Averaging correlated signals, where sqrt-n breaks

Question

You average n = 100 signals, each with standard deviation \sigma = 10. But they are not independent: every pair is positively correlated with common correlation \rho = 0.2. What is the standard deviation of the average? Compare it to the independent case.

Accepted Answer

The usual \sigma/\sqrt{n} rule assumes the observations are independent so their noise cancels. When everything leans the same way, a shared component survives the averaging and never washes out. For n observations that each have variance \sigma^2 and share a common pairwise correlation ho, the variance of the average is 	ext{Var}(\bar{X}) = \frac{\sigma^2}{n}\Big[1 + (n-1)ho\Big]. The first term is the familiar independent piece; the second is the penalty from correlation. Plug in \sigma^2 = 100, n = 100, ho = 0.2: 	ext{Var}(\bar{X}) = \frac{100}{100}\Big[1 + 99 	imes 0.2\Big] = 1 \…

Averaging correlated signals, where sqrt-n breaks

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