In regression, which distribution will have the largest standard deviation? 
*

*the predicted scores on the outcome variable, Ŷ

*the residuals

*the observed scores on the outcome variable, Y

*the standardized regression coefficients 



My understanding is that the question wants to see which of these regression distributions has the greatest spread.
My first approach was to try and order these options on a spectrum of lowest to highest standard deviation. I think observed scores would have greater standard deviation versus predicted scores, as regression tries to minimize the residuals. I have no idea how to compare the other two distributions though. (It may have to do with the formulas for these, but I find them very confusing!)
What do you think?
 A: Without more information, I don't believe you can actually order these four.  Fortunately, the question only asks you which is largest; which is, e.g., second largest is not required.  Let's see if some hints will help you determine which must be largest, and why:  


*

*Consider that if $X$ and $Y$ are perfectly uncorrelated, not only in the population, but also in your sample, then the SD of the residuals will be identical to the SD of $Y$.  I suppose the question is assuming that $X$ and $Y$ are not perfectly uncorrelated in your sample (which would be incredibly unlikely), even if they are in the population.  In that case what would be the ordinal relationship between them?  

*Next, how are the observed scores, $y_i$, related to the predicted scores, $\hat y_i$, and the residuals, $\hat e_i$?  (This can be inferred from the full regression equation, or from the equation that lets you calculate the residuals given the fitted model, for example.)  Given the way these three are related, which must have the largest SD?  

*The SD of the standardized regression coefficient is a little trickier.  How is the standard error of the estimated coefficient related to the standard deviation of the residuals?  Given what you had figured out in #1 & 2, what does this imply?  
