In the realm of Lean Six Sigma, a methodology focused on improving process efficiency and quality, hypothesis testing plays a crucial role in making data-driven decisions. When the data under investigation does not follow a normal distribution—a common scenario in real-world applications—the selection of appropriate hypothesis tests becomes pivotal. This article delves into the framework for selecting suitable tests for hypothesis testing with non-normal data, ensuring practitioners can confidently infer about their processes.

Understanding Non-Normal Data

Before diving into the selection of tests, it's essential to recognize what constitutes non-normal data. Data that do not follow the bell-shaped curve of a normal distribution may exhibit skewness, kurtosis, or follow entirely different distributions such as exponential, uniform, or binomial. Non-normality can arise due to natural process variations, measurement errors, or the influence of external factors. Identifying the nature of your data's distribution is the first step in choosing the correct hypothesis test.

Hypothesis Testing Framework

The hypothesis testing framework provides a structured approach to making decisions about the process or system being studied. It involves the following steps:

1. Define the Null and Alternative Hypotheses: The null hypothesis (H0) usually states that there is no effect or difference, whereas the alternative hypothesis (H1) suggests a significant effect or difference.

2. Select the Significance Level (α): This is the probability of rejecting the null hypothesis when it is actually true, typically set at 0.05 or 5%.

3. Choose the Appropriate Test: The selection depends on the data's distribution, sample size, and whether you are comparing means, variances, or proportions.

4. Calculate the Test Statistic: This involves using your data to compute a value that will be compared against a critical value or used to calculate a p-value.

5. Make a Decision: Based on the p-value or comparison with the critical value, decide whether to reject or fail to reject the null hypothesis.

   
   

Selection of Appropriate Tests for Non-Normal Data

When dealing with non-normal data, the choice of hypothesis test is guided by the type of data and the question at hand. Here are some commonly used tests for non-normal datasets:

• Mann-Whitney U Test: Useful for comparing two independent samples when you are interested in differences in their medians. It's a non-parametric alternative to the t-test.
  

• Wilcoxon Signed-Rank Test: Serves for comparing two paired or matched samples, similar to the paired t-test, but for non-normal data.

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• Kruskal-Wallis H Test: An extension of the Mann-Whitney U Test for comparing more than two independent samples. It's the non-parametric counterpart to the one-way ANOVA.

  
  

• Friedman Test: Used for comparing more than two paired or matched samples, acting as a non-parametric alternative to the repeated measures ANOVA.

  
  

• Chi-Square Test of Independence: Applies when you want to examine the relationship between two categorical variables.

  
  

• Fisher's Exact Test: Ideal for small sample sizes, this test examines the association between two categorical variables, similar to the Chi-Square test but more accurate for small datasets.

Conclusion

Selecting the appropriate hypothesis test for non-normal data is a crucial aspect of Lean Six Sigma projects, ensuring that decisions are based on solid statistical ground. By understanding the distribution of your data and aligning it with the objective of your analysis, you can choose the most fitting test. This structured approach not only enhances the reliability of your findings but also bolsters the credibility of the improvements you propose based on these insights. As with all statistical methods, the key lies in a thorough understanding of the underlying assumptions and limitations of each test, ensuring they align with the specifics of your data and research questions.

Which test statistic should you use ?

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Please understand that the chart below is essential knowledge for any Black Belt.

It is imperative to select the appropriate test for your data.

Attempting to pass your exam without this understanding is a guaranteed path to failure.

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Source:https://onishlab.colostate.edu/summer-statistics-workshop-2019/which_test_flowchart/

Please note that McNemar's test is not encompassed within the Black Belt Body of Knowledge.Even more complete chart here:

(Not required Knowledge for the Black Belt Body of Knowledge.)https://statsandr.com/blog/what-statistical-test-should-i-do/