Sample Wilcoxon

The 1 Sample Wilcoxon test—also known as the Wilcoxon signed‑rank test—builds on the logic of the Sign test but incorporates additional information to increase statistical power. Like the Sign test, it compares a sample to a known or target value. However, instead of considering only the direction of differences, it also considers their magnitude, making it more sensitive to meaningful shifts in the process. 

The test begins by calculating the difference between each observation and the target value. Differences equal to zero are discarded. The absolute differences are then ranked from smallest to largest, and each rank is assigned a positive or negative sign based on whether the original observation was above or below the target. The test statistic is the sum of the signed ranks. 

This approach allows the Wilcoxon test to detect smaller shifts than the Sign test, especially when sample sizes are modest. It remains robust to outliers because ranks, rather than raw values, drive the analysis. This makes it ideal for skewed data, ordinal data, or situations where the mean is not a reliable measure of central tendency. 

The Wilcoxon test assumes that the distribution of differences is symmetric. While this assumption is less restrictive than normality, it is still important to verify visually using histograms or boxplots. When the assumption is reasonably met, the Wilcoxon test provides a powerful, flexible alternative to the 1‑sample t‑test. 

In the Analyze phase, the 1 Sample Wilcoxon test is particularly useful when evaluating whether a process has shifted relative to a target, especially in environments where data is messy or non‑normal. It provides a balanced combination of robustness and sensitivity, helping you draw meaningful conclusions even when traditional assumptions fail.