# Student's t-test

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The t-test is any statistical hypothesis test in which the test statistic follows a Student's t-distribution under the null hypothesis.

A statistical hypothesis, sometimes called confirmatory data analysis, is a hypothesis that is testable on the basis of observing a process that is modeled via a set of random variables. A statistical hypothesis test is a method of statistical inference. Commonly, two statistical data sets are compared, or a data set obtained by sampling is compared against a synthetic data set from an idealized model. A hypothesis is proposed for the statistical relationship between the two data sets, and this is compared as an alternative to an idealized null hypothesis that proposes no relationship between two data sets. The comparison is deemed statistically significant if the relationship between the data sets would be an unlikely realization of the null hypothesis according to a threshold probability—the significance level. Hypothesis tests are used when determining what outcomes of a study would lead to a rejection of the null hypothesis for a pre-specified level of significance.

A test statistic is a statistic used in statistical hypothesis testing. A hypothesis test is typically specified in terms of a test statistic, considered as a numerical summary of a data-set that reduces the data to one value that can be used to perform the hypothesis test. In general, a test statistic is selected or defined in such a way as to quantify, within observed data, behaviours that would distinguish the null from the alternative hypothesis, where such an alternative is prescribed, or that would characterize the null hypothesis if there is no explicitly stated alternative hypothesis.

In probability and statistics, Student's t-distribution is any member of a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and the population standard deviation is unknown. It was developed by William Sealy Gosset under the pseudonym Student.

## Contents

A t-test is most commonly applied when the test statistic would follow a normal distribution if the value of a scaling term in the test statistic were known. When the scaling term is unknown and is replaced by an estimate based on the data, the test statistics (under certain conditions) follow a Student's t distribution. The t-test can be used, for example, to determine if the means of two sets of data are significantly different from each other.

In probability theory, the normaldistribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate.

In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution.

Data is a set of values of subjects with respect to qualitative or quantitative variables.

## History

The t-statistic was introduced in 1908 by William Sealy Gosset, a chemist working for the Guinness brewery in Dublin, Ireland. "Student" was his pen name. [1] [2] [3] [4]

In statistics, the t-statistic is the ratio of the departure of the estimated value of a parameter from its hypothesized value to its standard error. It is used in hypothesis testing via Student's t-test. For example, it is used in estimating the population mean from a sampling distribution of sample means if the population standard deviation is unknown.

William Sealy Gosset was an English statistician, chemist and brewer who served as Head Brewer of Guinness and Head Experimental Brewer of Guinness and was a pioneer of modern statistics. He pioneered small sample experimental design and analysis with an economic approach to the logic of uncertainty. Gosset published under the pen name Student and developed most famously Student's t-distribution – originally called Student's "z" – and "Student's test of statistical significance".

Guinness is a dark Irish dry stout that originated in the brewery of Arthur Guinness at St. James's Gate, Dublin, Ireland, in 1759. It is one of the most successful beer brands worldwide, brewed in almost 50 countries, and available in over 120. Sales in 2011 amounted to 850 million litres (220,000,000 US gal). It is popular with the Irish, both in Ireland and abroad. In spite of declining consumption since 2001, it is still the best-selling alcoholic drink in Ireland where Guinness & Co. Brewery makes almost €2 billion worth annually.

Gosset had been hired owing to Claude Guinness's policy of recruiting the best graduates from Oxford and Cambridge to apply biochemistry and statistics to Guinness's industrial processes. [2] Gosset devised the t-test as an economical way to monitor the quality of stout. The t-test work was submitted to and accepted in the journal Biometrika and published in 1908. [5] Company policy at Guinness forbade its chemists from publishing their findings, so Gosset published his statistical work under the pseudonym "Student" (see Student's t-distribution for a detailed history of this pseudonym, which is not to be confused with the literal term student ).

Biochemistry, sometimes called biological chemistry, is the study of chemical processes within and relating to living organisms. Biochemical processes give rise to the complexity of life.

Statistics is the discipline that concerns the collection, organization, displaying, analysis, interpretation and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments. See glossary of probability and statistics.

Stout is a dark, top-fermented beer with a number of variations, including dry stout, Baltic porter, milk stout, and imperial stout.

Guinness had a policy of allowing technical staff leave for study (so-called "study leave"), which Gosset used during the first two terms of the 1906–1907 academic year in Professor Karl Pearson's Biometric Laboratory at University College London. [6] Gosset's identity was then known to fellow statisticians and to editor-in-chief Karl Pearson. [7]

Karl Pearson was an English mathematician and biostatistician. He has been credited with establishing the discipline of mathematical statistics. He founded the world's first university statistics department at University College, London in 1911, and contributed significantly to the field of biometrics and meteorology. Pearson was also a proponent of social Darwinism and eugenics. Pearson was a protégé and biographer of Sir Francis Galton. He edited and completed both William Kingdon Clifford's Common Sense of the Exact Sciences (1885) and Isaac Todhunter's History of the Theory of Elasticity, Vol. 1 (1886-1893) & Vol. 2 (1893) following their deaths.

University College London, which has operated under the official name of UCL since 2005, is a public research university located in London, United Kingdom. It is a member institution of the federal University of London, and is the third largest university in the United Kingdom by total enrolment, and the largest by postgraduate enrolment.

## Uses

Among the most frequently used t-tests are:

• A one-sample location test of whether the mean of a population has a value specified in a null hypothesis.
• A two-sample location test of the null hypothesis such that the means of two populations are equal. All such tests are usually called Student's t-tests, though strictly speaking that name should only be used if the variances of the two populations are also assumed to be equal; the form of the test used when this assumption is dropped is sometimes called Welch's t-test. These tests are often referred to as "unpaired" or "independent samples" t-tests, as they are typically applied when the statistical units underlying the two samples being compared are non-overlapping. [8]

A location test is a statistical hypothesis test that compares the location parameter of a statistical population to a given constant, or that compares the location parameters of two statistical populations to each other. Most commonly, the location parameter of interest are expected values, but location tests based on medians or other measures of location are also used.

In inferential statistics, the null hypothesis is a general statement or default position that there is nothing new happening, like there is no association among groups, or no relationship between two measured phenomena. Testing the null hypothesis—and thus concluding that there are or are not grounds for believing that there is a relationship between two phenomena —is a central task in the modern practice of science; the field of statistics gives precise criteria for rejecting a null hypothesis.

In probability theory, the expected value of a random variable is a key aspect of its probability distribution. Intuitively, a random variable's expected value represents the average of a large number of independent realizations of the random variable. For example, the expected value of rolling a six-sided die is 3.5, because the average of all the numbers that come up converges to 3.5 as the number of rolls approaches infinity. The expected value is also known as the expectation, mathematical expectation, mean, or first moment.

## Assumptions

Most test statistics have the form t = Z/s, where Z and s are functions of the data.

Z may be sensitive to the alternative hypothesis (i.e., its magnitude tends to be larger when the alternative hypothesis is true), whereas s is a scaling parameter that allows the distribution of t to be determined.

As an example, in the one-sample t-test

${\displaystyle t={\frac {Z}{s}}={\frac {{\bar {X}}-\mu }{{\widehat {\sigma }}/{\sqrt {n}}}}}$

where X is the sample mean from a sample X1, X2, …, Xn, of size n, s is the standard error of the mean, ${\textstyle {\widehat {\sigma }}}$ is the estimate of the standard deviation of the population, and μ is the population mean.

The assumptions underlying a t-test in its simplest form are that

• X follows a normal distribution with mean μ and variance σ2/n
• s2 follows a χ2 distribution with n  1 degrees of freedom. This assumption is met when the observations used for estimating s2 come from a normal distribution (and i.i.d for each group).
• Z and s are independent.

In the t-test comparing the means of two independent samples, the following assumptions should be met:

• Mean of the two populations being compared should follow a normal distribution. Under weak assumptions, this follows in large samples from the central limit theorem, even when the distribution of observations in each group is non-normal. [9]
• If using Student's original definition of the t-test, the two populations being compared should have the same variance (testable using F-test, Levene's test, Bartlett's test, or the Brown–Forsythe test; or assessable graphically using a Q–Q plot). If the sample sizes in the two groups being compared are equal, Student's original t-test is highly robust to the presence of unequal variances. [10] Welch's t-test is insensitive to equality of the variances regardless of whether the sample sizes are similar.
• The data used to carry out the test should be sampled independently from the two populations being compared. This is in general not testable from the data, but if the data are known to be dependently sampled (that is, if they were sampled in clusters), then the classical t-tests discussed here may give misleading results.

Most two-sample t-tests are robust to all but large deviations from the assumptions. [11]

For exactness, the t-test and Z-test require normality of the sample means, and the t-test additionally requires that the sample variance follows a scaled χ2 distribution, and that the sample mean and sample variance be statistically independent. Normality of the individual data values is not required if these conditions are met. By the central limit theorem, sample means of moderately large samples are often well-approximated by a normal distribution even if the data are not normally distributed. For non-normal data, the distribution of the sample variance may deviate substantially from a χ2 distribution. However, if the sample size is large, Slutsky's theorem implies that the distribution of the sample variance has little effect on the distribution of the test statistic.

## Unpaired and paired two-sample t-tests

Two-sample t-tests for a difference in mean involve independent samples (unpaired samples) or paired samples. Paired t-tests are a form of blocking, and have greater power than unpaired tests when the paired units are similar with respect to "noise factors" that are independent of membership in the two groups being compared. [12] In a different context, paired t-tests can be used to reduce the effects of confounding factors in an observational study.

### Independent (unpaired) samples

The independent samples t-test is used when two separate sets of independent and identically distributed samples are obtained, one from each of the two populations being compared. For example, suppose we are evaluating the effect of a medical treatment, and we enroll 100 subjects into our study, then randomly assign 50 subjects to the treatment group and 50 subjects to the control group. In this case, we have two independent samples and would use the unpaired form of the t-test.

### Paired samples

Paired samples t-tests typically consist of a sample of matched pairs of similar units, or one group of units that has been tested twice (a "repeated measures" t-test).

A typical example of the repeated measures t-test would be where subjects are tested prior to a treatment, say for high blood pressure, and the same subjects are tested again after treatment with a blood-pressure lowering medication. By comparing the same patient's numbers before and after treatment, we are effectively using each patient as their own control. That way the correct rejection of the null hypothesis (here: of no difference made by the treatment) can become much more likely, with statistical power increasing simply because the random interpatient variation has now been eliminated. However, an increase of statistical power comes at a price: more tests are required, each subject having to be tested twice. Because half of the sample now depends on the other half, the paired version of Student's t-test has only n/2 − 1 degrees of freedom (with n being the total number of observations).[ citation needed ] Pairs become individual test units, and the sample has to be doubled to achieve the same number of degrees of freedom. Normally, there are n − 1 degrees of freedom (with n being the total number of observations). [13]

A paired samples t-test based on a "matched-pairs sample" results from an unpaired sample that is subsequently used to form a paired sample, by using additional variables that were measured along with the variable of interest. [14] The matching is carried out by identifying pairs of values consisting of one observation from each of the two samples, where the pair is similar in terms of other measured variables. This approach is sometimes used in observational studies to reduce or eliminate the effects of confounding factors.

Paired samples t-tests are often referred to as "dependent samples t-tests".

## Calculations

Explicit expressions that can be used to carry out various t-tests are given below. In each case, the formula for a test statistic that either exactly follows or closely approximates a t-distribution under the null hypothesis is given. Also, the appropriate degrees of freedom are given in each case. Each of these statistics can be used to carry out either a one-tailed or two-tailed test.

Once the t value and degrees of freedom are determined, a p-value can be found using a table of values from Student's t-distribution. If the calculated p-value is below the threshold chosen for statistical significance (usually the 0.10, the 0.05, or 0.01 level), then the null hypothesis is rejected in favor of the alternative hypothesis.

### One-sample t-test

In testing the null hypothesis that the population mean is equal to a specified value μ0, one uses the statistic

${\displaystyle t={\frac {{\bar {x}}-\mu _{0}}{s/{\sqrt {n}}}}}$

where ${\displaystyle {\bar {x}}}$ is the sample mean, s is the sample standard deviation and n is the sample size. The degrees of freedom used in this test are n − 1. Although the parent population does not need to be normally distributed, the distribution of the population of sample means ${\displaystyle {\bar {x}}}$ is assumed to be normal.

By the central limit theorem, if the observations are independent and the second moment exists, then ${\displaystyle t}$ will be approximately normal N(0;1).

### Slope of a regression line

Suppose one is fitting the model

${\displaystyle Y=\alpha +\beta x+\varepsilon }$

where x is known, α and β are unknown, and ε is a normally distributed random variable with mean 0 and unknown variance σ2, and Y is the outcome of interest. We want to test the null hypothesis that the slope β is equal to some specified value β0 (often taken to be 0, in which case the null hypothesis is that x and y are uncorrelated).

Let

{\displaystyle {\begin{aligned}{\widehat {\alpha }},{\widehat {\beta }}&={\text{least-squares estimators}},\\SE_{\widehat {\alpha }},SE_{\widehat {\beta }}&={\text{the standard errors of least-squares estimators}}.\end{aligned}}}

Then

${\displaystyle t_{\text{score}}={\frac {\left({\widehat {\beta }}-\beta _{0}\right)}{SE_{\widehat {\beta }}}}\sim {\mathcal {T}}_{n-2}}$

has a t-distribution with n − 2 degrees of freedom if the null hypothesis is true. The standard error of the slope coefficient:

${\displaystyle SE_{\widehat {\beta }}={\frac {\sqrt {{\dfrac {1}{n-2}}\displaystyle \sum _{i=1}^{n}\left(y_{i}-{\widehat {y}}_{i}\right)^{2}}}{\sqrt {\displaystyle \sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{2}}}}}$

can be written in terms of the residuals. Let

{\displaystyle {\begin{aligned}{\widehat {\varepsilon }}_{i}&=y_{i}-{\widehat {y}}_{i}=y_{i}-\left({\widehat {\alpha }}+{\widehat {\beta }}x_{i}\right)={\text{residuals}}={\text{estimated errors}},\\{\text{SSR}}&=\sum _{i=1}^{n}{{\widehat {\varepsilon }}_{i}}^{2}={\text{sum of squares of residuals}}.\end{aligned}}}

Then tscore is given by:

${\displaystyle t_{\text{score}}={\frac {\left({\widehat {\beta }}-\beta _{0}\right){\sqrt {n-2}}}{\sqrt {{\text{SSR}}\left/\displaystyle \sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{2}\right.}}}.}$

Another way to determine the tscore is:

${\displaystyle t_{\text{score}}={\frac {r{\sqrt {n-2}}}{\sqrt {1-r^{2}}}},}$

where r is the Pearson correlation coefficient.

The tscore, intercept can be determined from the tscore, slope:

${\displaystyle t_{\text{score,intercept}}={\frac {\alpha }{\beta }}{\frac {t_{\text{score,slope}}}{\sqrt {s_{\text{x}}^{2}+{\bar {x}}^{2}}}}}$

where sx2 is the sample variance.

### Independent two-sample t-test

#### Equal sample sizes, equal variance

Given two groups (1, 2), this test is only applicable when:

• the two sample sizes (that is, the number n of participants of each group) are equal;
• it can be assumed that the two distributions have the same variance;

Violations of these assumptions are discussed below.

The t statistic to test whether the means are different can be calculated as follows:

${\displaystyle t={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}}{s_{p}{\sqrt {\frac {2}{n}}}}}}$

where

${\displaystyle s_{p}={\sqrt {\frac {s_{X_{1}}^{2}+s_{X_{2}}^{2}}{2}}}.}$

Here sp is the pooled standard deviation for n = n1 = n2 and s 2
X1
and s 2
X2
are the unbiased estimators of the variances of the two samples. The denominator of t is the standard error of the difference between two means.

For significance testing, the degrees of freedom for this test is 2n − 2 where n is the number of participants in each group.

#### Equal or unequal sample sizes, equal variance

This test is used only when it can be assumed that the two distributions have the same variance. (When this assumption is violated, see below.) The previous formulae are a special case of the formulae below, one recovers them when both samples are equal in size: n = n1 = n2.

The t statistic to test whether the means are different can be calculated as follows:

${\displaystyle t={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}}{s_{p}\cdot {\sqrt {{\frac {1}{n_{1}}}+{\frac {1}{n_{2}}}}}}}}$

where

${\displaystyle s_{p}={\sqrt {\frac {\left(n_{1}-1\right)s_{X_{1}}^{2}+\left(n_{2}-1\right)s_{X_{2}}^{2}}{n_{1}+n_{2}-2}}}}$

is an estimator of the pooled standard deviation of the two samples: it is defined in this way so that its square is an unbiased estimator of the common variance whether or not the population means are the same. In these formulae, ni − 1 is the number of degrees of freedom for each group, and the total sample size minus two (that is, n1 + n2 − 2) is the total number of degrees of freedom, which is used in significance testing.

#### Equal or unequal sample sizes, unequal variances

This test, also known as Welch's t-test, is used only when the two population variances are not assumed to be equal (the two sample sizes may or may not be equal) and hence must be estimated separately. The t statistic to test whether the population means are different is calculated as:

${\displaystyle t={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}}{s_{\bar {\Delta }}}}}$

where

${\displaystyle s_{\bar {\Delta }}={\sqrt {{\frac {s_{1}^{2}}{n_{1}}}+{\frac {s_{2}^{2}}{n_{2}}}}}.}$

Here si2 is the unbiased estimator of the variance of each of the two samples with ni = number of participants in group i (1 or 2). In this case s2
Δ
is not a pooled variance. For use in significance testing, the distribution of the test statistic is approximated as an ordinary Student's t-distribution with the degrees of freedom calculated using

${\displaystyle \mathrm {d.f.} ={\frac {\left({\frac {s_{1}^{2}}{n_{1}}}+{\frac {s_{2}^{2}}{n_{2}}}\right)^{2}}{{\frac {\left(s_{1}^{2}/n_{1}\right)^{2}}{n_{1}-1}}+{\frac {\left(s_{2}^{2}/n_{2}\right)^{2}}{n_{2}-1}}}}.}$

This is known as the Welch–Satterthwaite equation. The true distribution of the test statistic actually depends (slightly) on the two unknown population variances (see Behrens–Fisher problem).

### Dependent t-test for paired samples

This test is used when the samples are dependent; that is, when there is only one sample that has been tested twice (repeated measures) or when there are two samples that have been matched or "paired". This is an example of a paired difference test.

${\displaystyle t={\frac {{\bar {X}}_{D}-\mu _{0}}{\frac {s_{D}}{\sqrt {n}}}}.}$

For this equation, the differences between all pairs must be calculated. The pairs are either one person's pre-test and post-test scores or between pairs of persons matched into meaningful groups (for instance drawn from the same family or age group: see table). The average (XD) and standard deviation (sD) of those differences are used in the equation. The constant μ0 is zero if we want to test whether the average of the difference is significantly different. The degree of freedom used is n − 1, where n represents the number of pairs.

Example of repeated measures
NumberNameTest 1Test 2
1Mike35%67%
2Melanie50%46%
3Melissa90%86%
4Mitchell78%91%
Example of matched pairs
PairNameAgeTest
1John35250
1Jane36340
2Jimmy22460
2Jessy21200

## Worked examples

Let A1 denote a set obtained by drawing a random sample of six measurements:

${\displaystyle A_{1}=\{30.02,\ 29.99,\ 30.11,\ 29.97,\ 30.01,\ 29.99\}}$

and let A2 denote a second set obtained similarly:

${\displaystyle A_{2}=\{29.89,\ 29.93,\ 29.72,\ 29.98,\ 30.02,\ 29.98\}}$

These could be, for example, the weights of screws that were chosen out of a bucket.

We will carry out tests of the null hypothesis that the means of the populations from which the two samples were taken are equal.

The difference between the two sample means, each denoted by Xi, which appears in the numerator for all the two-sample testing approaches discussed above, is

${\displaystyle {\bar {X}}_{1}-{\bar {X}}_{2}=0.095.}$

The sample standard deviations for the two samples are approximately 0.05 and 0.11, respectively. For such small samples, a test of equality between the two population variances would not be very powerful. Since the sample sizes are equal, the two forms of the two-sample t-test will perform similarly in this example.

### Unequal variances

If the approach for unequal variances (discussed above) is followed, the results are

${\displaystyle {\sqrt {{\frac {s_{1}^{2}}{n_{1}}}+{\frac {s_{2}^{2}}{n_{2}}}}}\approx 0.04849}$

and the degrees of freedom

${\displaystyle {\text{d.f.}}\approx 7.031.}$

The test statistic is approximately 1.959, which gives a two-tailed test p-value of 0.09077.

### Equal variances

If the approach for equal variances (discussed above) is followed, the results are

${\displaystyle s_{p}\approx 0.04849}$

and the degrees of freedom

${\displaystyle {\text{d.f.}}=10.}$

The test statistic is approximately equal to 1.959, which gives a two-tailed p-value of 0.07857.

### Alternatives to the t-test for location problems

The t-test provides an exact test for the equality of the means of two i.i.d. normal populations with unknown, but equal, variances. (Welch's t-test is a nearly exact test for the case where the data are normal but the variances may differ.) For moderately large samples and a one tailed test, the t-test is relatively robust to moderate violations of the normality assumption. [15] In large enough samples, the t-test asymptotically approaches the z-test, and becomes robust even to large deviations from normality. [9]

If the data are substantially non-normal and the sample size is small, the t-test can give misleading results. See Location test for Gaussian scale mixture distributions for some theory related to one particular family of non-normal distributions.

When the normality assumption does not hold, a non-parametric alternative to the t-test may have better statistical power. However, when data are non-normal with differing variances between groups, a t-test may have better type-1 error control than some non-parametric alternatives. [16] Furthermore, non-parametric methods, such as the Mann-Whitney U test discussed below, typically do not test for a difference of means, so should be used carefully if a difference of means is of primary scientific interest. [9] For example, Mann-Whitney U test will keep the type 1 error at the desired level alpha if both groups have the same distribution. It will also have power in detecting an alternative by which group B has the same distribution as A but after some shift by a constant (in which case there would indeed be a difference in the means of the two groups). However, there could be cases where group A and B will have different distributions but with the same means (such as two distributions, one with positive skewness and the other with a negative one, but shifted so to have the same means). In such cases, MW could have more than alpha level power in rejecting the Null hypothesis but attributing the interpretation of difference in means to such a result would be incorrect.

In the presence of an outlier, the t-test is not robust. For example, for two independent samples when the data distributions are asymmetric (that is, the distributions are skewed) or the distributions have large tails, then the Wilcoxon rank-sum test (also known as the Mann–Whitney U test) can have three to four times higher power than the t-test. [15] [17] [18] The nonparametric counterpart to the paired samples t-test is the Wilcoxon signed-rank test for paired samples. For a discussion on choosing between the t-test and nonparametric alternatives, see Lumley, et al. (2002). [9]

One-way analysis of variance (ANOVA) generalizes the two-sample t-test when the data belong to more than two groups.

### A design which includes both paired observations and independent observations

When both paired observations and independent observations are present in the two sample design, assuming data are missing completely at random (MCAR), the paired observations or independent observations may be discarded in order to proceed with the standard tests above. Alternatively making use of all of the available data, assuming normality and MCAR, the generalized partially overlapping samples t-test could be used. [19]

### Multivariate testing

A generalization of Student's t statistic, called Hotelling's t-squared statistic, allows for the testing of hypotheses on multiple (often correlated) measures within the same sample. For instance, a researcher might submit a number of subjects to a personality test consisting of multiple personality scales (e.g. the Minnesota Multiphasic Personality Inventory). Because measures of this type are usually positively correlated, it is not advisable to conduct separate univariate t-tests to test hypotheses, as these would neglect the covariance among measures and inflate the chance of falsely rejecting at least one hypothesis (Type I error). In this case a single multivariate test is preferable for hypothesis testing. Fisher's Method for combining multiple tests with alpha reduced for positive correlation among tests is one. Another is Hotelling's T2 statistic follows a T2 distribution. However, in practice the distribution is rarely used, since tabulated values for T2 are hard to find. Usually, T2 is converted instead to an F statistic.

For a one-sample multivariate test, the hypothesis is that the mean vector (μ) is equal to a given vector (μ0). The test statistic is Hotelling's t2:

${\displaystyle t^{2}=n({\bar {\mathbf {x} }}-{{\boldsymbol {\mu }}_{0}})'{\mathbf {S} }^{-1}({\bar {\mathbf {x} }}-{{\boldsymbol {\mu }}_{0}})}$

where n is the sample size, x is the vector of column means and S is an m × m sample covariance matrix.

For a two-sample multivariate test, the hypothesis is that the mean vectors (μ1, μ2) of two samples are equal. The test statistic is Hotelling's two-sample t2:

${\displaystyle t^{2}={\frac {n_{1}n_{2}}{n_{1}+n_{2}}}\left({\bar {\mathbf {x} }}_{1}-{\bar {\mathbf {x} }}_{2}\right)'{\mathbf {S} _{\text{pooled}}}^{-1}\left({\bar {\mathbf {x} }}_{1}-{\bar {\mathbf {x} }}_{2}\right).}$

## Software implementations

Many spreadsheet programs and statistics packages, such as QtiPlot, LibreOffice Calc, Microsoft Excel, SAS, SPSS, Stata, DAP, gretl, R, Python, PSPP, Matlab and Minitab, include implementations of Student's t-test.

Language/ProgramFunctionNotes
Microsoft Excel pre 2010TTEST(array1, array2, tails, type)See
Microsoft Excel 2010 and laterT.TEST(array1, array2, tails, type)See
LibreOffice Calc TTEST(Data1; Data2; Mode; Type)See
Google Sheets TTEST(range1, range2, tails, type)See
Python scipy.stats.ttest_ind(a, b, axis=0, equal_var=True)See
Matlab ttest(data1, data2)See
Mathematica TTest[{data1,data2}]See
R t.test(data1, data2, var.equal=TRUE)See
SAS PROC TTESTSee
Java tTest(sample1, sample2)See
Julia EqualVarianceTTest(sample1, sample2)See
Stata ttest data1 == data2See

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In statistics, the Pearson correlation coefficient, also referred to as Pearson's r, the Pearson product-moment correlation coefficient (PPMCC) or the bivariate correlation, is a measure of the linear correlation between two variables X and Y. According to the Cauchy–Schwarz inequality it has a value between +1 and −1, where 1 is total positive linear correlation, 0 is no linear correlation, and −1 is total negative linear correlation. It is widely used in the sciences. It was developed by Karl Pearson from a related idea introduced by Francis Galton in the 1880s and for which the mathematical formula was derived and published by Auguste Bravais in 1844. The naming of the coefficient is thus an example of Stigler's Law.

The power of a binary hypothesis test is the probability that the test rejects the null hypothesis (H0) when a specific alternative hypothesis (H1) is true. The statistical power ranges from 0 to 1, and as statistical power increases, the probability of making a type II error (wrongly failing to reject the null hypothesis) decreases. For a type II error probability of β, the corresponding statistical power is 1 − β. For example, if experiment 1 has a statistical power of 0.7, and experiment 2 has a statistical power of 0.95, then there is a stronger probability that experiment 1 had a type II error than experiment 2, and experiment 2 is more reliable than experiment 1 due to the reduction in probability of a type II error. It can be equivalently thought of as the probability of accepting the alternative hypothesis (H1) when it is true—that is, the ability of a test to detect a specific effect, if that specific effect actually exists. That is,

In statistics, an effect size is a quantitative measure of the magnitude of a phenomenon. Examples of effect sizes are the correlation between two variables, the regression coefficient in a regression, the mean difference, or even the risk with which something happens, such as how many people survive after a heart attack for every one person that does not survive. For most types of effect size, a larger absolute value always indicates a stronger effect, with the main exception being if the effect size is an odds ratio. Effect sizes complement statistical hypothesis testing, and play an important role in power analyses, sample size planning, and in meta-analyses. They are the first item (magnitude) in the MAGIC criteria for evaluating the strength of a statistical claim. Especially in meta-analysis, where the purpose is to combine multiple effect sizes, the standard error (S.E.) of the effect size is of critical importance. The S.E. of the effect size is used to weigh effect sizes when combining studies, so that large studies are considered more important than small studies in the analysis. The S.E. of the effect size is calculated differently for each type of effect size, but generally only requires knowing the study's sample size (N), or the number of observations in each group.

In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation. Typically the standard deviations of residuals in a sample vary greatly from one data point to another even when the errors all have the same standard deviation, particularly in regression analysis; thus it does not make sense to compare residuals at different data points without first studentizing. It is a form of a Student's t-statistic, with the estimate of error varying between points.

In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by the two parameters, a and b, which are its minimum and maximum values. The distribution is often abbreviated U(a,b). It is the maximum entropy probability distribution for a random variable X under no constraint other than that it is contained in the distribution's support.

Sample size determination is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a population from a sample. In practice, the sample size used in a study is usually determined based on the cost, time, or convenience of collecting the data, and the need for it to offer sufficient statistical power. In complicated studies there may be several different sample sizes: for example, in a stratified survey there would be different sizes for each stratum. In a census, data is sought for an entire population, hence the intended sample size is equal to the population. In experimental design, where a study may be divided into different treatment groups, there may be different sample sizes for each group.

In statistics, simple linear regression is a linear regression model with a single explanatory variable. That is, it concerns two-dimensional sample points with one independent variable and one dependent variable and finds a linear function that, as accurately as possible, predicts the dependent variable values as a function of the independent variables. The adjective simple refers to the fact that the outcome variable is related to a single predictor.

In statistics, an empirical distribution function is the distribution function associated with the empirical measure of a sample. This cumulative distribution function is a step function that jumps up by 1/n at each of the n data points. Its value at any specified value of the measured variable is the fraction of observations of the measured variable that are less than or equal to the specified value.

As with other probability distributions with noncentrality parameters, the noncentral t-distribution generalizes a probability distribution – Student's t-distribution – using a noncentrality parameter. Whereas the central distribution describes how a test statistic t is distributed when the difference tested is null, the noncentral distribution describes how t is distributed when the null is false. This leads to its use in statistics, especially calculating statistical power. The noncentral t-distribution is also known as the singly noncentral t-distribution, and in addition to its primary use in statistical inference, is also used in robust modeling for data.

In statistics, bootstrapping is any test or metric that relies on random sampling with replacement. Bootstrapping allows assigning measures of accuracy to sample estimates. This technique allows estimation of the sampling distribution of almost any statistic using random sampling methods. Generally, it falls in the broader class of resampling methods.

In probability theory and statistics, the normal-gamma distribution is a bivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and precision.

In statistics, D’Agostino’s K2 test, named for Ralph D'Agostino, is a goodness-of-fit measure of departure from normality, that is the test aims to establish whether or not the given sample comes from a normally distributed population. The test is based on transformations of the sample kurtosis and skewness, and has power only against the alternatives that the distribution is skewed and/or kurtic.

Tukey's range test, also known as the Tukey's test, Tukey method, Tukey's honest significance test, or Tukey's HSD test, is a single-step multiple comparison procedure and statistical test. It can be used to find means that are significantly different from each other.

In statistics, Tukey's test of additivity, named for John Tukey, is an approach used in two-way ANOVA to assess whether the factor variables are additively related to the expected value of the response variable. It can be applied when there are no replicated values in the data set, a situation in which it is impossible to directly estimate a fully general non-additive regression structure and still have information left to estimate the error variance. The test statistic proposed by Tukey has one degree of freedom under the null hypothesis, hence this is often called "Tukey's one-degree-of-freedom test."

In statistics, almost sure hypothesis testing or a.s. hypothesis testing utilizes almost sure convergence in order to determine the validity of a statistical hypothesis with probability one. This is to say that whenever the null hypothesis is true, then an a.s. hypothesis test will fail to reject the null hypothesis w.p. 1 for all sufficiently large samples. Similarly, whenever the alternative hypothesis is true, then an a.s. hypothesis test will reject the null hypothesis with probability one, for all sufficiently large samples. Along similar lines, an a.s. confidence interval eventually contains the parameter of interest with probability 1. Dembo and Peres (1994) proved the existence of almost sure hypothesis tests.

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