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I have a correlation between two variables of 0.30, hence a medium effect. but it is not statistically significant and I want to report the reasons for that. looking at the test statistic I see that the sample size (as in all test statistics) has an influence. But what else could be the reason? Does the standard deviation or standard error have an influence?

edit: Thank you for your helpful and fast answers!

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It is the sample size. For a simple correlation between two variables, the coefficient and sample size are the only pieces of information that you need in order to compute the standard error, and hence assess statistical significance. So if the coefficient does not differ significantly from zero despite being a canonical "medium" sized effect, then the only other culprit can be the sample size.

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The size of the correlation coefficient doesn't tell you something about the significance of the effect. Look at the Wiki page about the Fisher transformation. As Jake Westfall pointed out, only the correlation coefficient $r$ and the sample size $n$ are used to calculate the standard error $SE=1/\sqrt{n-3}$ and the $z$-value which is then used to calculate the $p$-value. As you can see from the formula for the standard error, at least 4 observations are necessary to use Fisher's transformation.

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