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

Normalizing Scores with a Data Transformation Doesn’t Change the Construct Being Measured

In the presence of skewed data distributions, one or another of the commonly used data transformations (square-root, log10, or reciprocal) can help to normalize the data. Any of the texts by Tabachnick & Fidell are a good source of information about score transformations. Of course the transformed scores take on different numerical values than the original scores, sometimes even resulting in score reflection–the lowest original scores become the highest transformed scores and the highest original scores become the lowest transformed scores. Reflected scores can be (and should be) re-reflected (again, see Tabachnick and Fidell, 2013) to solve that problem, but nothing will change the fact that the transformed scores will still take on different values than the original scores. This can be vexing in some applications. For instance, where the mean of a set of original IQ score might have a value of 100 and be perfectly interpretable (“average”), the transformed mean might be 10, not a directly meaningful value. One should not make too much of these changed values, though. Assuming that one has re-reflected reflected scores, high transformed scores still reflect greater amounts of the attribute and low transformed scores still reflect lessor amounts of the attribute. This is a point that is missed by some, who make far too much of the fact that transformed scores take on different numerical values than the original scores, to the point that they believe that the transformed scores no longer measure the same construct that was measured by the original scores. The quickest way to dispel this misimpression is to examine the correlation between original and transformed scores. That correlation typically runs in the upper 0.90’s. It is axiomatic in statistics that, to the extent that two variables are correlated, they measure the same thing. Data transformations do not change the construct being measured; just the score values and the shape of the distribution of the scores.