# 1-Sample Wilcoxon

## Summary

Analyzes the observed difference between a process median and a known value. This test is similar to the 1-sample t-test, and is used as an alternative test for cases in which the data are not reasonably normal.

• Is the median of the process significantly different from a known standard (for example, a previous known value of the process mean or a nominal specification)?
When to Use Purpose
Pre-project Verify that the process is producing output significantly different from expectations, which validates the need for an improvement project.
Mid-project Test whether the output changes significantly when an input is controlled at a new setting or a previously uncontrolled setting is now controlled.
Mid-project Verify changes from the pre-project standard, throughout the course of making improvements.
End of project Verify that the median output from the controlled improved process is different from the pre-project median. This step assumes one of the goals of the project was to shift the process median.

### Data

Continuous Y (output).

## How-To

1. Verify the measurement system for the Y data is adequate.
2. Develop a data collection strategy (who should collect the data, as well as where and when; how many data values are needed; the preciseness of the data; how to record the data, and so on).
3. Collect process data, and enter the values into a single column in a Minitab worksheet.
4. Enter the standard or benchmark value to compare the process data against.
5. Determine your hypothesis. You are trying to prove the alternative hypothesis (Ha) with the data. In a 1-sample Wilcoxon test, the alternative hypothesis tests whether the process median is greater than less than, or not equal to a benchmark value. The null hypothesis (Ho) is the opposite of the alternative hypothesis.

## Guidelines

• Develop a sound data-collection strategy to ensure your conclusions are based on truly representative data.
• The 1-sample Wilcoxon test is from a category called nonparametric statistics. The test is meant to be an alternative to the parametric test (1-sample t-test) for cases where the normality assumption fails badly. This failure does not occur frequently because of the robustness of the parametric tests, so parametric tests are recommended for the majority of cases.
• The data must be continuous, and the distribution must be reasonably symmetric. The Wilcoxon test is less powerful than the t-test, as long as the data are reasonably normal. In fact, for skewed data, the Wilcoxon test is not a good alternative to the t-test, which is very robust for skewed data. A third alternative is the 1-sample sign test, which does not make any assumptions about the distribution of the data. However, the Wilcoxon test is more powerful than the sign test if the data are reasonably symmetric.
• For these reasons, you should use the:
• 1-sample t-test when the data are reasonably normal (or the sample size is large)
• 1-sample Wilcoxon test as an alternative to the 1-sample t-test as long as the data are reasonably symmetric
• 1-sample sign test as a last resort
• It is always good practice to graph your data whenever you use a statistical test. For the 1-sample Wilcoxon test, use a histogram or normal probability plot to evaluate for reasonable normality or reasonable symmetry. You can also use the histogram to identify outliers.
• If you have discrete numeric data from which you can obtain every equally spaced value, and you have measured at least 10 possible values, your data often are evaluated as though they are continuous.
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