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?

  • $\begingroup$ I think this was actually answered here: stats.stackexchange.com/questions/5747/…. $\endgroup$
    – eagle34
    Nov 29, 2021 at 2:06
  • $\begingroup$ 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. $\endgroup$
    – Glen_b
    Nov 29, 2021 at 2:48
  • $\begingroup$ 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 $\tau \ne 0.$ $\endgroup$
    – whuber
    Nov 29, 2021 at 14:58