Let $X_1, ..., X_n$ and $Y$ be random variables. Is it possible for the $X_i$'s to all have a high magnitude of correlation (absolute value of Pearson's $r$) but not be strongly correlated with each other?
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I think this was actually answered here: https://stats.stackexchange.com/questions/5747/if-a-and-b-are-correlated-with-c-why-are-a-and-b-not-necessarily-correlated. – eagle34 Nov 29 '21 at 02:06
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That link is relevant though I'm pretty sure there's even closer matches to the question to be found. I didn't spot one with a quick search but that might just be tiredness. – Glen_b Nov 29 '21 at 02:48
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Let $X_1,\ldots, X_n$ be uncorrelated random variables, each with unit variance, and let $\tau$ be any number. Define $$Z=\frac{\tau(X_1+\cdots+X_{n-1}) + X_n}{\sqrt{1 + (n-1)\tau^2}}.$$ Consider the set of random variables $(X_1,X_2,\ldots,X_{n-1},Z).$ The squared correlation coefficients cannot exceed $1/(n-1),$ so whether this is an example of "high magnitude" depends on what you mean by "high"--but it's certainly nonzero when $\tau \ne 0.$ – whuber Nov 29 '21 at 14:58