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 unequal variances. It consists of the statistic

t = {\overline{X}_1 - \overline{X}_2 \over \sqrt{ {s_1^2 \over N_1} + {s_2^2 \over N_2} }}\,

where the denominator is not based on a pooled variance estimate. The degrees of freedom ν associated with this variance estimate is approximated using the Welch-Satterthwaite equation.


\nu_{\overline{X}_1 - \overline{X}_2} = {{\left( {s_1^2 \over N_1} + {s_2^2 \over N_2}\right)^2 } \over {{s_1^4 \over N_1^2 \cdot \nu_1}+{s_2^4 \over N_2^2 \cdot \nu_2}}}\,

Here νi is Ni−1, the degress of freedom associated with the ith mean.

[edit] References

  • Welch, B. L. The generalization of "student's" problem when several different population variances are involved. Biometrika. 1947. 34: 28-35
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