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I recently had a conversation with a college of mine and this question popped into my mind and I am unsure how to solve it.

Let's say I have 100 variables. Some of the variables have a pairwise relationship but most of them do not and are just random.

My question is what amount of correlation can I expect that is just due to randomness?

How would I go about calculating an upper bound of correlation due to randomness?

Any help or pointing in a direction would be greatly appreciated.

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    $\begingroup$ Welcome to Cross Validated! There’s always some tiny chance that, just by some bad luck, your variables that should be uncorrelated wind up perfectly correlated, so the entire $[-1,1]$ range is in play. I’ll expand on this in an answer if no one beats me to discussing the sampling distribution of the correlation. // Your question might count as a duplicate, though I see a subtle difference between your question and the linked question. $\endgroup$
    – Dave
    Jun 25 at 15:36

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As discussed here, the sampling distribution of correlation for uncorrelated Gaussian variables covers the entire range of $[-1,1]$. While you have not assumed Gaussian variables, you have not ruled out Gaussians. Consequently, the upper bound is $1$, and the lower bound is $-1$.

Even if you do rule our Gaussian variables, the entire range of $[-1,1]$ is possible. Just due to bad luck, which we can quantify using an appropriate sampling distribution, it’s always possible to happen to have perfect positive or negative correlation in the sample, despite zero population correlation.

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    $\begingroup$ A useful interpretation of "upper bound" in this context would be a probability statement, perhaps of the form "given the variables are iid with distribution $F,$ there is $x\%$ chance that all the sample correlation coefficients will have magnitudes less than $r$" where an accurate mathematical relationship between $x,$ $r,$ the sample size $n,$ number of variables $k,$ and $F$ is provided. This has the potential of offering universal bounds for $r$ depending on $n$ and $k,$ regardless of what $F$ might be. $\endgroup$
    – whuber
    Jun 25 at 16:00

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