The wilcoxon signed rank test determine whether the median of the sample is equal to some specified value. The Wilcoxon signed rank test utilizes both the sign of the difference as well as the magnitude of the differences, hence it provides more information than the sign test, it is often more powerful non-parametric test.

## Assumptions Wilcoxon Signed ranked test

- The population from which the sample is drawn is symmetric.
- The observations are independent.
- The scale of measurement is at least interval.
- Data are paired and come from the same population.
- Each pair is chosen randomly.

### Hypothesis of Wilcoxon Signed ranked test

We want to test the following hypothesis :

### Test Procedure

Firstly we find the difference,

### Critical Region and Comment

### Wilcoxon Signed Rank Test for large sample (n >25)

Comment: If the calculated vale of z falls in the critical region then we may reject our null hypothesis /H0 ; Otherwise we may accept it.

**How to perform the test?**

Let, we have two scores: math score and physics score.

Step-1: Subtract physics score from math score to get the differences:

Step-2: place the differences in order and then rank them. Ignore the sign when placing in rank order.

Step-3: Note the sign of the difference.

Step-4: Calculate the sum of the ranks of the negative differences (W^{–} ).

Step-5: Calculate the sum of the ranks of the positive differences (W^{+} ).

And Finally, follow the above test procedure to find the result.

### Limitations

According to wikipedia, there are some limitation of wilcoxon signed rank test. They are: when the difference between the groups is zero, the observations are discarded. This is of particular concern if the samples are taken from a discrete distribution. In these scenarios the modification to the Wilcoxon test by Pratt 1959, provides an alternative which incorporates the zero differences. ^{}^{}This modification is more robust for data on an ordinal scale.

See other non-parametric Test: