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{title2=1911–1989}
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<Bernard Lewis Welch> developed inference for <mean>[means] when <sample variance>[sample variances] cannot be treated as estimates of one shared population <variance>. His 1947 paper explicitly generalized Student's problem to several different population <variances>.<The Generalization of Student's Problem when Several Different Population Variances are Involved by Bernard Lewis Welch>[Biometrika 34(1–2), 28–35 (1947), publisher title and byline]{ref}

The familiar two-group <Welch's t-test> estimates the <standard error> of a <mean> difference using $\sqrt{s_1^2/n_1+s_0^2/n_0}$, rather than pooling both <sample variance>[sample variances] under an equal-variance model. Its reference <probability distribution>[distribution] uses approximate statistical <statistical degrees of freedom>[degrees of freedom]. This makes the distinction between unequal population <variances> and unequal <sample size>[sample sizes] explicit: both enter the uncertainty calculation, and unequal group sizes do not by themselves justify assuming equal <variances>.


Bernard Lewis Welch (1911–1989)

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Bernard Lewis Welch (1911–1989)

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