What is the correct formula for the Mann-Whitney U test? I have noticed an inconsistency in the formula for the Mann-Whitney U test. Sites like wikipedia often report it as $$U_1={\frac{n_1(n_1+1)}2}-R_1$$ However, the original paper by Mann and Whitney reports it  as $$U_1=n_1n_2+{\frac{n_1(n_1+1)}2}-R_1$$ What difference does $n_1n_2$ make?
Or are these different tests?
 A: It's the same test, but you're actually reading it wrong. Wikipedia defines $U_1$ as:
$$
U_1 = R_1 - \frac{n_1(n_1+1)}{2}
$$
And, using the same notation, the Mann-Whitney paper defines $U_1$ as:
$$
U_1 = n_1n_2 + \frac{n_2(n_2+1)}{2}-R_2
$$
Note that aside from the $n_1n_2$ piece, the rest of the definition of $U_1$ is actually in terms of $n_2$ and $R_2$ ($m$ and $T$ in the paper). Actually you can do some rearranging to get $U_1$ directly in terms of $U_2$:
$$
U_1 = n_1n_2 - (R_2-\frac{n_2(n_2+1)}{2})
$$
The bracketed term is of course just $U_2$, so:
$$
U_1 = n_1n_2 - U_2
$$
You can see that this is true by considering the fact that the sum of all ranks is just $\frac{(n_1+n_2)(n_1+n_2+1)}{2}$, so:
$$
R_1+R_2=\frac{(n_1+n_2)(n_1+n_2+1)}{2}
$$
Putting $R_1$ and $R_2$ in terms of $U_1$ and $n_1$ and $U_2$ and $n_2$ yields:
$$
U_1+\frac{(n_1)(n_1+1)}{2}+U_1+\frac{(n_2)(n_2+1)}{2}=\frac{(n_1+n_2)(n_1+n_2+1)}{2}
$$
Then you can do some algebra and see the relationship between $U_1$ and $U_2$:
$$U_1=n_1n_2-U_2$$
