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Tabachnick and Fidell (2012) recommend examining the normality (outliers, skewness, kurtosis) of a variable separately/by group/sub-group if one is planning to do a group-based analysis (e.g., t-test, ANOVA).

If one finds that a particular variable is normal for one group (e.g., older adults) but non-normal for another group (e.g., significant skewness and/or kurtosis for younger adults) should both groups data be transformed to (a) address the skewness and/or kurtosis of the younger adults group and (b) allow the groups to be compared in the t-test/ANOVA?

Presumably only transforming one of these groups would lead to a meaningless significant difference between the groups because, of course, the mean of the square root group will be different than the mean of the raw data group.

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    $\begingroup$ Could you tell us a little about how you are assessing normality? Perhaps the situation is not as complicated as it seems. It is often the case that a common transformation will symmetrize the distributions in the worst groups without badly damaging the symmetries in the other groups. When such a common transformation cannot be found, heed the recommendation at the end of oce's answer: since you have thereby discovered an interesting difference among groups, investigate it! $\endgroup$
    – whuber
    Commented Feb 18, 2015 at 18:10
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    $\begingroup$ I do not understand how a test to compare skewness or kurtosis across groups would be directly relevant to this question. $\endgroup$
    – whuber
    Commented Feb 23, 2015 at 23:25
  • $\begingroup$ Thanks for your comments oce and whuber! I had asked the question for a friend who found significant skewness and kurtosis (S and K divided by appropriate standard error was significant at p<.001) for questionnaire scale and sub-scale totals differed between older adults and younger adults in a large sample of several hundred people. Therefore the power to find 'significant' non-normality (due to the diminished standard error) is large. Tabachnick and Fidell suggest that with large samples and visually not-too non-normal distributions should be ok, and this is what my friend decided to do. $\endgroup$
    – Patrick
    Commented Feb 23, 2015 at 23:29
  • $\begingroup$ Sorry whuber, my comment had accidentally posted before I finished writing it (though your comment may still hold for my completed comment that I re-posted). The tests just examined if the data was significantly kurtotic or skewed within a group - and did not compare if the groups differed from each other significantly. Just in several cases, the older adults data would be significantly skewed and/or kurtotic for a particular scale/sub-scale while the younger adults would not be (or vice versa). maybe this crude look at differences in normality between the groups is not the best way to go. $\endgroup$
    – Patrick
    Commented Feb 23, 2015 at 23:34
  • $\begingroup$ Thank you for your patient clarification, Patrick. It would be well to distinguish between a significant difference and an important one. As discussed extensively elsewhere on this site, it is very easy to identify tiny, unimportant deviations from normality in medium to large datasets. What you should focus on is how and by how much the groups depart from appearing normal. $\endgroup$
    – whuber
    Commented Feb 23, 2015 at 23:53

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(This should be a comment but I can't comment so put as an answer)

I agree you should transform both group as you have suggested.I don't know what you variable is, but normally, if one group of data is normal, and the other is not, this sounds a bit worrying if the data collection procedure is okay. It may be the sample were not random enough, or it could be data collection method issue, like ceiling effect (say if you are testing how many word one can remember out of 7, and most young ppl can remember all, so this could cause the non-normal data) So before you transform data, I would suggest you to look into why one group is normal while the other is not.

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  • $\begingroup$ Thanks oce, I have passed on your suggestions to my friend! The data was not random - it was a convenience sample of older and younger adults - and that may be partially responsible for the discrepancy. But, as I noted above in my comments to whuber, after exploring whether common transformations could help the data (they tended to make things worse :/) my friend decided to leave the data untransformed since there was a large sample (several hundred respondents) and the skewness and kurtosis tended to be not tooooooo bad (maybe z scores of 3.5 to 9 at the worst). Hopefully this is defensible. $\endgroup$
    – Patrick
    Commented Feb 23, 2015 at 23:40

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