Wilcoxon Test with n=4 per group not significant, why? I have the price per bottle Preis Normal of a certain drink, the difference is the number in each packaging. the Magnum is double the size of the normal bottle, so Preis Magnum / 2 should give the same value as Preis Normal, which isn't the case basically.  Each row represents a certain point in time, where the prices were gathered.
       Preis Normal` `Preis Magnum / 2` `Preis Magnum`
           <dbl>              <dbl>          <dbl>
1           372.               470            940 
2           857.              1109.          2218.
3           661.               864.          1728.
4           813.               813.          1627.


First I boxplotted the the two columns, to get a first graphical overview.

They differ substantially.
Using the R-command Wilcox-Test the following way wilcox.test(PichonMag$`Preis Normal`, PichonMag$`Preis Magnum / 2`, paired = FALSE, exact = TRUE, alternative = "less") gives me the following output:
Wilcoxon rank sum test

data:  PichonMag$`Preis Normal` and PichonMag$`Preis Magnum / 2`
W = 5, p-value = 0.2429
alternative hypothesis: true location shift is less than 0

With this p-value I can't reject H0, which means, that prices don't differ.
I still don't get why, as they differ substantially (at least with my real life experience). Did I make a mistake here?
 A: The 2-sample Wilcoxon Rank Sum considers the ranks of the two samples.
In a two-sided test, it is (just barely) possible to get a significant result at the 5% level with two groups of four,
but only if all of the observations in one group are below any of the
observations in the other. Then the P-value of the test is $2/{8\choose 4} = 2/70 < 0.05 = 0.02857 < 5\%,$ as below.
wilcox.test(1:4, 5:8)

        Wilcoxon rank sum test

data:  1:4 and 5:8
W = 0, p-value = 0.02857
alternative hypothesis: true location shift is not equal to 0

Otherwise, with some overlap between the two groups (or with fewer than
4 observations in both groups), a P-value below 5% is not possible.
Your boxplots show overlapping data.
Only the ranks matter. With two samples of size four, the 2-sided, 2-sample Wilcoxon test will not
give a P-value below 5% if there is any overlap at all, not even if some of the values
in one group are hugely larger than values in the other group.
wilcox.test(c(1,2,3,5), c(4, 90, 200, 1000))

        Wilcoxon rank sum test

data:  c(1, 2, 3, 5) and c(4, 90, 200, 1000)
W = 1, p-value = 0.05714
alternative hypothesis: true location shift is not equal to 0

