmqlye-f9ns4d
Out[3]=3

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

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

Create a HypothesisTestData object for repeated property extraction:

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

The properties available for extraction:

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

Extract some properties from a HypothesisTestData object:

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

The -value and test statistic:

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

Extract any number of properties simultaneously:

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

The -value and test statistic:

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

Reporting  (3)

Tabulate the test results:

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

Retrieve the entries from a test table for customized reporting:

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

Tabulate -values or test statistics:

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

The -value from the table:

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

The test statistic from the table:

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

Options  (9)Common values & functionality for each option

AlternativeHypothesis  (3)

A two-sided test is performed by default:

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

Test versus :

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

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

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

Test versus :

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

Test versus :

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

Test versus :

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

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

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

Test versus :

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

Test versus :

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

MaxIterations  (2)

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

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

By default, 500 iterations are allowed:

In[2]:=2
✖
https://wolfram.com/xid/0enwmqlye-l6esri
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0enwmqlye-eb62z0
Out[3]=3

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

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

The -values are not equivalent:

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

Method  (3)

By default, -values are computed using the BinomialDistribution for univariate data:

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

Asymptotic methods can be used for univariate data:

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

For multivariate data, only the asymptotic result is available:

In[1]:=1
✖
https://wolfram.com/xid/0enwmqlye-b21vkx
In[2]:=2
✖
https://wolfram.com/xid/0enwmqlye-xj27d
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0enwmqlye-cwvw4
Out[3]=3

SignificanceLevel  (1)

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

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

Applications  (2)Sample problems that can be solved with this function

A new sleeping aid was tested on eight patients. The number of minutes taken for each subject to fall asleep was recorded for a night taking the medication and for a night with a placebo:

In[1]:=1
✖
https://wolfram.com/xid/0enwmqlye-d2f0zn
In[3]:=3
✖
https://wolfram.com/xid/0enwmqlye-zrbrs
Out[3]=3

The SignTest does not detect a difference in the sleep aid and placebo:

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

The datasets, while very small, do not fail a test for normality:

In[5]:=5
✖
https://wolfram.com/xid/0enwmqlye-bszeq9
Out[5]=5

A more powerful PairedTTest shows a significant reduction in time to sleep with the sleep aid:

In[6]:=6
✖
https://wolfram.com/xid/0enwmqlye-dva4k
Out[6]=6

A group of 10 students with low assessments in mathematics and science was asked to participate in tutoring program. A test similar to the original assessment was administered after the program. The students' scores on the math and science portions of both assessments are as follows:

In[1]:=1
✖
https://wolfram.com/xid/0enwmqlye-bpvtuq
In[2]:=2
✖
https://wolfram.com/xid/0enwmqlye-bs111p
In[3]:=3
✖
https://wolfram.com/xid/0enwmqlye-nxajtb
Out[3]=3

There is a significant improvement in scores overall:

In[4]:=4
✖
https://wolfram.com/xid/0enwmqlye-c1ru7a
Out[4]=4

The Bonferroni-corrected tests of the individual components suggest that math scores alone account for the detected improvement:

In[5]:=5
✖
https://wolfram.com/xid/0enwmqlye-em7p9w
In[6]:=6
✖
https://wolfram.com/xid/0enwmqlye-frjnr9
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0enwmqlye-ki65yt
Out[7]=7

Properties & Relations  (6)Properties of the function, and connections to other functions

Conceptually, the SignTest counts the number of positive signs in a dataset:

In[1]:=1
✖
https://wolfram.com/xid/0enwmqlye-dhszi9
In[2]:=2
✖
https://wolfram.com/xid/0enwmqlye-mv1i75
In[3]:=3
✖
https://wolfram.com/xid/0enwmqlye-b58j9i
In[4]:=4
✖
https://wolfram.com/xid/0enwmqlye-gbu3q8
Out[4]=4

For univariate data, the test statistic follows a BinomialDistribution, ignoring zeros:

In[5]:=5
✖
https://wolfram.com/xid/0enwmqlye-ecw34
In[6]:=6
✖
https://wolfram.com/xid/0enwmqlye-eogj
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0enwmqlye-7teuu
Out[7]=7

The SignTest is generally less powerful than other hypothesis tests for location:

In[1]:=1
✖
https://wolfram.com/xid/0enwmqlye-l487ko
In[2]:=2
✖
https://wolfram.com/xid/0enwmqlye-e4yza
In[3]:=3
✖
https://wolfram.com/xid/0enwmqlye-iykx13
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0enwmqlye-gahra
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0enwmqlye-05hmt
Out[5]=5

For multivariate data, spatial signs are used when computing the test statistic:

In[1]:=1
✖
https://wolfram.com/xid/0enwmqlye-iwe5v
In[2]:=2
✖
https://wolfram.com/xid/0enwmqlye-j2dthq

Spatial signs tend to cluster when the spatial median is nonzero:

In[3]:=3
✖
https://wolfram.com/xid/0enwmqlye-bxzy6
Out[3]=3

The amount of clustering is quantified by the test statistic:

In[4]:=4
✖
https://wolfram.com/xid/0enwmqlye-3w8ek
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0enwmqlye-do94ni
Out[5]=5

The test statistic follows a ChiSquareDistribution[p]:

In[6]:=6
✖
https://wolfram.com/xid/0enwmqlye-38ivn
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0enwmqlye-ff80um
Out[7]=7
In[8]:=8
✖
https://wolfram.com/xid/0enwmqlye-jvazn
Out[8]=8

The test statistic is affine invariant for multivariate data:

In[1]:=1
✖
https://wolfram.com/xid/0enwmqlye-gu9uw4
In[2]:=2
✖
https://wolfram.com/xid/0enwmqlye-8xxnw
In[3]:=3
✖
https://wolfram.com/xid/0enwmqlye-dqfr71
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0enwmqlye-lwz8r7
Out[4]=4

The sign test works with the values only when the input is a TimeSeries:

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

The sign test works with all the values together when the input is a TemporalData:

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

Test all the values only:

In[3]:=3
✖
https://wolfram.com/xid/0enwmqlye-t6toez
Out[3]=3

Test the difference of the medians of the two paths:

In[4]:=4
✖
https://wolfram.com/xid/0enwmqlye-kxzwhd
In[5]:=5
✖
https://wolfram.com/xid/0enwmqlye-jdvl55
Out[5]=5

Neat Examples  (2)Surprising or curious use cases

Compute the statistic when the null hypothesis is true:

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

The test statistic given a particular alternative:

In[4]:=4
✖
https://wolfram.com/xid/0enwmqlye-c5cy2n

Compare the distributions of the test statistics:

In[5]:=5
✖
https://wolfram.com/xid/0enwmqlye-87eb6q
Out[5]=5

The distribution of spatial signs in three dimensions shows that larger deviations from a zero mean vector produce more highly clustered spatial signs and larger sign statistics:

In[1]:=1
✖
https://wolfram.com/xid/0enwmqlye-clce4j
In[2]:=2
✖
https://wolfram.com/xid/0enwmqlye-ezx1n
In[3]:=3
✖
https://wolfram.com/xid/0enwmqlye-e98rvq
In[4]:=4
✖
https://wolfram.com/xid/0enwmqlye-s842b
Out[4]=4
Wolfram Research (2010), SignTest, Wolfram Language function, https://reference.wolfram.com/language/ref/SignTest.html.
✖
Wolfram Research (2010), SignTest, Wolfram Language function, https://reference.wolfram.com/language/ref/SignTest.html.

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_signtest, author="Wolfram Research", title="{SignTest}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/SignTest.html}", note=[Accessed: 29-March-2025 ]}

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@misc{reference.wolfram_2025_signtest, author="Wolfram Research", title="{SignTest}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/SignTest.html}", note=[Accessed: 29-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_signtest, organization={Wolfram Research}, title={SignTest}, year={2010}, url={https://reference.wolfram.com/language/ref/SignTest.html}, note=[Accessed: 29-March-2025 ]}

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@online{reference.wolfram_2025_signtest, organization={Wolfram Research}, title={SignTest}, year={2010}, url={https://reference.wolfram.com/language/ref/SignTest.html}, note=[Accessed: 29-March-2025 ]}