Editing Student's t-test (section)

===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|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.<ref name="Sawilowsky-Blair">{{cite journal |last1=Sawilowsky |first1=Shlomo S. |last2=Blair |first2=R. Clifford |year=1992 |title=A More Realistic Look at the Robustness and Type II Error Properties of the ''t'' Test to Departures From Population Normality |journal=Psychological Bulletin |volume=111 |issue=2 |pages=352–360 |doi=10.1037/0033-2909.111.2.352}}</ref> In large enough samples, the ''t''-test asymptotically approaches the [[Z-test|''z''-test]], and becomes robust even to large deviations from normality.<ref name=":0" />

If the data are substantially non-normal and the sample size is small, the ''t''-test can give misleading results. See [[location testing for Gaussian scale mixture distributions|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|type-1 error]] control than some non-parametric alternatives.<ref>{{Cite journal|last=Zimmerman|first=Donald W.|date=January 1998|title=Invalidation of Parametric and Nonparametric Statistical Tests by Concurrent Violation of Two Assumptions|journal=The Journal of Experimental Education|volume=67|issue=1|pages=55–68|doi=10.1080/00220979809598344|issn=0022-0973}}</ref> Furthermore, non-parametric methods, such as the [[Mann–Whitney U test|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.<ref name=":0"/> 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 [[skewness|skewed]]) or the distributions have large tails, then the Wilcoxon rank-sum test (also known as the [[Mann–Whitney U test|Mann–Whitney ''U'' test]]) can have three to four times higher power than the ''t''-test.<ref name="Sawilowsky-Blair"/><ref>{{cite journal |last1=Blair |first1=R. Clifford |last2=Higgins |first2=James J. |journal=Journal of Educational Statistics |year=1980 |pages=309–335 | volume=5 |issue=4 |title=A Comparison of the Power of Wilcoxon's Rank-Sum Statistic to That of Student's ''t'' Statistic Under Various Nonnormal Distributions |doi=10.2307/1164905 |jstor=1164905}}</ref><ref>{{cite journal |last1=Fay |first1=Michael P. |last2=Proschan |first2=Michael A. |journal=Statistics Surveys |year=2010 |pages=1–39 |volume=4 |url=http://www.i-journals.org/ss/viewarticle.php?id=51 |title=Wilcoxon–Mann–Whitney or ''t''-test? On assumptions for hypothesis tests and multiple interpretations of decision rules |doi=10.1214/09-SS051 |pmid=20414472 |pmc=2857732}}</ref> 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).<ref name=":0" />

One-way [[analysis of variance]] (ANOVA) generalizes the two-sample ''t''-test when the data belong to more than two groups.