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On wiki page about Fisher transformation I read that variance of sample correlation coefficient becomes smaller as population correlation coefficient (in absolute value) gets closer to 1. Could anybody, please explain why this happens? Thanks a lot for any relevant formulas or intuitive explanations!

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This tends to happen with nearly any bounded variable as its expectation approaches a bound. I'll give a loose argument that conveys an intuitive sense of why.

Consider that as the population quantity approaches the bound, the average of the distribution of the sample quantity will also tend to approach the bound. [This doesn't assume a sample estimate is necessarily unbiased, but that it will tend on average to be close to the population quantity, and more specifically that as the population quantity moves closer to the bound the mean of the sampling distribution of the sample quantity will move closer to it as well, even though it may be biased.]

So as the population quantity approaches the bound, the ability of the sample quantity to be between the bound and the population quantity (and also between the bound and its own mean) is reduced more and more (it gets squeezed between the bound and the population statistic).

This then limits the ability of the sample quantity to be very far on the other side of the mean (and the nearby population quantity) as well, because if it when it was below it tended to be far below, while it could never be far above when it's above, the mean of the sample distribution would tend to be pulled further and further away from the population quantity, which contradicts our original statement that the two would remain close.

So as the population quantity squeezes up against the bound, the variance will tend to reduce - there's less and less room for it to vary above, so it tends (in order to keep the mean moving up as well) to have to reduce the variation below, meaning the variance decreases.

This argument applies to pretty much any quantity as is approaches a bound; particular exceptions may well exist, but it conveys a sense of why we would tend to typically see it with bounded variables.

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