Welch's t-test ($t=(\overline{X}-\overline{Y})/\sqrt{s_X^2/n_X+s_Y^2/n_Y}$)

For two independent samples from approximately normal distributions, Welch's t-test compares their means without assuming equal unknown variances. The displayed test statistic uses sample variances $s_X^2,s_Y^2$. Under the equal-mean null hypothesis, its reference Student t-distribution has approximate statistical degrees of freedomA p-value from this approximation addresses the specified statistical error model; it cannot distinguish a material effect from an unmeasured experimenter bias when sample and experimenter are confounded.

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 satisfyIndeed 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.