The is a powerful tool for comparing two when data isn't normally distributed or is ordinal. It's like a nonparametric version of the t-test, helping us spot differences between groups without assuming normal distributions.
This test combines and ranks data from both samples, calculates , and uses the to determine if there's a . For larger samples, we can use z-scores, while smaller samples rely on .
Nonparametric Tests for Two Independent Samples
Appropriateness of Mann-Whitney U Test
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Compares two independent samples not related or paired in any way ()
Serves as a to the independent samples t-test when data is ordinal or assumptions of normality are violated (, presence of outliers)
Tests the that the two populations have the same distribution against the that they have different distributions (location, shape, or variability)
Execution of Mann-Whitney U Test
Combines observations from both samples into a single ordered array and assigns ranks from lowest to highest (1, 2, 3...)
Averages the ranks for (2.5 for two observations tied for 2nd and 3rd rank)
Calculates the for each sample (R1 for sample 1 and R2 for sample 2)
Determines the of the two groups (n1 for sample 1 and n2 for sample 2)
Calculates the U statistic for each sample using the formulas:
U1=n1n2+2n1(n1+1)−R1
U2=n1n2+2n2(n2+1)−R2
Selects the smaller U value (Umin) for further analysis
Calculation of U statistic
Follows a known distribution for sample sizes greater than 20, allowing for
Relies on tables to find critical values for smaller sample sizes (n < 20)
Calculates the : μU=2n1n2
Calculates the : σU=12n1n2(n1+n2+1)
For large sample sizes, calculates the : z=σUUmin−μU
Compares the calculated z-score or Umin to the critical value at the desired (0.05, 0.01)
Interpretation of Mann-Whitney results
Rejects the null hypothesis if the calculated z-score or Umin is less than the critical value, concluding a significant difference between the two populations ()
Fails to reject the null hypothesis if the calculated z-score or Umin is greater than the critical value, concluding to suggest a significant difference (no treatment effect)
Reports results using appropriate language and statistical terminology, including the U statistic, sample sizes, significance level, and conclusion drawn from the test
Discusses the implications of the findings in the context of the research question or problem being addressed (, limitations, future research)