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.