What happens when the assumptions of parametric tests like normality and equal variances which are not satisfied?
Do we stop analyzing?
Of course not. That’s where Non-Parametric Tests step in. These tests don’t rely heavily on distributional assumptions, making them useful when working with small samples, ordinal data, or skewed distributions.
Purpose of Non-Parametric Tests
Parametric tests such as the t-test and ANOVA are powerful, but they require assumptions like normally distributed data and equal variances. When these assumptions break down, non-parametric tests rely on the ranking of data rather than the raw values, which reduces the influence of outliers and non-normality.
Mann–Whitney U Test
Concept and Use-Case
The Mann–Whitney U test is often called the Wilcoxon rank-sum test. It’s the non-parametric alternative to the independent samples t-test. Use it when you want to compare two independent groups but can’t assume normality.
Example scenario: Comparing the satisfaction scores of two groups of employees working under different supervisors, where scores are given in ranks (ordinal data).
Procedure
- Combine data from both groups and rank them (lowest = 1, highest = n).
- Calculate the sum of ranks for each group (say, R₁ and R₂).
- Compute the test statistic:
U = n₁n₂ + (n₁(n₁+1))/2 – R₁
where n₁ and n₂ are sample sizes of the two groups, and R₁ is the sum of ranks for group 1. - Compare the calculated U with critical values from the Mann–Whitney table or use normal approximation for large samples.
Worked Example
Group A scores: 3, 1, 4
Group B scores: 2, 5
Ranks: 1 (score 1), 2 (2), 3 (3), 4 (4), 5 (5)
Group A ranks = 3+1+4 = 8
Group B ranks = 2+5 = 7
U = n₁n₂ + (n₁(n₁+1))/2 – R₁ = (3×2) + (3×4/2) – 8 = 6 + 6 – 8 = 4.
Interpretation depends on comparison with critical value. If U is less than or equal to critical U, we reject H₀ (no difference).
Kruskal–Wallis H Test
Concept and Use-Case
The Kruskal–Wallis test is the non-parametric alternative to one-way ANOVA. It’s used when comparing three or more independent groups on an ordinal or non-normal scale.
Example: Testing whether customer satisfaction differs across three branches of a retail store, where satisfaction is measured on a ranking scale.
Procedure
- Rank all observations across groups.
- Calculate the sum of ranks for each group (Rᵢ).
- Use the formula:
H = (12 / (N(N+1))) Σ (Rᵢ² / nᵢ) – 3(N+1)
where N = total number of observations, nᵢ = sample size of group i, Rᵢ = sum of ranks of group i. - Compare calculated H with chi-square critical value (df = k–1, where k = number of groups).
Worked Example
Suppose we have 3 groups with satisfaction scores (ranks assigned):
- Group 1: 2, 5
- Group 2: 3, 6
- Group 3: 1, 4
After ranking all six observations, compute Rᵢ for each group, then plug into formula. The decision comes from comparing H to chi-square critical value at chosen significance level.
Rank Correlation Test (Spearman’s Rank)
Concept
Correlation measures the strength and direction of a relationship. Spearman’s Rank Correlation is a non-parametric alternative to Pearson’s correlation. It uses ranks instead of raw values, making it useful when data are ordinal or not normally distributed.
Formula
ρ = 1 – (6 Σd²) / (n(n²–1))
Where:
- d = difference between the two ranks for each observation
- n = number of observations
Worked Example
Suppose we have two rankings of 5 salespersons based on performance and customer feedback:
| Salesperson | Performance Rank (X) | Feedback Rank (Y) | d = X–Y | d² |
|---|---|---|---|---|
| A | 1 | 2 | -1 | 1 |
| B | 2 | 1 | 1 | 1 |
| C | 3 | 3 | 0 | 0 |
| D | 4 | 5 | -1 | 1 |
| E | 5 | 4 | 1 | 1 |
Σd² = 4, n = 5
ρ = 1 – (6×4) / (5(25–1)) = 1 – 24/120 = 0.8
Interpretation: A strong positive relationship exists between performance and feedback.
Limitations of Non-Parametric Tests
- Less powerful than parametric tests if assumptions of parametric tests are actually satisfied.
- Results are based on ranks, which may lose information from actual data values.
- Interpretation is often limited to median differences or ordinal relationships.
Non-parametric tests are the safety net of statistics. When normality or equal variance assumptions break, they provide reliable alternatives without halting analysis. Mann–Whitney U helps compare two groups. Kruskal–Wallis extends the idea to multiple groups. Spearman’s rank correlation captures relationships without requiring linearity or normality. Together, they expand the toolkit of a business researcher who faces imperfect, real-world data every day.
So, when your data doesn’t behave the way theory wants, will you give up? Or will you switch to tools that respect the data as it is? That’s the reflection every serious student of statistics should carry forward.