Non-parametric tests are statistical methods used when data do not meet the conditions for common parametric tests, especially normality. This cheat sheet helps students choose the right test for ranked, ordinal, categorical, or strongly skewed data. It is useful for comparing groups, checking associations, and analyzing small samples.
These tests focus on medians, ranks, signs, or counts instead of relying heavily on means and standard deviations.
The main idea is to convert data into ranks, signs, or frequency counts and then compare the observed pattern to what would be expected by chance. Common tests include the sign test, Wilcoxon signed-rank test, Mann-Whitney test, Kruskal-Wallis test, chi-square tests, and Spearman rank correlation. Many non-parametric hypotheses use medians or distributions rather than population means.
Students should match the test to the number of groups, whether samples are paired or independent, and whether the data are ordinal, quantitative, or categorical.
Key Facts
- Use a non-parametric test when data are ordinal, strongly skewed, have major outliers, or do not reasonably satisfy normality assumptions.
- For a sign test with nonzero paired differences, the number of positive signs follows under the null hypothesis.
- The Wilcoxon signed-rank test ranks the absolute paired differences and uses the sum of signed ranks, often written as or .
- For the Mann-Whitney test, one test statistic is , where is the rank sum for group .
- For the Kruskal-Wallis test, , where is the rank sum for group .
- For a chi-square goodness-of-fit or independence test, , where is observed count and is expected count.
- For a chi-square test of independence in an table, the degrees of freedom are .
- Spearman rank correlation measures monotonic association and can be computed by when there are no tied ranks.
Vocabulary
- Non-parametric test
- A statistical test that does not require a specific population distribution such as a normal distribution.
- Rank
- A rank is the position of a data value after all values are ordered from smallest to largest or largest to smallest.
- Ordinal data
- Ordinal data are values with a meaningful order, such as ratings, rankings, or survey levels, but unequal or unknown spacing.
- Mann-Whitney U test
- The Mann-Whitney test compares two independent groups by ranking all observations together and comparing rank sums.
- Wilcoxon signed-rank test
- The Wilcoxon signed-rank test compares paired measurements by ranking the sizes of the nonzero differences and keeping their signs.
- Chi-square test
- A chi-square test compares observed categorical counts with expected counts using .
Common Mistakes to Avoid
- Using a non-parametric test only because the sample is small is wrong because the data type, design, and assumptions still matter.
- Using the Mann-Whitney test for paired data is wrong because Mann-Whitney assumes independent groups, while paired data usually need the sign test or Wilcoxon signed-rank test.
- Ignoring tied ranks is wrong because ties can change rank sums and may require assigning average ranks or using a corrected procedure.
- Interpreting every non-parametric test as a test of means is wrong because many of these tests compare medians, distributions, ranks, signs, or counts.
- Using a chi-square test when expected counts are too small is wrong because the chi-square approximation may be unreliable when many expected counts are below .
Practice Questions
- 1 Two independent groups have sample sizes and . If the rank sum for group is , calculate .
- 2 In a chi-square goodness-of-fit test, one category has observed count and expected count . Find that category's contribution to .
- 3 For a Spearman rank correlation with and rank differences of , calculate .
- 4 A student wants to compare pain ratings before and after treatment for the same patients using an ordinal scale from to . Which non-parametric test is most appropriate, and why?
Understanding Non-parametric Tests
A rank test begins by putting all relevant values in order, from smallest to largest. The actual gaps between values are then mostly ignored. A score of eighty is not treated as being exactly twice a score of forty.
This is useful when the scale itself is uncertain, such as ratings from strongly disagree to strongly agree. It also reduces the influence of one unusually large value.
The trade-off is that ranking throws away some numerical detail. When data really do follow the assumptions of a mean-based test, a parametric test can often detect a smaller effect more efficiently.
The first decision is whether observations are linked. Linked observations come from the same person measured twice, or from deliberately matched pairs such as twins. In this setting, each pair gives a direction of change and sometimes a size of change.
The sign test uses only the direction, so it is simple but may miss useful evidence. The Wilcoxon signed-rank test gives more weight to larger changes after ranking their sizes. Independent samples come from separate people or objects.
Mann-Whitney compares the positions of two groups within one combined ranking. It is often described as a test of medians, but that interpretation is safest when the two groups have similarly shaped distributions.
Kruskal-Wallis extends the comparison idea to three or more independent groups. A significant result says that at least one group differs in its typical rank pattern. It does not identify which groups differ.
Follow-up pair comparisons are needed, with an adjustment for making several comparisons. Chi-square tests work differently because they use counts in categories. For example, a school survey could compare transport method with year group.
Expected counts must be large enough for the chi-square approximation to be trustworthy. Very small expected counts can make the reported probability inaccurate, so categories may need combining or an exact method may be better.
Spearman correlation is useful when two variables move consistently upward or downward without forming a straight-line pattern. For example, hours spent practising may generally rise with music performance rank, even if the improvement slows at high practice levels. A correlation does not show that one variable causes the other.
It can be distorted by tied ranks, a restricted range of values, or separate clusters in the data. In every test, inspect a graph or table before calculating anything.
State the null hypothesis clearly, check whether the sample was collected fairly, report the probability value, and describe the size and practical meaning of the pattern. A statistically significant result can still be too small to matter in real life.