Basic Concepts

  1. The slope represents the the average change in Y associated with a change of one unit on X. If the relationship between X and Y is causal, then the slope lets you know how much you can change the average value of Y by increasing X.
  2. The standard error of the estimate reflects the degree to which the points diverge from the regression line. It reflects the accuracy of the prediction.
  3. The standard deviation of X is related to the variance explained. If a variable has a very small standard deviation then it will not be able to explain much variance.
  4. Pearson's r is determined by the three independent components: the slope, the standard error of the estimate, and the standard deviation of X. The same value of r can be composed of different combinations of these components. The following table shows three ways to product an r of .707.
  5. Slope

    SEest

    sdx

    r

    1

    10

    10

    .707

    .5

    5

    10

    .707

    1

    5

    5

    .707

  6. r2 is the ration of variance explained to total variance.

    The variance explained is equal to (slope2)(sdx2)

    The variance unexplained is equal to SEest2.
    r2 = explained/(explained + unexplained).

  7. The standard deviaiton of Y is simply the square root of the total variance as described above.

Credits

The simulations were developed as part of a grant from NSF to David Lane of Rice University. Partial support for this work was provided by the National Science Foundation's Division of Undergraduate Education through grant DUE 9751307. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.