= Welch degrees of freedom upper bound
{c}
{title2=$\nu\le n_1+n_2-2$}
For two samples of sizes $n_1,n_2>1$, write $a=s_1^2/n_1$ and $b=s_2^2/n_2$, with $a+b>0$. The reference <statistical degrees of freedom> in <Welch's t-test> satisfy
$$
\nu=\frac{(a+b)^2}{a^2/(n_1-1)+b^2/(n_2-1)}\le n_1+n_2-2.
$$
Indeed the <Cauchy-Schwarz inequality> applied to $(a/\sqrt{n_1-1},b/\sqrt{n_2-1})$ and $(\sqrt{n_1-1},\sqrt{n_2-1})$ gives the inequality after division. Thus an independent two-sample <Welch's t-test> on $N$ complete observations cannot have reference <statistical degrees of freedom> exceeding $N-2$. The <exact pooled two-sample t statistic> also uses $N-2$, under its stronger equal-variance assumptions.
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