If variable X and Y has correlation 0.1, how much does it help to predict Y based on X? In the simplest binary case, the probability (p) to correctly predict Y based on X is a linear function of correlation (c), i.e.
$$p=\frac{c+1}{2}$$
That means, a 0.1 correlation will increase the prediction probability from 0.5 to 0.55, a 5% increase. This could be significant if you are betting a lot of money 🙂
Hi Professor Xu, thank you for your two posts on the correlation coefficient. Regarding your formula on this page for (p) as a function of (c), I’m sorry, I don’t see why this is necessarily a linear function?
I see that your function satisfies the boundary conditions for (c) = 0 & (c) = 1. However, looking at the distribution curve of the correlation coefficient (c) on your other page, I see that (c) is a highly non-linear curve. The linearity vs. non-linearity is confusing to me.
If you have a moment, please, would you mind please re-explaining the relationship between (p) & (c) with reference to the distribution curve? If the curve is not relevant, however, would you mind please explaining why not?
Thank you very much for your consideration,
Brad