The Wilcoxon Signed-Rank Test is a non-parametric statistical hypothesis test used within the Lean Six Sigma framework, especially when analyzing data from processes in need of improvement. This test plays a pivotal role in the Analyze phase of the DMAIC (Define, Measure, Analyze, Improve, Control) methodology, where it helps to identify differences between before-and-after improvements or to compare two related samples when the data does not meet the assumptions necessary for the parametric paired t-test.

Introduction to Wilcoxon Signed-Rank Test

The Wilcoxon Signed-Rank Test is designed for situations where the data is paired and comes from a non-normally distributed population. It is particularly useful in Lean Six Sigma projects because it does not assume the data to be normally distributed, making it more flexible and widely applicable across different types of data and industries.

When to Use the Wilcoxon Signed-Rank Test

This test is appropriate under several conditions:

• Data Pairing: When you have two sets of data that are related or paired, such as measurements taken before and after a process improvement on the same set of items.

• Non-Normal Distribution: When the data does not follow a normal distribution, which is a common scenario in real-world process data.

• Ordinal or Continuous Data: It is suitable for ordinal data (data that can be ranked) or continuous data.
  

Conducting the Wilcoxon Signed-Rank Test

The test involves several steps to calculate and assess the significance of the observed differences between paired samples:

1. Determine the Differences: Calculate the difference between the pairs of observations.

2. Rank the Differences: Assign ranks to these differences, ignoring the signs (positive or negative) and excluding pairs with no difference.

3. Sum the Ranks: Calculate the sum of ranks for positive differences and the sum of ranks for negative differences.

4. Test Statistic: The test statistic is the smaller of the two sums of ranks.

5. Significance: Determine the significance of the result using critical values from the Wilcoxon signed-rank table or a p-value, comparing it to a predetermined significance level (commonly 0.05).
   

Interpretation

The decision rule for the Wilcoxon Signed-Rank Test depends on the calculated test statistic and its comparison to critical values or a p-value:

• Reject the Null Hypothesis: If the test statistic is less than the critical value or if the p-value is less than the significance level, there is sufficient evidence to reject the null hypothesis, indicating a significant difference between the paired samples.

• Fail to Reject the Null Hypothesis: If the test statistic is greater than the critical value or the p-value is higher than the significance level, there is insufficient evidence to suggest a significant difference.
  

Application in Lean Six Sigma

In Lean Six Sigma projects, the Wilcoxon Signed-Rank Test is invaluable for analyzing the effectiveness of process improvements, especially when the normality of data cannot be assumed. It provides a robust method for comparing pre- and post-improvement metrics, thus helping teams to make data-driven decisions regarding the impact of their improvement efforts.

Conclusion

The Wilcoxon Signed-Rank Test is a powerful tool in the Lean Six Sigma practitioner's arsenal, allowing for the analysis of paired samples without the strict requirement of normal distribution. By understanding and applying this test appropriately, professionals can gain deeper insights into their process improvements and substantiate their decisions with statistical evidence.

Wilcoxon Signed-Rank Test: Practical Example

The Wilcoxon Signed-Rank Test is a non-parametric statistical hypothesis test used when comparing two related samples or repeated measurements on a single sample to assess whether their population mean ranks differ. It is an alternative to the paired Student's t-test when the population cannot be assumed to be normally distributed.

Scenario: Customer Satisfaction Improvement Program

A retail company launched a new customer service training program for its staff to improve customer satisfaction. To assess the effectiveness of the program, the company measured customer satisfaction scores before and after the training for a random sample of 10 transactions.

Data

The satisfaction scores (out of 10) for the 10 transactions before and after the training were as follows:

Objective

To determine if the customer service training program significantly improved customer satisfaction scores.

Steps

1. Calculate the Differences: Subtract the "Before Training" scores from the "After Training" scores.

2. Rank the Differences: Ignore zeros and rank the absolute values of the differences.

Note: Transactions with a difference of 2 are assigned the average rank (since they are tied).

3. Sum of Ranks: Calculate the sum of positive and negative ranks. In this case, all differences are positive, indicating improvements, so we only have a sum of positive ranks.

Sum of Ranks = 4.5+4.5+4.5+4.5+2+4.5+4.5+2+4.5+4.5=36 4. Test Statistic: The smaller of the sum of positive or negative ranks is used as the test statistic. Here, since all ranks are positive, our test statistic is 36.

5. Critical Value: Determine the critical value from the Wilcoxon signed-rank table for n=10 (after excluding ties) and a chosen level of significance (e.g., α=0.05). Let's assume the critical value from the table is 8 (see below).

6. Decision Rule: If the test statistic W is less than or equal to the critical value, reject the null hypothesis. Otherwise, do not reject it.

Conclusion:

Given our test statistic of 36 exceeds the critical value of 8, we do not reject the null hypothesis at the 0.05 significance level. This means there is not enough evidence to conclude that the customer service training program significantly improved the customer satisfaction scores, based on the Wilcoxon signed-rank test. However, the positive differences across nearly all transactions suggest a trend toward improvement, even if not statistically significant by this test's criteria.