A question for the experts

[Girchuck]

I have a variable A correlate with variable B and the correlation coefficient R_square is 0.1.
What other information do I need to estimate the probability that variable A does not depend on variable B?

[pzhon]

If the correlation between two variables is 0, this is far weaker than saying that the variables are independent. It's like knowing that f(0)=g(0) doesn't tell you that f(x)=g(x) for all x.

If you know the correlation is nonzero, the variables can't be independent.

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What other information do I need to estimate the probability that variable A does not depend on variable B?

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To talk about such a probability, you need to have a probability distribution on a set of random variables from which A and B are drawn.

[Siegmund]

All of what pzhon said is true.

That said... I think what you MIGHT be asking is "I have n observations of two variables. I don't know if they are really correlated or not. I have observed r^2=.1 on the data I have collected."

If the two variables really are uncorrelated, r * sqrt(n-2) / sqrt(1-r^2) is supposed to have a t-distribution with n-2 degrees of freedom. (Approximately normal, for large n.)

r^2=.1 corresponds to |r|/(sqrt(1-r^2))=.333, so, working backwards, we see that at 95% significance level, this is a significant correlation if observed in a data set of 37 or more observations.

Notice, this proves non-independence if r is sufficiently far from zero, it does NOT prove independence if r is sufficiently small.