# Correlation of three variables: Why is Var1 not correlated with Var2 even though Var1 is correlated with Var3 and Var2 is correlated with Var3?

I do have three variables (Var1, Var2, and Var3) for which I have conducted a two-tailed pearson correlation (see also picture below). I am looking for a good explanation as to why:

• Var 1 is not correlated with Var 2
• Even though, Var 1 is correlated with Var3
• Under the light that Var2 is correlated with Var3

I would have assumed that Var1 should also be correlated with Var2, as Var2 is correlated with Var3...

Kind Regards and thanks for the help

• Suppose you started with $X_1$ and $X_2$ and $\epsilon$ all independent and then set $X_3=X_2-X_1+\epsilon$. Then you might get something like what you have seen with correlations between $X_3$ and the other two, but not between $X_1$ and $X_2$ – Henry Feb 10 at 8:17
• Thanks for the answer Henry: You need to help me out a little bit. Unfortunately, I am not a mathematician. Is there a simple explanation for the behaviour which I described above? Thx – Raphael Feb 10 at 8:28
• A scatter plot matrix with all pairwise scatter plots should also help, – Nick Cox Feb 10 at 10:00

A correlation expresses an association between two variables. One can find the variance that's shared between two variables by squaring the correlation coefficient (e.g., shared variance = $$r^2$$.) So, the shared variance between variable 1 and 3 in your diagram is 0.033, and the shared variance between variable 3 and 2 is 0.045. That means, there's 1-0.033 (.967) of the variance in variables 1 and 3 left unexplained by their relationship, and 1-0.045 (0.955) of the variance in variables 2 and 3 left unexplained by their relationship. Given the tiny components of explained variance, it's very possible that there is no overlap between variables 1 and 2, they're independent.