# Estimating values knowing their Pearson's r and their means and standard deviations

I apologize if my question is exceedingly simple. Imagine, for example, I am studying a paper which explains that two variables correlate with a certain Pearson's r (no p-value, no confidence interval, no sample size provided).

Can I estimate the value of one of the two variables if I know the other one? How?

I found some examples but all dealt with the knowledge of all the data, not just the final Pearson's r.

(I use R)

Can I estimate the value of one of the two variables if I know the other one and their means and standard deviations? How?

• you can estimate the p value and confidence interval if you know the correlation and sample size. – TPM Jul 9 '18 at 15:37
• Even if the relationship is linear, you'd need more than just the correlation; you'd need the means and standard deviations of both variables. – Glen_b Jul 9 '18 at 15:40
• Thank you for your comment TPM. Let me be clearer: I do not want to estimate p-value, confidence interval or sample size. Instead, I would like to "convert" a value from one variable to the other one. Thank you for your comment Glen_b. So, simply speaking, I can't do what I would like to do? – statisticianwannabe Jul 9 '18 at 15:40
• Ah, sorry for the confusion. Unfortunately, then, the answer is no. You would need to know the means and standard deviations of the variables to do this. – TPM Jul 9 '18 at 15:48
• Thank you for you comments. I added another question so if you want to elaborate a bit and make an answer, I would be glad to upvote them – statisticianwannabe Jul 9 '18 at 16:02

In simple linear regression, using the model $Y_i = \beta_0 + \beta_1x_i + e_i,$ where $e_i$ are independently $\mathsf{Norm}(0,\sigma),$ one can find estimates of $\beta_0,$ $\beta_1,$ and $\sigma$ knowing $n, \bar x, \bar Y, S_x, S_y,$ and $r.$ Most of the formulas can be found in any elementary text that covers simple linear regression. For example, the slope is estimated as $\hat \beta_1 = rS_y/S_x.$ From there, $\hat \beta_0$ can be found by noting that the regression line must pass through $(\bar x, \bar Y).$
With $\hat \beta_0$ and $\hat \beta_1,$ one can find the point estimate $\hat Y_i$ corresponding to $x_i.$ Knowing $\hat \sigma^2 = S_{Y|x}^2,$ one can find a confidence interval corresponding to that point estimate.
Some additional formulas for confidence and prediction intervals also require $$S_{Y|x}^2 = \frac{n-1}{n-2}S_Y^2(1 - r^2),$$ where, by definition, $S_{Y|x}^2 = \frac{1}{n-2}\sum_{i=1}^n(\hat Y_i - Y_i)^2.$