Groups can still be compared when means and normal distributions are poor summaries.
Suppose two distribution centers are compared on delivery delay. Most delays are between 5 and 15 minutes, but an accident can create a delay of two hours or more.
In such data, means and standard deviations can be dominated by a few extreme values. Nonparametric tests often compare ranks or signs instead of relying directly on the raw scale.
Compare ranks instead of raw values
When delivery delays are strongly skewed, ranking prevents one very large value from dominating the comparison.
| Observed value | 3 | 4 | 5 | 6 | 7 | 9 | 11 | 45 |
|---|---|---|---|---|---|---|---|---|
| Rank | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
Key question
Can we compare which group tends to have larger values without relying heavily on the exact distances between them?
Why replace values with ranks?
Parametric tests are built around means and variances. Rank tests order the observations and ask which group occupies the higher ranks more often.
This can make them more stable under severe skew or outliers. The tradeoff is that some information about the original distances is discarded, so a well-specified parametric test can be more powerful.
Nonparametric does not mean assumption-free
Independence, study design, and the shapes of the distributions still matter.
Which nonparametric test fits the design?
| Data structure | Parametric method | Nonparametric alternative |
|---|---|---|
| Two independent groups | Independent t-test | Mann–Whitney U |
| Same people measured twice | Paired t-test | Wilcoxon signed-rank |
| Three or more independent groups | One-way ANOVA | Kruskal–Wallis |
| Three or more repeated conditions | Repeated-measures ANOVA | Friedman |
The core choice
Use Mann–Whitney U for two independent groups and Wilcoxon signed-rank for paired observations. Kruskal–Wallis and Friedman extend the same logic to three or more conditions.
What do rank tests actually compare?
Mann–Whitney U tests whether the two rank distributions differ. When the distributions have similar shapes, the result is often interpreted as a location or median difference.
If one distribution is narrow and the other is much wider, significance may reflect a shape difference rather than a simple shift in medians.
Mann–Whitney U test
Delivery delay at centers A and B
Median A
6 min
Median B
11 min
U
214
p-value
.012
Reporting results
Report medians and interquartile ranges with the test statistic and p-value. A rank-biserial correlation or another suitable effect size can also be included.
When are they appropriate?
A normality-test p-value below 0.05 is not, by itself, a reason to abandon a t-test or ANOVA. Sample size, outliers, distribution shape, and the target estimand should be considered together.
| Situation | Method to consider |
|---|---|
| Large sample, mild skew | t-test or ANOVA may be sufficiently robust |
| Small sample, severe skew | Rank-based or robust method |
| Single ordinal item | Nonparametric test |
| Mean difference is the research target | Robust mean test or transformation |
Change an outlier and watch the ranks
Increase one observation below and compare how much the mean difference and the rank-based summary move.
Mean difference
6.13
Median difference
2.00
Avg rank A
6.75
Avg rank B
10.25
A
3 · 4 · 5 · 6 · 7 · 8 · 9 · 10
B
5 · 6 · 7 · 8 · 9 · 10 · 11 · 45
Key lesson
A nonparametric test does not delete outliers. It places more emphasis on order than on exact distance.
Nonparametric tests are not merely backups for failed normality. They are useful when ranks match the scientific question better than raw distances.