# Relationship between Mann-Kendall and Kendall Tau-b

Do the Mann-Kendall and Kendall Tau-b use very similar test statistics? It seems that everytime I perform both tests, they always provide the same p-value and same conclusion.

If $$S$$ is the usual Mann-Kendall statistic, then $$\tau=\frac{S}{D}$$, where $$D^2= {n\choose 2}({n\choose 2}-\sum_i {t_i \choose 2})$$, where $$t_i$$ is the number of values tied at the $$i$$-th set of tied values.
Since (presumably) the times are all distinct (i.e. cannot be tied), the formula for $$\tau_b$$ will also take this form.
i.e. $$S$$ is merely a scaled $$\tau_b$$, where the scaling constant that depends only on the sample size and the pattern of ties (how many ties there are for each set of tied values, not the values taken).