Intro to Business Statistics: Nonparametric Tests When to Use Nonparametric Tests Ordinal Data NonNormal Distributions Small Samples

What This Is

Nonparametric tests are used when data does not meet the assumptions of parametric tests, such as normal distribution or equal variances. For example, a company wants to compare the satisfaction ratings of its customers with those of its competitors, but the ratings are on a scale of 1-5, which is ordinal data. In this case, a nonparametric test, such as the Wilcoxon rank-sum test, can be used to compare the median satisfaction ratings.

Key Formulas & Symbols

• Wilcoxon Rank-Sum Test: W = ∑Ri where Ri = rank of the ith observation in the combined sample.
• Mann-Whitney U Test: U = n1n2 + (n1n2 + 1)/2 - W where n1 and n2 are the sample sizes.
• Kruskal-Wallis H Test: H = (12/n(k-1)) ∑(Ri^2/ni - (n+1)^2/12) where n is the total sample size, k is the number of groups, and Ri is the sum of ranks for the ith group.
• Sign Test: S = ∑sgn(xi - μ) where xi is the ith observation, μ is the median, and sgn is the sign function.
• Friedman Test: F = (12/nk) ∑(Ri^2/ni - (n+1)^2/12) where n is the total sample size, k is the number of groups, and Ri is the sum of ranks for the ith group.
• Kendall's Tau: τ = (2P - n(n+1)/2) / (n(n-1)/2) where P is the number of concordant pairs and n is the total sample size.
• Spearman's Rank Correlation Coefficient: ρ = 1 - (6 ∑d^2) / (n(n^2 - 1)) where d is the difference in ranks and n is the total sample size.

Step-by-Step Procedure

1. State hypotheses: State the null and alternative hypotheses in terms of the population parameter.
2. Choose test: Choose the appropriate nonparametric test based on the data and research question.
3. Compute test statistic: Compute the test statistic using the data and the chosen test.
4. Find p-value or critical value: Find the p-value or critical value using the test statistic and the chosen test.
5. Compare to α: Compare the p-value or critical value to the significance level (α = 0.05).
6. Conclude: Conclude whether to reject the null hypothesis or fail to reject it.

Common Mistakes

• Mistake: Misinterpreting the p-value as the probability that the null hypothesis is true.
• Correction: The p-value is the probability of observing the data (or more extreme) if the null hypothesis is true.
• Mistake: Using a parametric test when the data is ordinal or non-normal.
• Correction: Use a nonparametric test when the data does not meet the assumptions of a parametric test.
• Mistake: Ignoring the assumptions of the nonparametric test.
• Correction: Check the assumptions of the nonparametric test before conducting the test.

Quick Practice Problems

1. A company wants to compare the satisfaction ratings of its customers with those of its competitors. The ratings are on a scale of 1-5, which is ordinal data. The sample mean of the company's customers is 3.5 and the sample mean of the competitors is 4.2. What is the p-value of the Wilcoxon rank-sum test? Answer: 0.01, The p-value is calculated using the Wilcoxon rank-sum test statistic and the sample sizes.
2. A company wants to compare the quality of its products with those of its competitors. The quality ratings are on a scale of 1-5, which is ordinal data. The sample mean of the company's products is 3.5 and the sample mean of the competitors is 4.2. What is the confidence interval of the difference in means? Answer: (0.5, 1.5), The confidence interval is calculated using the sample means and the sample sizes.
3. A company wants to compare the sales of its products with those of its competitors. The sales data is not normally distributed and the sample size is small. What is the appropriate nonparametric test to use? Answer: Wilcoxon rank-sum test, The Wilcoxon rank-sum test is used when the data is ordinal or non-normal and the sample size is small.

Last-Minute Cram Sheet

• ⚠️ p-value is NOT the probability that H₀ is true – it’s the probability of observing the data (or more extreme) if H₀ is true.
• Wilcoxon rank-sum test is used for comparing two independent samples.
• Mann-Whitney U test is used for comparing two independent samples.
• Kruskal-Wallis H test is used for comparing three or more independent samples.
• Sign test is used for comparing paired samples.
• Friedman test is used for comparing three or more related samples.
• Kendall's Tau is used for measuring the strength of association between two ordinal variables.
• Spearman's rank correlation coefficient is used for measuring the strength of association between two ordinal variables.
• Nonparametric tests are used when the data is ordinal or non-normal.
• Nonparametric tests are used when the sample size is small.
• Wilcoxon rank-sum test assumes that the data is continuous and the sample sizes are equal.
• Mann-Whitney U test assumes that the data is continuous and the sample sizes are equal.
• Kruskal-Wallis H test assumes that the data is continuous and the sample sizes are equal.
• Sign test assumes that the data is continuous and the sample sizes are equal.
• Friedman test assumes that the data is continuous and the sample sizes are equal.
• Kendall's Tau assumes that the data is ordinal and the sample sizes are equal.
• Spearman's rank correlation coefficient assumes that the data is ordinal and the sample sizes are equal.