Student's ttest
The ttest is any statistical hypothesis test in which the test statistic follows a Student's tdistribution under the null hypothesis.
A ttest 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 ttest can be used, for example, to determine if two sets of data are significantly different from each other.
Contents
History
The tstatistic 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]}
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 ttest as an economical way to monitor the quality of stout. The ttest 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 tdistribution for a detailed history of this pseudonym, which is not to be confused with the literal term student).
Guinness had a policy of allowing technical staff leave for study (socalled "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 editorinchief Karl Pearson.^{[7]}
Uses
Among the most frequently used ttests are:
 A onesample location test of whether the mean of a population has a value specified in a null hypothesis.
 A twosample location test of the null hypothesis such that the means of two populations are equal. All such tests are usually called Student's ttests, 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 ttest. These tests are often referred to as "unpaired" or "independent samples" ttests, as they are typically applied when the statistical units underlying the two samples being compared are nonoverlapping.^{[8]}
 A test of the null hypothesis that the difference between two responses measured on the same statistical unit has a mean value of zero. For example, suppose we measure the size of a cancer patient's tumor before and after a treatment. If the treatment is effective, we expect the tumor size for many of the patients to be smaller following the treatment. This is often referred to as the "paired" or "repeated measures" ttest:^{[8]}^{[9]} see paired difference test.
 A test of whether the slope of a regression line differs significantly from 0.
Assumptions
Most ttest statistics have the form t = Z/s, where Z and s are functions of the data. Typically, Z is designed to 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 onesample ttest
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, σ is the population standard deviation of the data, and μ is the population mean.
The assumptions underlying a ttest are that
 X follows a normal distribution with mean μ and variance σ^{2}
 ps^{2} follows a χ^{2} distribution with p degrees of freedom under the null hypothesis, where p is a positive constant
 Z and s are independent.
In a specific type of ttest, these conditions are consequences of the population being studied, and of the way in which the data are sampled. For example, in the ttest comparing the means of two independent samples, the following assumptions should be met:
 Each of the two populations being compared should follow a normal distribution. This can be tested using a normality test, such as the Shapiro–Wilk or Kolmogorov–Smirnov test, or it can be assessed graphically using a normal quantile plot.
 If using Student's original definition of the ttest, the two populations being compared should have the same variance (testable using Ftest, 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 ttest is highly robust to the presence of unequal variances.^{[10]} Welch's ttest 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 ttests discussed here may give misleading results.
Most twosample ttests are robust to all but large deviations from the assumptions.^{[11]}
Unpaired and paired twosample ttests
Twosample ttests for a difference in mean involve independent samples(unpaired samples) or paired samples. Paired ttests 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 ttests can be used to reduce the effects of confounding factors in an observational study.
Independent (unpaired) samples
The independent samples ttest 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 ttest. The randomization is not essential here – if we contacted 100 people by phone and obtained each person's age and gender, and then used a twosample ttest to see whether the mean ages differ by gender, this would also be an independent samples ttest, even though the data are observational.
Paired samples
Paired samples ttests typically consist of a sample of matched pairs of similar units, or one group of units that has been tested twice (a "repeated measures" ttest).
A typical example of the repeated measures ttest 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 bloodpressure 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 betweenpatient variation has now been eliminated. Note however that 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 ttest 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 ttest based on a "matchedpairs 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 ttests are often referred to as "dependent samples ttests".
Calculations
Explicit expressions that can be used to carry out various ttests are given below. In each case, the formula for a test statistic that either exactly follows or closely approximates a tdistribution 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 onetailed or twotailed test.
Once the t value and degrees of freedom are determined, a pvalue can be found using a table of values from Student's tdistribution. If the calculated pvalue 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.
Onesample ttest
In testing the null hypothesis that the population mean is equal to a specified value μ_{0}, one uses the statistic
where is the sample mean, s is the sample standard deviation of the sample 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 sampling of the parent population is independent and the second moment of the parent population exists then the sample means will be approximately normal in the large sample limit.^{[15]} (The degree of approximation will depend on how close the parent population is to a normal distribution and the sample size, n.)
Slope of a regression line
Suppose one is fitting the model
where x is known, α and β are unknown, and 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 tdistribution 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.
Independent twosample ttest
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:
where
Here s_{p} is the pooled standard deviation for n = n_{1} = n_{2} 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.) Note that 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, n_{1} + n_{2} − 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 ttest, 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}^{2} is the unbiased estimator of the variance of each of the two samples with n_{i} = number of participants in group i (1 or 2). Note that in this case s^{2}
_{Δ} is not a pooled variance. For use in significance testing, the distribution of the test statistic is approximated as an ordinary Student's tdistribution with the degrees of freedom calculated using
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 ttest 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.
For this equation, the differences between all pairs must be calculated. The pairs are either one person's pretest and posttest scores or between pairs of persons matched into meaningful groups (for instance drawn from the same family or age group: see table). The average (X_{D}) and standard deviation (s_{D}) of those differences are used in the equation. The constant μ_{0} is nonzero if we want to test whether the average of the difference is significantly different from μ_{0}. 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
Worked examples
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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 twosample 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 twosample ttest will perform similarly in this example.
Unequal variances
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 twotailed test pvalue of 0.09077.
Equal variances
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 twotailed pvalue of 0.07857.
Alternatives to the ttest for location problems
The ttest provides an exact test for the equality of the means of two normal populations with unknown, but equal, variances. (Welch's ttest 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 ttest is relatively robust to moderate violations of the normality assumption.^{[16]}
For exactness, the ttest and Ztest require normality of the sample means, and the ttest 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 wellapproximated by a normal distribution even if the data are not normally distributed. For nonnormal 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. If the data are substantially nonnormal and the sample size is small, the ttest can give misleading results. See Location test for Gaussian scale mixture distributions for some theory related to one particular family of nonnormal distributions.
When the normality assumption does not hold, a nonparametric alternative to the ttest can often have better statistical power. Similarly, in the presence of an outlier, the ttest 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 ranksum test (also known as the Mann–Whitney U test) can have three to four times higher power than the ttest.^{[16]}^{[17]}^{[18]} The nonparametric counterpart to the paired samples ttest is the Wilcoxon signedrank test for paired samples. For a discussion on choosing between the ttest and nonparametric alternatives, see Sawilowsky (2005).^{[19]}
Oneway analysis of variance (ANOVA) generalizes the twosample ttest when the data belong to more than two groups.
Multivariate testing
A generalization of Student's t statistic, called Hotelling's tsquared 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 ttests 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 onesample 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 twosample multivariate test, the hypothesis is that the mean vectors (μ_{1}, μ_{2}) of two samples are equal. The test statistic is Hotelling's twosample t^{2}:
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 ttest.
Language/Program  Function  Notes 

Microsoft Excel pre 2010  TTEST(array1, array2, tails, type) 
See [1] 
Microsoft Excel 2010 and later  T.TEST(array1, array2, tails, type) 
See [2] 
LibreOffice  TTEST(Data1; Data2; Mode; Type) 
See [3] 
Google Sheets  TTEST(range1, range2, tails, type) 
See [4] 
Python  scipy.stats.ttest_ind(a, b, axis=0, equal_var=True) 
See [5] 
Matlab  ttest(data1, data2) 
See [6] 
Mathematica  TTest[{data1,data2}] 
See [7] 
R  t.test(data1, data2, var.equal=TRUE) 
See [8] 
SAS  PROC TTEST 
See [9] 
Java  tTest(sample1, sample2) 
See [10] 
Julia  EqualVarianceTTest(sample1, sample2) 
See [11] 
Stata  ttest data1 == data2 
See [12] 
See also
 Conditional change model
 Ftest
 Noncentral tdistribution in power analysis
 Student's tstatistic
 Ztest
 Mann–Whitney U test
 Šidák correction for ttest
 Welch's ttest
 Analysis of variance (ANOVA)
Notes
 ^ Mankiewicz, Richard (2004). The Story of Mathematics (Paperback ed.). Princeton, NJ: Princeton University Press. p. 158. ISBN 9780691120461.
 ^ ^{a} ^{b} O'Connor, John J.; Robertson, Edmund F., "William Sealy Gosset", MacTutor History of Mathematics archive, University of St Andrews.
 ^ Fisher Box, Joan (1987). "Guinness, Gosset, Fisher, and Small Samples". Statistical Science. 2 (1): 45–52. doi:10.1214/ss/1177013437. JSTOR 2245613.
 ^ http://www.aliquote.org/cours/2012_biomed/biblio/Student1908.pdf
 ^ "The Probable Error of a Mean" (PDF). Biometrika. 6 (1): 1–25. 1908. doi:10.1093/biomet/6.1.1. Retrieved 24 July 2016.
 ^ Raju, T. N. (2005). "William Sealy Gosset and William A. Silverman: Two "students" of science". Pediatrics. 116 (3): 732–5. doi:10.1542/peds.20051134. PMID 16140715.
 ^ Dodge, Yadolah (2008). The Concise Encyclopedia of Statistics. Springer Science & Business Media. pp. 234–235. ISBN 9780387317427.
 ^ ^{a} ^{b} Fadem, Barbara (2008). HighYield Behavioral Science. HighYield Series. Hagerstown, MD: Lippincott Williams & Wilkins. ISBN 0781782589.
 ^ Zimmerman, Donald W. (1997). "A Note on Interpretation of the PairedSamples t Test". Journal of Educational and Behavioral Statistics. 22 (3): 349–360. doi:10.3102/10769986022003349. JSTOR 1165289.
 ^ Markowski, Carol A.; Markowski, Edward P. (1990). "Conditions for the Effectiveness of a Preliminary Test of Variance". The American Statistician. 44 (4): 322–326. doi:10.2307/2684360. JSTOR 2684360.
 ^ Bland, Martin (1995). An Introduction to Medical Statistics. Oxford University Press. p. 168. ISBN 9780192624284.
 ^ Rice, John A. (2006). Mathematical Statistics and Data Analysis (3rd ed.). Duxbury Advanced.^{[ISBN missing]}
 ^ Weisstein, Eric. "Student's tDistribution". mathworld.wolfram.com.
 ^ David, H. A.; Gunnink, Jason L. (1997). "The Paired t Test Under Artificial Pairing". The American Statistician. 51 (1): 9–12. doi:10.2307/2684684. JSTOR 2684684.
 ^ Box, George; Hunter, William; Hunter, J. Stuart. Statistics for Experimenters. pp. 66–67. ISBN 9780471093152.
 ^ ^{a} ^{b} Sawilowsky, Shlomo S.; Blair, R. Clifford (1992). "A More Realistic Look at the Robustness and Type II Error Properties of the t Test to Departures From Population Normality". Psychological Bulletin. 111 (2): 352–360. doi:10.1037/00332909.111.2.352.
 ^ Blair, R. Clifford; Higgins, James J. (1980). "A Comparison of the Power of Wilcoxon's RankSum Statistic to That of Student's t Statistic Under Various Nonnormal Distributions". Journal of Educational Statistics. 5 (4): 309–335. doi:10.2307/1164905. JSTOR 1164905.
 ^ Fay, Michael P.; Proschan, Michael A. (2010). "Wilcoxon–Mann–Whitney or ttest? On assumptions for hypothesis tests and multiple interpretations of decision rules". Statistics Surveys. 4: 1–39. doi:10.1214/09SS051. PMC 2857732 . PMID 20414472.
 ^ Sawilowsky, Shlomo S. (2005). "Misconceptions Leading to Choosing the t Test Over The Wilcoxon Mann–Whitney Test for Shift in Location Parameter". Journal of Modern Applied Statistical Methods. 4 (2): 598–600. Retrieved 20140618.
References
 O'Mahony, Michael (1986). Sensory Evaluation of Food: Statistical Methods and Procedures. CRC Press. p. 487. ISBN 0824773373.

Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (1992). https://web.archive.org/web/20151128053615/http://numerical.recipes/
archiveurl=
missing title (help). Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press. p. 616. ISBN 0521431085. Archived from the original (PDF) on 20151128.
Further reading
 Boneau, C. Alan (1960). "The effects of violations of assumptions underlying the t test". Psychological Bulletin. 57 (1): 49–64. doi:10.1037/h0041412.
 Edgell, Stephen E.; Noon, Sheila M. (1984). "Effect of violation of normality on the t test of the correlation coefficient". Psychological Bulletin. 95 (3): 576–583. doi:10.1037/00332909.95.3.576.
External links
Wikiversity has learning resources about ttest 
 Hazewinkel, Michiel, ed. (2001) [1994], "Student test", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104
 A conceptual article on the Student's ttest
 Econometrics lecture (topic: hypothesis testing) on YouTube by Mark Thoma