# Solved – Equivalence of Mann-Whitney U-test and t-test on ranks

What does it mean when people say that a t-test performed on ranked data is equivalent to a Mann-Whitney U-test? Does that mean they just test the same hypothesis/are useful in the same situations or are they are supposed to give the exact same p-values? The reason I ask is I tried both in `R` and compared two groups with very small sample sizes (3 and 4). I got completely different answers: one significant and one not.

The two groups are A=(1,2,3) and B=(4,5,6,7).

t-test: p = 0.01

Mann-Whitney U-test: p = 0.06

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Does that mean they just test the same hypothesis/are useful in the same situations or are they are supposed to give the exact same p-values?

It means:

(i) the test statistics will be monotonic transformations of each other.

(ii) they give the same p-values, if you work out the p-values correctly.

Your problem is that the t-test on ranks doesn't have the distribution you use when you use t-tables (though in large samples they'll be close). You need to calculate its true distribution to correctly calculate the p-value. It matters most for small samples … but they're also the ones where you can calculate the actual distribution most easily.

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