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PairedTTest
✖
PairedTTest

✖
PairedTTest[data]

tests whether the mean of data is zero.

✖
PairedTTest[{data1,data2}]

tests whether the mean of data1– data2 is zero.

✖
PairedTTest[dspec,μ0]

tests a location measure against μ0.

✖
PairedTTest[dspec,μ0,"property"]

returns the value of "property".

Details and Options

  • PairedTTest tests the null hypothesis against the alternative hypothesis :
  • data
    {data1,data2}
  • where μ is the population mean for data and μ12 is the mean 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.
  • PairedTTest[dspec,μ0,"HypothesisTestData"] returns a HypothesisTestData object htd that can be used to extract additional test results and properties using the form htd["property"].
  • PairedTTest[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
  • PairedTTest is more powerful than the TTest when samples are matched.
  • For univariate samples, PairedTTest performs the Student test for matched pairs. The test statistic is assumed to follow a StudentTDistribution.
  • For multivariate samples, PairedTTest performs Hotelling's test for matched pairs. The test statistic is assumed to follow a HotellingTSquareDistribution.
  • The following options can be used:
  • AlternativeHypothesis "Unequal"the inequality for the alternative hypothesis
    SignificanceLevel 0.05cutoff for diagnostics and reporting
    VerifyTestAssumptions Automaticwhat assumptions to verify
  • For the PairedTTest, 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. This value is also used in diagnostic tests of assumptions, including tests for normality, equal variance, and symmetry. By default, is set to 0.05.
  • Named settings for VerifyTestAssumptions in PairedTTest include:
  • "Normality"verify that all data is normally distributed

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

Test whether the mean of a population is zero:

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-g7mcn
In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-yfec
Out[2]=2

The full test table:

In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-bbkq82
Out[3]=3

Test whether the means of two dependent populations differ:

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-lq5ldd

The mean of the differences:

In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-soqc9
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-bolh4v
Out[3]=3

At the 0.05 level, the mean of the differenced data is not significantly different from 0:

In[4]:=4
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-csendg
Out[4]=4

Compare the locations of dependent multivariate populations:

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-2lnxj

The mean of the differences:

In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-nj57pj
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-b6etdn
Out[3]=3

At the 0.05 level, the mean of the differenced data is not significantly different from 0:

In[4]:=4
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-j55r4w
Out[4]=4

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

Testing  (10)

Test versus :

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-mk1rwj

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

In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-dcnktx
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-b15esj
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-bbn60w
In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-cv9w6t
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-nloyf
Out[3]=3

Test versus :

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-ccr6o1

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

In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-fk03hb
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-e810zu
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-bqd543
In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-hr2vr
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-cze22
Out[3]=3

Test whether the mean of differenced datasets is zero:

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-ekpnsf

The -values are generally small when the mean is not zero:

In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-bud71n
Out[2]=2

The -values are generally large when the mean is zero:

In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-nnksx4
Out[3]=3

Test whether the mean of differenced data is 3:

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-exhx0q

The order of the datasets affects the test results:

In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-f3e5kf
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-jqamzo
Out[3]=3

Test whether the mean vector of differenced multivariate data is the zero vector:

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-cbbn53
In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-f9ns4d
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-hkcfuq
Out[3]=3

Create a HypothesisTestData object for repeated property extraction:

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-0x5m5
In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-cc8eh1

The properties available for extraction:

In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-frvg20
Out[3]=3

Extract some properties from a HypothesisTestData object:

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-c03go
In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-bpn9dr

The -value, test statistic, and degrees of freedom:

In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-365dq
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-bn5rjv
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-c7xo2s
Out[5]=5

Extract any number of properties simultaneously:

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-bycagv
In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-dmu6hk

The -value, test statistic, and degrees of freedom:

In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-i6fwj7
Out[3]=3

Reporting  (3)

Tabulate the test results:

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-ba6zb1
In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-hb1mu5
In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-hh3kq
Out[3]=3

Retrieve the entries from a test table for customized reporting:

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-cg07e5
In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-98x7a
In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-fforak
Out[3]=3

Tabulate -values or test statistics:

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-fr3ezf
In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-blo8x
In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-g8i1dt
Out[3]=3

The -value from the table:

In[4]:=4
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-o0wuj
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-dw7vzl
Out[5]=5

The test statistic from the table:

In[6]:=6
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-bitsqd
Out[6]=6

Options  (8)Common values & functionality for each option

AlternativeHypothesis  (3)

A two-sided test is performed by default:

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-bgoxnx

Test versus :

In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-he0w0s
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-jqt2u8
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-kwmm8d

Test versus :

In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-dy0fuc
Out[2]=2

Test versus :

In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-g8h639
Out[3]=3

Test versus :

In[4]:=4
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-f19ykz
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-cay5xk
In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-v01gr
Out[2]=2

Test versus :

In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-ci67wa
Out[3]=3

Test versus :

In[4]:=4
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-m3sbc
Out[4]=4

SignificanceLevel  (2)

Set the significance level for diagnostic tests:

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-fn2ahc
In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-dwfi1
Out[2]=2

By default, 0.05 is used:

In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-cod4ca
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-bhkod7
In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-lasldz
In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-hykroc
In[4]:=4
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-bvt7nt
In[5]:=5
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-hpqqgh
In[6]:=6
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-flavjg
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-m2oyg2
Out[7]=7

VerifyTestAssumptions  (3)

By default, normality is tested:

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-db7e1z

Here normality is assumed:

In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-dnfybp
Out[2]=2

The result is the same, but a warning is issued:

In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-fx7k7n
Out[3]=3

Alternatively, use All:

In[4]:=4
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-djhg9b
Out[4]=4

Set the normality assumption to True:

In[5]:=5
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-f03ye
Out[5]=5

Bypassing diagnostic tests can save compute time:

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-hrliot
In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-mmqfh1
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-bglfb1
Out[3]=3

It is often useful to bypass diagnostic tests for simulation purposes:

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-kfpeh6
In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-cxjgtt
Out[2]=2

The assumptions of the test hold by design, so a great deal of time can be saved:

In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-cmfh98
Out[3]=3

The results are identical:

In[4]:=4
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-btr43n
Out[4]=4

Applications  (1)Sample problems that can be solved with this function

A group of 15 students is convinced that they were more prepared for the mathematics portion of the ACT than that of the SAT. Test this claim assuming that national ACT and SAT scores are normally distributed with means 21.7 and 528.5 and standard deviations 4.1 and 117.2, respectively:

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-ffl9kw
In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-k935ra

For comparison, the data should be normalized:

In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-ksaw0d

Density estimates for the individual and differenced scores:

In[5]:=5
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-d8ui3
Out[5]=5

can be rejected at the 5% level. The students' claim is not refuted:

In[6]:=6
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-dud2zh
Out[6]=6

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

PairedTTest is equivalent to a TTest for a single dataset:

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-tv5gf
In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-cs4lki
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-izj5k0
Out[3]=3

For two datasets, the PairedTTest is equivalent to a TTest of the paired differences:

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-f2r3wk
In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-ilce0f
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-c9a7mg
Out[3]=3

If the variance of the population is known, the more powerful PairedZTest can be used:

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-de5rmm
In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-2aqwh
In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-ez3vrg

The probability that the PairedZTest will return a -value smaller than the PairedTTest:

In[4]:=4
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-eknsou
Out[4]=4

If the data can be paired, the PairedTTest is more powerful than the TTest:

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-j7yihz

The paired test detects a significant difference where the unpaired test does not:

In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-jsty9
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-e5l7f9
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-7py5sj
In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-57bf57
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-vvd88
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-qry3ls
In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-b1l4g0
Out[2]=2

Test all the values only:

In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-t6toez
Out[3]=3

Test the difference of the means of the two paths:

In[4]:=4
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-kxzwhd
In[5]:=5
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-jdvl55
Out[5]=5

Possible Issues  (1)Common pitfalls and unexpected behavior

PairedTTest requires that the data be normally distributed:

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-gs49tr
In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-18i20
Out[2]=2

Use a median-based test instead:

In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-dsqwzr
Out[3]=3

Neat Examples  (1)Surprising or curious use cases

Compute the statistic when the null hypothesis is true:

In[1]:=1
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-2qqg3c
In[2]:=2
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-ywy3ty

The test statistic given a particular alternative:

In[3]:=3
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-c5cy2n

Compare the distributions of the test statistics:

In[4]:=4
✖
https://wolfram.com/xid/0bc6at4mouuyx7a-87eb6q
Out[4]=4
Wolfram Research (2010), PairedTTest, Wolfram Language function, https://reference.wolfram.com/language/ref/PairedTTest.html.
✖
Wolfram Research (2010), PairedTTest, Wolfram Language function, https://reference.wolfram.com/language/ref/PairedTTest.html.

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_pairedttest, author="Wolfram Research", title="{PairedTTest}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/PairedTTest.html}", note=[Accessed: 21-May-2025 ]}

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@misc{reference.wolfram_2025_pairedttest, author="Wolfram Research", title="{PairedTTest}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/PairedTTest.html}", note=[Accessed: 21-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_pairedttest, organization={Wolfram Research}, title={PairedTTest}, year={2010}, url={https://reference.wolfram.com/language/ref/PairedTTest.html}, note=[Accessed: 21-May-2025 ]}

✖
@online{reference.wolfram_2025_pairedttest, organization={Wolfram Research}, title={PairedTTest}, year={2010}, url={https://reference.wolfram.com/language/ref/PairedTTest.html}, note=[Accessed: 21-May-2025 ]}