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I have a group which is not normally distributed. I need to check whether the difference between left and right hand within the group is significant. Initially, I calculated the absolute difference (I don't really care which one is better, I just need to check if the difference is significant) and then I did the Kolmogorov-Smirnov test but I am not sure that this is the right one to use. If the group was normal I would have done a one sample t-test on the difference but now I am quite confused.

Please, could you please advise me which test is better to use?

Thank you in advance!

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A Kolmogorov-Smirnov one-sample test on the absolute differences does not make much sense (what would be the reference distribution anyway).

Most prominent non-parametric alternative to the one-sample t-test on the differences $D_i$ is Wilcoxon's signed-rank test. It has never much lower power than the classic t-test but has less assumptions.

Other options include

  • One-sample median test (checks the null hypothesis of a median difference of 0 through a binomial test)

  • Permutation version of the one-sample t-test based on randomly switching signs of the differences

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  • $\begingroup$ Note that this test may reject the null due to different variances and not means - it is a test on distribution not on mean $\endgroup$
    – Xavier Bourret Sicotte
    Commented May 10, 2019 at 1:48
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One-sample Wilcoxon test may be what you are looking for.

It test if non-independent samples differ and it is non-parametric.

Nice tutorial with R is here.

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