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?


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 $U_2$:



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