What is the correlation between the value of a random variable and the average value of a sum of random variables (of which the variable is one)?

Question

There are $n$ random variables, which are all independent of each other. At time $t$, I press a button and gain a value $w_{i,t}$ for each random variable $i$. From this, I calculate the average $\bar{w}_{t}$ of all the random variables at time $t$. I do this repeatedly $T$ times.

What is the correlation between the values of $w_i$ and $\bar{w}$ through time?

I had assumed it would just be $1/n$, as the variable $w_i$ contributes $1/n$ to the average value $\bar{w}$ at each time. But from a quick use of a spreadsheet in Excel, it turns out that when $n = 10$, the correlation between any specific $w_i$ and $\bar{w}$ is about 0.31. What's going on here?

Answer 1

Covariance of $w_i$ and $\bar w$ is indeed $\frac1n\cdot Var(w_1)$: $$ \text{cov}(w_i,\bar w)=\text{cov}\left(w_i, \frac{w_1+\ldots+w_n}{n}\right)=\frac{\text{cov}(w_i,w_1)+\ldots+\text{cov}(w_i,w_n)}{n}=\frac{\text{cov}(w_i,w_i)}n = \frac{Var(w_i)}{n}. $$ Since correlation is covariance divided by standard deviation, and variance of $\bar w$ is $Var(w_i)/n$, then $$ \text{corr}(w_i,\bar w)=\frac{Var(w_i)/n}{\sqrt{Var(w_i)Var(\bar w)}} = \frac{Var(w_i)/n}{\sqrt{Var(w_i)Var(w_i)/n}} = \frac{1}{\sqrt{n}}. $$ For $n=10$, $$\frac1{\sqrt{n}}=0,316227766.$$

Answer 2

You seem to have assumes that the $n$ independent variables $W_i$ are identically distributed or at least have the same variance, let's say $\sigma^2$. Then you can say

• $W_1$ has variance $\sigma^2$ and standard deviation $\sigma$
• $\sum W_i$ has variance $n \sigma^2$ and standard deviation $\sqrt{n} \sigma$
• $\bar W$ has variance $\frac1n \sigma^2$ and standard deviation $\frac1{\sqrt{n}} \sigma$
• $W_1$ and $\sum W_i$ have covariance $\sigma^2$ and correlation $\frac{\sigma^2}{\sqrt{n}\sigma^2} = \frac1{\sqrt{n}}$
• $W_1$ and $\bar W$ have covariance $\frac1n\sigma^2$ and correlation $\frac{\sigma^2/n}{\sigma^2/\sqrt{n}} = \frac1{\sqrt{n}}$