Gaussian forced-choice sensitivity relation ($P(\mathrm{correct})=\Phi (d^{\prime }/\sqrt{2})$)

Compare independent Gaussian samples $R_s,R_n$ with means ${\mu }_s,{\mu }_n$ and common variance ${\sigma }^2$. Choosing the larger is correct when $R_s-R_n>0$. The difference has mean ${\mu }_s-{\mu }_n$ and variance $2{\sigma }^2$, so standardization proves the displayed relation. Correlated samples, unequal variances or a different response rule need a different calculation.