Wilcoxon Signed Rank Test Last Updated : 03 Apr, 2025 Comments Improve Suggest changes Like Article Like Report The Wilcoxon Signed Rank Test is a non-parametric statistical test used to compare two related groups. It is often applied when the assumptions for the paired t-test (such as normality) are not met. This test evaluates whether there is a significant difference between two paired observations, making it especially useful for non-normally distributed or ordinal data. It is widely used in fields like medicine, psychology, and social sciences, where normality assumptions are often violated.Key Features of the Wilcoxon Signed Rank TestNon-parametric Nature: The test does not assume that the data follows any specific distribution, making it ideal for non-normal data or ordinal variables.W-Statistic: It calculates the W-statistic, which approximates a normal distribution for larger samples (n > 10).When to Use the Wilcoxon Signed Rank Test?The test is appropriate when:The data consists of paired (dependent) observations.The differences between pairs are continuous or ordinal.The distribution of differences is not normal but symmetric.ExampleTo evaluate the effectiveness of a new teaching method, researchers measure literacy levels of 20 children before and after applying the method. Since the data is non-normal, the Wilcoxon Signed Rank Test is ideal.HypothesesNull Hypothesis (H₀): The median difference between paired samples is zero (no significant difference).Alternative Hypothesis (H₁): The median difference is not zero (there is a significant difference).Steps to Perform the Wilcoxon Signed Rank TestFormulate Hypotheses:H0: There is no significant difference between the pairs.H1: There is a significant difference between the pairs.Find the Difference (D) between paired observations (i.e., D = B − A).Calculate the Absolute Difference (|D|) for each pair.Assign Ranks to the absolute differences from lowest to highest. For tied ranks (duplicate absolute differences), assign them the average rank.Example for ties:If two absolute differences are both 3, assign them a rank of \frac{3 + 4}{2} = 3.5.Calculate the Sum of Ranks:T+: Sum of ranks for positive differences.T-: Sum of ranks for negative differences.Find the Wilcoxon Rank Statistic (Wcalc): Wcalc = min (T+ ,T−)Compare Wcalc to Wtable:Use the sample size n and the significance level α (commonly 0.05) to find the critical value Wtable from a table of critical values for the Wilcoxon Signed Rank Test.Interpret the Result:If Wcalc < Wtable , reject H0.If Wcalc > Wtable , accept H0.Interpretation of ResultsRejecting H0: If Wcalc is less than the critical value Wtable, it indicates that the two groups are not identically distributed, and there is a significant difference.Accepting H0: If Wcalc is greater than Wtable, it means the two groups are identically distributed, and there is no significant difference.Example: Smog Concentration ComparisonLet's consider an example where we compare smog concentrations in India from May to December. The data is presented for 13 states with measurements in both months.StatesSmog in May(A)Smog in December(B)Difference[D]Absolute Difference[Abs-D]Rank Delhi13.311.1-2.22.25Mumbai10.016.26.26.29Chennai16.515.3-1.21.23Kerala 7.919.912.012.011Karnataka9.510.51.01.02Tamil Nadu8.315.57.27.210Orissa12.612.70.10.11UP8.914.25.35.37MP13.615.62.02.04Rajasthan 8.120.412.312.312Gujarat18.312.7-5.65.68West Bengal8.111.23.13.16Jammu 13.436.823.423.413Step 5: Calculate T+ and T−Sum of positive ranks (T+): T+ = 9 + 11 + 2 + 10 + 1 + 7 + 4 + 12 + 8 + 6 + 13 = 82Sum of negative ranks (T-): T- = 5 + 3 = 8Step 6: Calculate WcalcWcalc = min(T+ ,T−) = min(82 ,8) = 8Step 7: Find WtableUsing the sample size n = 13 and α = 0.05 for a two-tailed test, we look up the critical value Wtable in a Wilcoxon signed rank table. For n=13, the critical value is 17.Step 8: Interpret the ResultSince Wcalc = 8 is less than Wtable = 17, we reject the null hypothesis H0. This means that there is a significant difference in the smog concentrations between May and December.ConclusionThe Wilcoxon Signed Rank Test shows that the smog concentration has significantly changed from May to December.Difference Between the Wilcoxon Signed Rank Test and Wilcoxon Rank-Sum TestAspectWilcoxon Signed Rank TestWilcoxon Rank-Sum Test (Mann-Whitney U test)Data TypePaired (related) dataIndependent (unrelated) dataPurposeCompares two related groups or measurementsCompares two independent groups to determine if they come from the same distributionAssumptionData is paired and dependent (e.g., before and after measurement)Data is independent (e.g., comparing two different groups)Test StatisticBased on the difference between pairs of data pointsBased on the ranks of data points from two groups combinedNull HypothesisThe median difference between paired samples is zeroThe two groups come from the same distributionExample Comparing the effect of a drug on patients before and after treatmentComparing the test scores of two independent groups of studentsTest TypeNon-parametric test for paired data Non-parametric test for independent data Comment More infoAdvertise with us Next Article Kruskal Wallis Test P prakharr0y Follow Improve Article Tags : Machine Learning AI-ML-DS Practice Tags : Machine Learning Similar Reads Maths for Machine Learning Mathematics is the foundation of machine learning. 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