# Using paired t test if data is not normally distributed but the data set is large

I need to test some before and after data, to see if there is a significant shift (n = 114) but the data differences are not normally distributed (or, at least, a histogram would seem to indicate that). Can I use a paired t-test?

• 114 is not that large.
– Gala
Sep 12, 2013 at 8:50

Have you thought about a non-parametric test such as the Mann-Wilcoxon? You could test the before and after distribution to see if the average value has shifted significantly in one direction or the other.

### Edit

@Scortchi is correct in that in your specific case, the Wilcoxon signed rank test would be a more appropriate non-parametric test.

• Yes, I have considered using the Wilcoxon Signed Rank Test, but without a data analysis package like SPSS that is very tedious. I am hoping to use t tests as they are easy to complete using an Excel formula Sep 12, 2013 at 8:03
• If you have R it is easy. The coin package for R has a lot of non-parametric tests with both exact and asymptotic power calculations. I've done MWW tests in Excel; it isn't that difficult using RANK. If you have ties, you'll have to use some IF statements. Sep 12, 2013 at 8:11
• @Avraham: Note they're paired observations: you'd want the Wilcoxon signed-rank test to test the null hypothesis that the mean difference within each pair is zero. And the OP might want to consider relying on the central limit theorem to justify the $t$-test approximately - it's quite a large sample size. Sep 12, 2013 at 8:28
• @Scortchi Yup; I should have been more precise in the specific non-parametric test. Personally when I use them, it is in the context of demonstrating changes in claim severity/frequency or aggregate loss distributions to indicate if prior experience for a given insurer may no longer be so credible vis-a-vis reinsurance pricing. As such, I'm not dealing with paired observations. Also, while the sample size may make the $t$ test still OK, the OP says the data does not look normal, and if e.g. there is heavy skew, the t wouldn't work so well, but non-parametric tests would still be appropriate. Sep 12, 2013 at 15:00