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. As such, it is an approximate solution to the Behrens-Fisher problem.

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[edit] Formulas for t, ν

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


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

where \overline{X}_{i}, s_{i}^{2} and Ni 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 ν associated with this variance estimate is approximated using the Welch-Satterthwaite equation:


\nu   = 
 {{\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 degrees of freedom associated with the ith variance estimate.

[edit] Statistical test

Once t and ν 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.

[edit] See also

[edit] References

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