Research Design and Statistical Consulting
George M. Diekhoff, Ph.D.

The Method of Modified z-Scores: “You can get any mean you want with…..” (to the tune of Alice’s Restaurant)

Some years ago, when I wrote my text, Statistics for the Social and Behavioral Sciences: Univariate, Bivariate, and Multivariate , I included a description of a method by which one can recode the values of a continuous variable so as to have any desired mean and standard deviation. This is useful in a surprisingly large number of situations. For instance, supposed you have a measure of Spiritual Well Being (SWB) that seems also to reflect a certain amount of plain old depression. To remove that “contamination” from the SWB variable, you can regress SWB on a measure of depression (like the Beck Depression Inventory), and save the residuals. Those residuals represent SWB from which the influence of depression has been removed–“pure” SWB. The problem is, the residual scores look VERY different than did the original SWB scores. It doesn’t really matter, because strong positive values still represent large amounts of SWB and strong negative values still represent small amounts of SWB. Even so, it can be disconcerting to work with numbers that look so unfamiliar. Solution: recode the residual scores to have whatever mean and standard deviation you want them to have–maybe something familiar like a mean of 100 and standard deviation of 15–like IQ scores. 

How do you do that? The method of modified z-scores. (1) Standardize the residualized SWB variable, i.e., convert the residuals to z-score form. See my comment immediately above for advice on standardizing residuals. (2) Create your new variable with a compute statement as follows: NewVar = (desired standard deviation * z) + desired mean. (3) Finally, always be sure to check the mean and standard deviation of your new variable to be sure it has the desired values.