The Anderson-Darling test is a powerful and versatile statistical test that plays a crucial role in the Lean Six Sigma methodology, particularly within the Analyze phase of the DMAIC (Define, Measure, Analyze, Improve, Control) framework. This test is used to assess whether a given sample of data follows a specific probability distribution, such as the normal, exponential, or Weibull distribution. It is especially useful in the context of hypothesis testing, where understanding and validating the distribution of data is critical for making informed decisions.

Introduction to Anderson-Darling Test

The Anderson-Darling test is named after Theodore W. Anderson and Donald A. Darling, who proposed this test in 1952. The test is a modification of the Cramér-von Mises criterion and is designed to give more weight to the tails of the distribution. This sensitivity to the tails makes it particularly effective for detecting departures from the specified distribution.

Why It Matters in Lean Six Sigma

Lean Six Sigma projects often involve analyzing process data to identify variations and understand their causes. The assumption of normality underpins many statistical methods used in these projects, such as control charts, capability analysis, and hypothesis testing. Before applying these techniques, it's crucial to confirm that the data follows a normal distribution. The Anderson-Darling test helps verify this assumption, ensuring the appropriate application of statistical tools and the validity of the project's findings.

How the Anderson-Darling Test Works

The Anderson-Darling test calculates a test statistic based on the difference between the observed cumulative distribution function (CDF) of the sample data and the expected CDF of the specified theoretical distribution. The test statistic is then compared to critical values from the Anderson-Darling distribution to determine whether to reject the null hypothesis. The null hypothesis (H0) posits that the data follow the specified distribution, while the alternative hypothesis (H1) suggests otherwise.

The graph above visually represents the concept behind the Anderson-Darling Test by comparing the empirical cumulative distribution function (CDF) of a sample (in blue) against the theoretical CDF of a normal distribution (red dashed line). The focus of the Anderson-Darling Test, particularly on the tails of the distribution, is highlighted in yellow.

How the Anderson-Darling Test Works in details (Not required to know for the Black belt)

The Anderson-Darling test is based on the calculation of a test statistic, A2, which quantifies the discrepancy between the empirical cumulative distribution function (ECDF) of the sample data and the cumulative distribution function (CDF) of a specified theoretical distribution. The ECDF represents the proportion of sample observations that are less than or equal to a certain value, while the CDF represents the theoretical probability of observing a value less than or equal to that value under the specified distribution.

The test statistic A2 is defined as:

where:

• n is the sample size,

• yi​ are the ordered sample observations,

• F is the CDF of the specified theoretical distribution.
  

This formula essentially weights the squares of the distances between the ECDF and the CDF, with more emphasis placed on the tails of the distribution. The weighting factor (2i−1) increases the sensitivity of the test to deviations in the tail ends of the distribution, which are often of significant interest in quality control and process improvement projects.

Decision-Making Process

After calculating the A2 statistic, the next step is to compare it to critical values from the Anderson-Darling distribution. These critical values depend on the significance level (α) chosen for the test, which is typically 0.05 (5%) for many applications. The significance level represents the probability of rejecting the null hypothesis when it is actually true (Type I error).

• If A2 is less than the critical value, we do not have sufficient evidence to reject the null hypothesis (H0​), and we conclude that the data do not significantly deviate from the specified distribution.

• If A2 is greater than the critical value, we reject the null hypothesis (H0​) in favor of the alternative hypothesis (H1​), indicating that the data significantly deviate from the specified distribution.

  
  

Null and Alternative Hypotheses:

• Null Hypothesis (H0​): The data follow the specified distribution.

• Alternative Hypothesis (H1​): The data do not follow the specified distribution.

Rejecting the null hypothesis suggests that the data distribution is significantly different from the theoretical distribution, prompting further investigation or consideration of alternative distributions.

Advantages of the Anderson-Darling Test

1. Sensitivity to Tail Differences: Its emphasis on the tail regions makes it more sensitive to outliers and extreme values, which are crucial in assessing the quality of processes.

2. Versatility: It can be applied to various distributions, making it a flexible tool for different types of data analysis in Lean Six Sigma projects.

3. Objective Decision-Making: By providing a clear statistical basis for accepting or rejecting the normality assumption, it aids in more objective decision-making.
   

Implementation in Lean Six Sigma

In the context of Lean Six Sigma, the Anderson-Darling test is typically applied during the Analyze phase. For instance, when a team is investigating the root causes of defects in a manufacturing process, they may collect data on the dimensions of the manufactured parts. Before using the data to conduct further analysis, the team would use the Anderson-Darling test to check if the data on dimensions follow a normal distribution. This step is crucial for applying the correct statistical tools thereafter.

Conclusion

The Anderson-Darling test is an essential tool in the Lean Six Sigma toolkit for hypothesis testing. Its ability to accurately assess the fit of a sample to a specified distribution supports the rigorous analysis of process data. By ensuring that the assumptions underlying statistical tests are met, the Anderson-Darling test helps Lean Six Sigma practitioners make informed decisions, ultimately leading to more effective process improvements and enhanced quality outcomes. Whether you are a Green Belt, Black Belt, or Master Black Belt, understanding and applying the Anderson-Darling test is key to the success of your Lean Six Sigma projects.