Center-of-momentum frame

In statistics, Welch's t test is an adaptation of Student's t-test intended for use with two samples having possibly unequal variances. As such, it is an approximate solution to the Behrens–Fisher problem.

Formulas

Welch's t-test defines the statistic t by the following formula:

$t\quad =\quad {\;{\overline {X}}_{1}-{\overline {X}}_{2}\; \over {\sqrt {\;{s_{1}^{2} \over N_{1}}\;+\;{s_{2}^{2} \over N_{2}}\quad }}}\,$

where ${\overline {X}}_{i}$, $s_{i}^{2}$ and $N_{i}$ are the $i$^th sample mean, sample variance and sample size, respectively. Unlike in Student's t-test, the denominator is not based on a pooled variance estimate.

The degrees of freedom $\nu$  associated with this variance estimate is approximated using the Welch–Satterthwaite equation:

$\nu \quad \approx \quad {{\left(\;{s_{1}^{2} \over N_{1}}\;+\;{s_{2}^{2} \over N_{2}}\;\right)^{2}} \over {\quad {s_{1}^{4} \over N_{1}^{2}\nu _{1}}\;+\;{s_{2}^{4} \over N_{2}^{2}\nu _{2}}\quad }}$

Here $\nu _{i}$ = $N_{i}-1$, the degrees of freedom associated with the $i$^th variance estimate.

Statistical test

Once t and $\nu$ 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.

References

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Further reading

• Daniel Borcard, 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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