# Is a log transformation a valid technique for t-testing non-normal data?

In reviewing a paper, the authors state, "Continuous outcome variables exhibiting a skewed distribution were transformed, using the natural logarithms, before t tests were conducted to satisfy the prerequisite assumptions of normality."

Is this an acceptable way to analyze non-normal data, particularly if the underlying distribution is not necessarily lognormal?

This may be a quite uncommon question, but I have not seen this done before....

• Well, if the initial distribution is not log-normal, then the transformed data does not satisfy the prerequisite assumptions of normality, so what is being gained by the transformation? Apr 2 '12 at 23:43
• @Macro - true enough! (+1) - they probably just wanted to get the distributions closer to symmetric, which is not a bad thing to want to do for t-testing, but, unless they checked and wrote it up, we don't know if the log transform induced a negative skew that might have made matters worse... Apr 3 '12 at 0:23
• We might infer that because it was done to satisfy normality, and normality was checked int the first place, that normality was checked afterwards. It's strongly implicit in the language here.
– John
Apr 3 '12 at 1:25
• Generally speaking if the assumptions required to carry out a t-test are not met, then it would be more appropriate to use a non-parametric test. Apr 3 '12 at 2:06
• A t-test for the logarithms is neither the same as a t-test for the untransformed data nor a nonparametric test. The t-test on the logs compares geometric means, not the (usual) arithmetic means. This is one of several important considerations in deciding whether using the logarithms is acceptable (which it can be, depending on the application).
– whuber
Apr 3 '12 at 6:04

Transformations to normality should always be followed by an investigation of the normality assumption, to assess whether the transformed data looks "normal enough". This can be done using for instance histograms, QQ-plots and tests for normality. The t-test is particularly sensitive to deviations from normality in form of skewness and therefore a test for normality that is directed towards skew alternatives would be preferable. Pearson's sample skewness $\frac{n^{-1}\sum_{i=1}^n(x_i-\bar{x})^3}{(n^{-1}\sum_{i=1}^n(x_i-\bar{x})^2)^{3/2}}$ is a suitable test statistic in this case.