I have 4 sets of data points and for each I want to see how much these values vary from zero.

I used the tool for calculating the results of the test on this page but I'm having trouble understanding the results.

The results for my 4 sets of data points are:

W+ = 3,       W- = 12,      N = 5,   p <= 0.3125     for 101 data points
W+ = 12,      W- = 3,       N = 5,   p <= 0.3125     for 101 data points
W+ = 451,     W- = 1089,    N = 55,  p <= 0.007618   for 297 data points
W+ = 950.50,  W- = 427.50,  N = 52,  p <= 0.01746    for 297 data points

Please could you tell me which values I should be looking at and what the mean in terms of the above results?

Do the low p values mean that the variance from zero is not significant?


  • $\begingroup$ The Wilcoxon signed rank test you linked to is for paired data. You don't seem to have paired data. Are you trying to see if the mean is different from 0? Or the median? Or something else? $\endgroup$
    – Peter Flom
    Sep 8, 2011 at 11:08
  • $\begingroup$ @Peter, my data is formed of pairs, so by one data point I mean one pair of values. For a pair (x, y) I have the actual value for X and y is always zero (the expected value). I want to know how significantly the x values vary from zero. $\endgroup$
    – Griffin
    Sep 8, 2011 at 11:20
  • 2
    $\begingroup$ @PeterFlom In fact the signed rank test tests whether a univariate (one-sample) median is different from zero, assuming symmetry under the null. When applied to differences among pairs, it's a paired test that satisfies the symmetry-in-differences by assuming the pairs have the same distributional shape. This is the same issue as we have with the sign test - it's actually a univariate test for a median of zero, but when applied to differences between pairs becomes a paired test. I'm guessing you already knew all that - but it might mislead people with less statistical background. $\endgroup$
    – Glen_b
    Mar 5, 2013 at 9:48

1 Answer 1


As you can read here, for example, the null hypothesis (or one version of it) for the signed-rank test is that the median difference is zero.

The p-value always means the probability that the observed test statistic (which is calculated based on your specific observations) would be as extreme as observed (or more extreme), if the null hypothesis were true.

So, a small p-value means that the observed result is highly unlikely if the null hypothesis were true. Consequently, we conclude (since we do observe this result) that the null hypothesis is unlikely to be true, and reject it.

Conclusion (disregarding other issues with your data and multiple testing correction): for the tests where the p-value is below a certain threshold (typically 0.05), we conclude that the median difference is significantly different from 0. For the others, we have no proof that it is significantly different (but it might still be in the population).

  • $\begingroup$ So, for my first two sets of results, where p=0.3125, the median difference is not significantly different from zero? $\endgroup$
    – Griffin
    Sep 8, 2011 at 12:20
  • $\begingroup$ As per my last sentence: you do not have proof that it is, nor that it doesn't. The phrase typically used for this is "we cannot reject (at ... significance level)" that the median difference is zero. $\endgroup$
    – Nick Sabbe
    Sep 8, 2011 at 13:10

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