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How should the tau number from a Mann-Kendall test be interpreted? I've read that for trends to be detected by the test they must be monotonic.

If I have data across years, group them by calendar month and then run the Mann-Kendall test and get a value of .3 for Jan, does that mean that all the previous Jans had a lower average value year on year? And does the 0.3 have any real mean in terms of its value, by that I mean can I infer that each year that month's mean raises by 3%?

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How should the tau number from a Mann-Kendall test be interpreted?

(You don't make it clear which of several versions of this statistic you refer to. I assume it's the ordinary one rather than say the seasonal one.)

In that case, you can interpret the coefficient much as you would interpret a Kendall-$\tau$ in other situations; it's the proportion of up-movements against time vs the proportion of down-movements, looking at all possible pairwise time-differences.

I've read that for trends to be detected by the test they must be monotonic.

Can you say where you read this?

It can detect any kind of trend where there's a tendency to increase more often than it decreases or vice-versa. It doesn't have to be necessarily monotonic, though that's when it has the most power.

It is, however, a measure of the degree to which the trend is monotonic.

If I have data across years, group them by calendar month and then run the Mann-Kendall test and get a value of .3 for Jan, does that mean that all the previous Jans had a lower average value year on year?

From what you said there, it's not clear what calculation you're actually doing. Whatever you did, the answer is likely to be 'no'. Could you clarify what calculation was done, perhaps with a small example? Then I can try to explain more clearly what it is telling you for whatever calculation was done.

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