Welch's t test

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In statistics, Welch's t test is an adaptation of Student's t-test intended for use with two samples having possibly unequal variances.[1] As such, it is an approximate solution to the Behrens–Fisher problem.


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 ith 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 ith variance estimate.

Statistical test[edit]

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 alternative 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.


  1. ^ Welch, B. L. (1947). "The generalization of "Student's" problem when several different population variances are involved". Biometrika 34 (1–2): 28–35. doi:10.1093/biomet/34.1-2.28. MR 19277. 
Further reading