The ** t-test** is any statistical hypothesis test in which the test statistic follows a Student's

- History
- Uses
- Assumptions
- Unpaired and paired two-sample t-tests
- Independent (unpaired) samples
- Paired samples
- Calculations
- One-sample t-test
- Slope of a regression line
- Independent two-sample t-test
- Dependent t-test for paired samples
- Worked examples
- Unequal variances
- Equal variances
- Related statistical tests
- Alternatives to the t-test for location problems
- A design which includes both paired observations and independent observations
- Multivariate testing
- Software implementations
- See also
- References
- Citations
- Sources
- Further reading
- External links

A *t*-test is the 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.

The term "*t*-statistic" is abbreviated from "hypothesis test statistic".^{ [1] } In statistics, the t-distribution was first derived as a posterior distribution in 1876 by Helmert ^{ [2] }^{ [3] }^{ [4] } and Lüroth.^{ [5] }^{ [6] }^{ [7] } The t-distribution also appeared in a more general form as Pearson Type IV distribution in Karl Pearson's 1895 paper.^{ [8] } However, the T-Distribution, also known as Student's t-distribution, gets its name from William Sealy Gosset who first published it in English in 1908 in the scientific journal Biometrika using his pseudonym "Student"^{ [9] }^{ [10] } because his employer preferred staff to use pen names when publishing scientific papers instead of their real name, so he used the name "Student" to hide his identity.^{ [11] } Gosset worked at the Guinness Brewery in Dublin, Ireland, and was interested in the problems of small samples – for example, the chemical properties of barley with small sample sizes. Hence a second version of the etymology of the term Student is that Guinness did not want their competitors to know that they were using the t-test to determine the quality of raw material (see Student's *t*-distribution for a detailed history of this pseudonym, which is not to be confused with the literal term * student *). Although it was William Gosset after whom the term "Student" is penned, it was actually through the work of Ronald Fisher that the distribution became well known as "Student's distribution"^{ [12] } and "Student's t-test".

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.^{ [13] } 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.^{ [14] }

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.^{ [15] } Gosset's identity was then known to fellow statisticians and to editor-in-chief Karl Pearson.^{ [16] }

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**, 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*-tests*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.^{ [17] }

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

where *X* is the sample mean from a sample *X*_{1}, *X*_{2}, …, *X*_{n}, of size *n*, *s* is the standard error of the mean, is the estimate of the standard deviation of the population, and *μ* is the population mean.

The assumptions underlying a *t*-test in the simplest form above are that:

*X*follows a normal distribution with mean*μ*and variance*σ*^{2}/*n**s*^{2}(*n*− 1)/*σ*^{2}follows a*χ*^{2}distribution with*n*− 1 degrees of freedom. This assumption is met when the observations used for estimating*s*^{2}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:

- The means of the two populations being compared should follow normal distributions. 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.
^{ [18] } - 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.^{ [19] }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 either be sampled independently from the two populations being compared or be fully paired. This is in general not testable from the data, but if the data are known to be dependent (e.g. paired by test design), a dependent test has to be applied. For partially paired data, the classical independent
*t*-tests may give invalid results as the test statistic might not follow a*t*distribution, while the dependent*t*-test is sub-optimal as it discards the unpaired data.^{ [20] }

Most two-sample *t*-tests are robust to all but large deviations from the assumptions.^{ [21] }

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 *χ*^{2} 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 *χ*^{2} 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.

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 (probability of avoiding a type II error, also known as a false negative) 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.^{ [22] } In a different context, paired *t*-tests can be used to reduce the effects of confounding factors in an observational study.

The independent samples *t*-test is used when two separate sets of independent and identically distributed samples are obtained, and one variable from each of the two populations is 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 *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). 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).^{ [23] }

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.^{ [24] } 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".

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.

In testing the null hypothesis that the sample mean is equal to a specified value *μ*_{0}, one uses the statistic

where 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 is assumed to be normal.

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

Suppose one is fitting the model

where *x* is known, *α* and *β* are unknown, *ε* is a normally distributed random variable with mean 0 and unknown variance *σ*^{2}, and *Y* is the outcome of interest. We want to test the null hypothesis that the slope *β* is equal to some specified value *β*_{0} (often taken to be 0, in which case the null hypothesis is that *x* and *y* are uncorrelated).

Let

Then

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

can be written in terms of the residuals. Let

Then *t*_{score} is given by:

Another way to determine the *t*_{score} is:

where *r* is the Pearson correlation coefficient.

The *t*_{score, intercept} can be determined from the *t*_{score, slope}:

where *s*_{x}^{2} is the sample 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:

where

Here *s _{p}* is the pooled standard deviation for

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

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* = *n*_{1} = *n*_{2}.

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

where

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, *n _{i}* − 1 is the number of degrees of freedom for each group, and the total sample size minus two (that is,

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:

where

Here *s _{i}*

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).

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. The *t* statistic is calculated as

where and are the average and standard deviation of the differences between all pairs. The pairs are e.g. 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 constant *μ*_{0} 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 Number Name Test 1 Test 2 1 Mike 35% 67% 2 Melanie 50% 46% 3 Melissa 90% 86% 4 Mitchell 78% 91% Example of matched pairs Pair Name Age Test 1 John 35 250 1 Jane 36 340 2 Jimmy 22 460 2 Jessy 21 200

This article may not properly summarize its corresponding main article. |

Let *A*_{1} denote a set obtained by drawing a random sample of six measurements:

and let *A*_{2} denote a second set obtained similarly:

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 *X*_{i}, which appears in the numerator for all the two-sample testing approaches discussed above, is

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.

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

and the degrees of freedom

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

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

and the degrees of freedom

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

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.^{ [25] } In large enough samples, the t-test asymptotically approaches the *z*-test, and becomes robust even to large deviations from normality.^{ [18] }

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.^{ [26] } 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.^{ [18] } 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.^{ [25] }^{ [27] }^{ [28] } 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).^{ [18] }

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

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.^{ [29] }

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 *T*^{2} statistic follows a *T*^{2} distribution. However, in practice the distribution is rarely used, since tabulated values for *T*^{2} are hard to find. Usually, *T*^{2} 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 (**μ**_{0}). The test statistic is Hotelling's *t*^{2}:

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 (**μ**_{1}, **μ**_{2}) of two samples are equal. The test statistic is Hotelling's two-sample *t*^{2}:

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/Program | Function | Notes |
---|---|---|

Microsoft Excel pre 2010 | `TTEST(` | See |

Microsoft Excel 2010 and later | `T.TEST(` | See |

LibreOffice Calc | `TTEST(` | See |

Google Sheets | `TTEST(range1, range2, tails, type)` | See |

Python | `scipy.stats.ttest_ind(` | See |

MATLAB | `ttest(data1, data2)` | See |

Mathematica | `TTest[{data1,data2}]` | See |

R | `t.test(data1, data2, var.equal=TRUE)` | See |

SAS | `PROC TTEST` | See |

Java | `tTest(sample1, sample2)` | See |

Julia | `EqualVarianceTTest(sample1, sample2)` | See |

Stata | `ttest data1 == data2` | See |

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

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The statistical **power** of a binary hypothesis test is the probability that the test correctly rejects the null hypothesis when a specific alternative hypothesis is true. It is commonly denoted by , and represents the chances of a "true positive" detection conditional on the actual existence of an effect to detect. Statistical power ranges from 0 to 1, and as the power of a test increases, the probability of making a type II error by wrongly failing to reject the null hypothesis decreases.

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In statistics, an **effect size** is a number measuring the strength of the relationship between two variables in a population, or a sample-based estimate of that quantity. It can refer to the value of a statistic calculated from a sample of data, the value of a parameter for a hypothetical population, or to the equation that operationalizes how statistics or parameters lead to the effect size value. Examples of effect sizes include the correlation between two variables, the regression coefficient in a regression, the mean difference, or the risk of a particular event happening. Effect sizes complement statistical hypothesis testing, and play an important role in power analyses, sample size planning, and in meta-analyses. The cluster of data-analysis methods concerning effect sizes is referred to as estimation statistics.

In statistics, a **studentized residual** is the quotient resulting from the division of a residual by an estimate of its standard deviation. It is a form of a Student's *t*-statistic, with the estimate of error varying between points.

**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, the number of **degrees of freedom** is the number of values in the final calculation of a statistic that are free to vary.

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 variable. The adjective *simple* refers to the fact that the outcome variable is related to a single predictor.

The **noncentral t-distribution** generalizes Student's

Although the subject of sexual dimorphism is not in itself controversial, the measures by which it is assessed differ widely. Most of the measures are used on the assumption that a random variable is considered so that probability distributions should be taken into account. In this review, a series of **sexual dimorphism measures** are discussed concerning both their definition and the probability law on which they are based. Most of them are sample functions, or statistics, which account for only partial characteristics, for example the mean or expected value, of the distribution involved. Further, the most widely used measure fails to incorporate an inferential support.

**Bootstrapping** is any test or metric that uses random sampling with replacement, and falls under the broader class of resampling methods. Bootstrapping assigns measures of accuracy to sample estimates. This technique allows estimation of the sampling distribution of almost any statistic using random sampling methods.

In statistics, **Welch's t-test**, or

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**Tukey's range test**, also known as **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, 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

In statistics, a **paired difference test** is a type of location test that is used when comparing two sets of measurements to assess whether their population means differ. A paired difference test uses additional information about the sample that is not present in an ordinary unpaired testing situation, either to increase the statistical power, or to reduce the effects of confounders.

The **Newman–Keuls** or **Student–Newman–Keuls (SNK)****method** is a stepwise multiple comparisons procedure used to identify sample means that are significantly different from each other. It was named after Student (1927), D. Newman, and M. Keuls. This procedure is often used as a post-hoc test whenever a significant difference between three or more sample means has been revealed by an analysis of variance (ANOVA). The Newman–Keuls method is similar to Tukey's range test as both procedures use studentized range statistics. Unlike Tukey's range test, the Newman–Keuls method uses different critical values for different pairs of mean comparisons. Thus, the procedure is more likely to reveal significant differences between group means and to commit type I errors by incorrectly rejecting a null hypothesis when it is true. In other words, the Neuman-Keuls procedure is more powerful but less conservative than Tukey's range 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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- O'Mahony, Michael (1986).
*Sensory Evaluation of Food: Statistical Methods and Procedures*. CRC Press. p. 487. ISBN 0-82477337-3. - Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (1992).
*Numerical Recipes in C: The Art of Scientific Computing*. Cambridge University Press. p. 616. ISBN 0-521-43108-5.

- Boneau, C. Alan (1960). "The effects of violations of assumptions underlying the
*t*test".*Psychological Bulletin*.**57**(1): 49–64. doi:10.1037/h0041412. PMID 13802482. - Edgell, Stephen E.; Noon, Sheila M. (1984). "Effect of violation of normality on the
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Wikiversity has learning resources about t-test |

- "Student test",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - Trochim, William M.K. "The T-Test",
*Research Methods Knowledge Base*, conjoint.ly - Econometrics lecture (topic: hypothesis testing) on YouTube by Mark Thoma

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