# How is this simplified pearson coefficient derived? [duplicate]

I came across a variation of the Pearson coefficient as seen here:

$$r=\left[1-\left(\frac{\sum_{i=1}^n (x_{ti}-x_{pi})^2}{\sum_{i=1}^n x^2_{ti}}\right)\right]^{1/2}\,,$$

where $$x_{ti}$$ is a target value and $$x_{pi}$$ is a predicted value.

I've seen the various forms of the r coefficient on Wikipedia, but can't figure out how you would get to this form. Any advice would be great. Thanks!

• If $x_t$ has mean zero, the formula you've posted is the root of the $R^2$ stat, which is equal to the pearson correlation coefficient (see eg here: stats.stackexchange.com/questions/83347/…). – CloseToC Oct 15 '19 at 14:28
• The formula is correct only for non-negative $r.$ – whuber Jan 11 at 13:44