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| {{DISPLAYTITLE:Welch's ''t''-test}}
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| In [[statistics]], '''Welch's ''t'' test''' is an adaptation of [[Student's t-test|Student's ''t''-test]] intended for use with two samples having possibly unequal [[variance]]s.<ref> {{Cite journal | last = Welch | first = B. L. | title = The generalization of "Student's" problem when several different population variances are involved | journal = [[Biometrika]] | volume = 34 |issue=1–2 | pages = 28–35 | year = 1947 |doi =10.1093/biomet/34.1-2.28 | mr = 19277 }}</ref> As such, it is an approximate solution to the [[Behrens–Fisher problem]].
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| ==Formulas==
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| Welch's t-test defines the statistic ''t'' by the following formula:
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| :<math>
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| t \quad = \quad {\; \overline{X}_1 - \overline{X}_2 \; \over \sqrt{ \; {s_1^2 \over N_1} \; + \; {s_2^2 \over N_2} \quad }}\,</math>
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| where <math>\overline{X}_{i}</math>, <math>s_{i}^{2}</math> and <math>N_{i}</math> are the <math>i</math><sup>th</sup> [[mean|sample mean]], [[variance|sample variance]] and [[sample size]], respectively. Unlike in [[Student's t test|Student's ''t''-test]], the denominator is ''not'' based on a [[pooled variance]] estimate.
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| The [[degrees of freedom]] <math>\nu</math> associated with this variance estimate is approximated using the [[Welch–Satterthwaite equation]]:
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| :<math>
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| \nu \quad \approx \quad
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| {{\left( \; {s_1^2 \over N_1} \; + \; {s_2^2 \over N_2} \; \right)^2 } \over
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| { \quad {s_1^4 \over N_1^2 \nu_1} \; + \; {s_2^4 \over N_2^2 \nu_2 } \quad }}
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| </math>
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| Here <math>\nu_i</math> = <math>N_i-1</math>, the degrees of freedom associated with the <math>i</math><sup>th</sup> variance estimate.
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| ==Statistical test==
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| Once ''t'' and ''<math>\nu</math>'' have been computed, these statistics can be used with the [[t-distribution]] to test the [[null hypothesis]] that the two population means are equal (using a [[two-tailed test]]), or the null hypothesis that one of the population means is greater than or equal to the other (using a one-tailed test). In particular, the test will yield a [[p-value]] which might or might not give evidence sufficient to reject the null hypothesis.
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| ==References==
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| {{Reflist}}
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| ;Further reading
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| * Daniel Borcard, ''[http://biol09.biol.umontreal.ca/BIO2041e/Correction_Welch.pdf Lecture Note Appendix: t-test with Welch correction''], excerpt from Legendre, P. and D. Borcard. ''Statistical comparison of univariate tests of homogeneity of variances''.
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| * {{cite journal |last=Sawilowsky |first=Shlomo S. |year=2002 |url=http://education.wayne.edu/jmasm/sawilowsky_behrens_fisher.pdf |title=Fermat, Schubert, Einstein, and Behrens–Fisher: The Probable Difference Between Two Means When σ<sub>1</sub> ≠ σ<sub>2</sub> |journal=Journal of Modern Applied Statistical Methods |volume=1 |number=2 |pages=461–472}}
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| {{statistics-stub}}
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| [[Category:Statistical approximations]] | |
| [[Category:Statistical tests]]
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