# One sided Wilcoxon test

I want to check whether a training program made an improvement on knowledge level on participants.

I applied a test before and after the training to the same people (this means I have 2-samples, paired, right?).

Data distribution is not normal so I used Wilcoxon test (SPSS)

Null hypothesis can be rejected but my question is:

How can I prove that there was an improvement on the knowledge level of participants?

My guess is that I need to change the alternative hypothesis so that median of differences is greater than zero, which means that scores on the second sample are greater than on the first sample, right?

How can I accomplish this?

• Please search this site. What the Wilcoxon test tests has been covered i detail. Hint: it’s not medians. Also, remember that nonparametric tests work great for normal distributions. Commented Nov 9, 2023 at 13:24

1. You say your data is not-normal: incorrect.What you should say is that you rejected normality based on some test. But what was your sample size? If small (<30?), then the normality tests are quite unreliable (both for rejection or acceptance of the null), and you really should not wirry about what they say.
2. But in your case, it is not the 2 paired samples which need to be normally distributed, it is the sampling distributon of the mean of the paired differences. Did you compute the paired differences, and is that data "normal enough" (i.e. normality test does not reject normality)?
3. If normality of the paired differences has to be rejected, but your sample size is large enough (>30?), then the CLT comes to the rescue. Because it is not your sample which needs to be "normal enough", it is the sampling distribution of the mean of the paired differences which needs to be. And with a large-ish sample size, the CLT guaranties this. And, as stated in 1), if your sample size is small (<30), then you can not trust the normality tests anyway...
4. Given the above, I would run a paired t-test. It will tell you about differences of the means. But yes, for a small sample size, and rejected normality on the paired differences, your confidence will not be the (traditional) 95%. Otherwise, you have your proof
5. The Wilcoxon Signed Rank test (WSRt) does not test the median. The true null is "the data is symmetrical around the hypothesized median" (hypothesized median being 0 in your case). So the alternate is "the data is either symmetrical around a significantly different median, or the data is significantly assymmetrical around 0, or some significant combination of both..." (in fact, some authors argue that the WSRt is a better test of symmetry than anything else). So, if you reject the null, you do not know what to conclude.
6. If you need a test of the median, use a simple Sign test. It is about the median, and does not depend on an assumption of symmetry (but it has lower power that a t test). So, with a low sample size, it is unlikely to give you a significant result.

Now, if in the end you reject normality of the paired differences, your sample size is too small to invoke the cLT, and the Sign test is not significant, you need to blame the small sample size...