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for a user study I want to compare some tasks. Each tasks has 2 conditions on a intervall scale. From each task I want to test the the conditions with:

  • a t-test if both samples are on a normal distributions
  • a wilcoxon, if both samples are not on a normal distribution

but what test I can choose, if one sample of a task is on normal distribution and the other is not?

I've read something about, that I can run samples from a normal distribution with a wilcoxon test, too. Are there some benefit of the t-test for normal distribution samples? Why should I not test all my samples with the wilcoxon?

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  • $\begingroup$ @Glen_b, Just a heads up, I think I know what your asking, but you might want to spell check this comment. $\endgroup$ – k6adams Apr 25 '15 at 14:48
  • $\begingroup$ I'll try that again: On what basis can you know that one sample is from a normal distribution? $\endgroup$ – Glen_b -Reinstate Monica Apr 25 '15 at 18:26
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The Wilcoxon is a non-parametric test which works on normal and non-normal data. However, we usually prefer not to use it if we can assume that the data is normally distributed. The non-parametric test comes with less statistical power, this is a price that one has to pay for more flexible assumptions. Use t-test if you can assume normal.

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You are correct, there is nothing "wrong" with using Wilcoxon when data is normally distributed. However, there are many advantages to that come with the normality assumption.

In general, because the Wilcoxon does not assume normality, it is a more conservative estimate. For example, say you conducted a t-test on normally distributed data, and the p-value of that test was marginally significant, $\approx .05$. Chances are if you were to conduct that same test using a Wilcoxon, then the p-value is likely to be $> .05$. You may concluded that there is no evidence to conclude that the two populations are different when in fact there is evidence.

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  • $\begingroup$ I personally encountered a situation in my Design of Experiments class where I wished the normality assumption had been met. The experiment involved microwaving a 1\2 cup of water for 30 seconds in the microwave and taking the temperature. There were 3 types of cups (ceramic, plastic, and glass), and three types of microwave conditions (clean, wet, dirty). In general wet, containers in clean microwaves reached higher temperatures than those in wet or dirty microwave conditions. $\endgroup$ – k6adams Apr 24 '15 at 22:46
  • $\begingroup$ The problem was that ceramic containers had much larger variance than their counterparts, and the homogeneity assumption was violated.That meant I needed to use Bonferroni-Weltch to make comparisons, and could not conclusively concluded that microwave cleanliness is a significant factor in cooking time, rather I could only state that a clean microwave with a glass container was different from a paper container with a dirty microwave. $\endgroup$ – k6adams Apr 24 '15 at 22:46
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    $\begingroup$ I'm a little confused by several things in your answer. What do you mean by 'a more conservative estimate'? (Estimate of what?) When you say 'generally speaking' what is assumed and what is being generalized over? $\endgroup$ – Glen_b -Reinstate Monica Apr 25 '15 at 5:52
  • $\begingroup$ @Student T has a great, slightly more technical answer. $\endgroup$ – k6adams Apr 25 '15 at 11:26
  • $\begingroup$ @Glen_b by conservative I mean less likely to take risk, (In this case the risk of assuming the underlying distribution is normal). By estimate, I mean the way in which we characterize the population we are trying to capture. By general, I am attempting to indicate that this is not going to be a super technical answer. $\endgroup$ – k6adams Apr 25 '15 at 14:46

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