## What is the Wilcoxon Signed Rank Test?

The Wilcoxon signed rank test is a nonparametric hypothesis test that can do the following:

- Evaluate the median difference between two paired samples.
- Compare a 1-sample median to a reference value.

In other words, it is the nonparametric alternative for both the 1-sample t-test and paired t-test.

To perform the 1-sample test, analyze the raw data values. For the paired version, calculate the differences between the paired values and analyze them.

Most frequently, analysts use the Wilcoxon signed rank test to evaluate paired samples, such as before and after treatment scores. For example, a medical study might assess medication effectiveness by comparing the pre-test and post-test median symptom scores.

Like all hypothesis tests, this one uses samples to draw conclusions about populations. Learn more about Populations vs. Samples.

If you need a nonparametric test for two independent groups, learn about the Mann Whitney U Test. For three or more groups, consider the Kruskal Wallis Nonparametric Test.

Learn more about Parametric vs. Nonparametric Tests and Hypothesis Testing Overview.

## Wilcoxon Signed Rank Test Assumptions

Statisticians often use the Wilcoxon signed rank test when their data do not follow the normal distribution. However, it has other advantages over t-tests, including the ability to analyze ordinal data and reduce the impact of outliers.

While the data don’t need to be normally distributed, they must follow a symmetrical distribution. When using the paired form, the distribution of the differences between the paired values must be symmetrical.

If the distribution is *asymmetric*, consider using the sign test. This nonparametric test is like the Wilcoxon signed rank test but can handle asymmetric distributions. However, the sign test is less powerful.

## Null and Alternative Hypotheses

Now, let’s delve into the hypotheses of the Wilcoxon signed rank test. There are two sets of hypotheses. Choosing the correct set depends on whether you perform the paired or one-sample test.

Depending on the form, you’ll either determine whether the median difference between paired observations differs from zero or determine whether the median differs from the benchmark value (one-sample).

### Paired Test

The following are the hypotheses for the paired Wilcoxon signed rank test:

**Null hypothesis:**The median of the paired differences equals zero in the population.**Alternative hypothesis:**The median of the paired differences does not equal zero in the population.

In the paired Wilcoxon signed rank test, a median difference of zero indicates no effect or difference between the paired observations. For example, when the pre-test and post-test medians are not significantly different, the treatment did not affect the outcomes.

However, if your p-value is less than or equal to your significance level, the results are statistically significant, and you reject the null hypothesis. You can conclude that the median difference is not zero. In other words, an effect exists in the population.

To better understand the paired test, read about the Paired T-Test, which evaluates the mean rather than the median.

### One-Sample Test

The following are the hypotheses for the one-sample Wilcoxon signed rank test:

**Null hypothesis:**The population median equals the benchmark value.**Alternative hypothesis:**The population median does not equal the benchmark value.

The one-sample form compares the sample median to a hypothetical population median. The hypothetical value can be a target or benchmark. When the results are statistically significant, reject the null hypothesis and conclude that the population median does not equal your benchmark value.

Consider this robust, nonparametric alternative when your data are misbehaving. Whether dealing with nonnormal distributions, ordinal data, or outliers, it’s a handy tool in your data analysis toolbox.

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