The Mann-Whitney U Test (also known as the Wilcoxon Rank-Sum Test) is a non-parametric statistical test used to compare two independent groups to determine whether there is a significant difference in their distributions. It is often used as an alternative to the independent samples t-test when the assumption of normality is violated.
1. Introduction
- It is used to compare two independent samples.
- It tests whether one population tends to have higher or lower values than the other.
- It does not assume normal distribution, making it useful for skewed or ordinal data.
- It ranks all observations and then analyzes differences based on ranks rather than raw scores.
2. Assumptions
- The dependent variable is either ordinal or continuous.
- The two groups are independent (no repeated measures).
- The observations should be randomly sampled from the populations.
- The shape of the distributions should be similar if comparing medians.
3.Hypotheses
- Null Hypothesis (H₀): The two groups have the same distribution (no difference).
- Alternative Hypothesis (H₁): The distributions of the two groups are different.
Depending on the research question, the alternative hypothesis may be:
- Two-tailed: The distributions are different in some way.
- One-tailed: One distribution tends to have larger values than the other.
4.Test statistics
For small sample size (n1 and n2 <=20)
\(U_1 = n_1n_2 + \frac{n_1(n_1+1)}{2}-R_1 \) ………….. i
\(U_2 = n_1n_2 + \frac{n_2(n_2+1)}{2}-R_2 \) ……………… ii
Where
- n1 = Number of observation in first sample.
- n2 = Number of observation in second sample.
- R1 = Sum of ranks of first sample.
- R2 = Sum of ranks of second sample.
Decision
If \(U_{min} > U_{tab}, H_0\) is accepted
For large sample size (n1 and n2 >20)
\(Z =\frac{U_{min}-\mu_u}{\sigma_u}\)
Where
\(\mu_u = \frac{n_1n_2}{2}\)
\(\sigma_u = \sqrt{\frac{n_1n_2(n_1+n_2+1)}{12}}\)