In the realm of Lean Six Sigma, where process improvement and reducing variability are paramount, the choice of statistical tools is critical. When facing the challenge of non-normal data, the Mann-Whitney U Test emerges as a beacon for hypothesis testing. This article focuses on unraveling the intricacies of the Mann-Whitney U Test, illustrating its application, strengths, and considerations in Lean Six Sigma projects.

Introduction to the Mann-Whitney U Test

The Mann-Whitney U Test, also known as the Wilcoxon rank-sum test, is a non-parametric statistical test used to assess whether there is a significant difference between two independent groups. It is particularly useful when the assumptions of normality required by parametric tests are not met. By comparing the ranks of two groups rather than their means, the Mann-Whitney U Test offers a robust alternative for analyzing ordinal or continuous data that deviates from a normal distribution.

Application in Lean Six Sigma

Lean Six Sigma practitioners often encounter non-normal data across various processes, making the Mann-Whitney U Test an essential tool in their analytical arsenal. Whether evaluating the efficiency of two different production lines, comparing customer satisfaction before and after process changes, or assessing the impact of a new tooling method on product quality, the Mann-Whitney U Test provides a reliable method for decision-making based on empirical evidence.

Conducting the Mann-Whitney U Test

1. Ranking the Data: Combine the observations from both groups and assign ranks from the smallest to the largest. In cases of tied values, assign the average rank to the tied observations.

2. Calculating the U Statistic: Sum the ranks for each group. The U statistic for each group is calculated using specific formulas that essentially measure the difference in rank sums between the two groups, adjusted for sample size.

3. Determining Significance: The significance of the observed U statistic is assessed against critical values from U distribution tables or through p-values obtained from statistical software. A result is considered statistically significant if it shows a less than 5% probability (p < 0.05) of occurring by chance.

Advantages of the Mann-Whitney U Test

• Versatility: It can be applied to a wide range of data types, including ordinal data and continuous data that are not normally distributed.

• Robustness: The test is less sensitive to outliers and skewed distributions, making it more reliable in real-world conditions.

• Efficiency: It is effective with small sample sizes, a common scenario in process improvement projects.

  
  

Considerations and Limitations

While the Mann-Whitney U Test is invaluable in non-parametric hypothesis testing, it is not without its limitations. The primary consideration is its relative power; while it is highly effective for non-normal distributions, it may be less powerful than parametric alternatives when the data actually follow a normal distribution. Additionally, the reliance on ranks rather than actual values means some information about the magnitude of differences is lost.

Conclusion

The Mann-Whitney U Test stands out in the landscape of statistical tools available for Lean Six Sigma practitioners, especially when dealing with the common challenge of non-normal data. Its ability to provide reliable, evidence-based insights into the differences between two independent groups, without the stringent assumptions required by parametric tests, makes it a critical component of the Lean Six Sigma toolkit. Understanding its application, strengths, and limitations allows practitioners to harness its full potential, leading to more informed decision-making and ultimately, more effective process improvements.

Mann-Whitney U Test Scenario

Imagine a company has introduced a new customer service protocol. They wish to compare the customer satisfaction scores before and after implementing this new protocol to determine if the change has significantly affected customer satisfaction. The satisfaction scores range from 1 to 10, where 10 indicates the highest level of satisfaction.

For simplicity, we have a small set of data representing satisfaction scores:

• Before Protocol (Group A): 4, 5, 6, 7, 6

• After Protocol (Group B): 8, 9, 7, 8, 9

Objective

Determine if there's a significant difference in customer satisfaction scores before and after the new protocol, using the Mann-Whitney U Test.

Step-by-Step Calculation

1. Combine and Rank the Data: First, we combine both groups and rank the scores from lowest to highest. In case of ties (repeated scores), we assign the average rank to the tied scores.

   
   

2. Calculate U for Each Group: We then calculate the U statistic for each group. The U statistic is the sum of ranks for each group minus a correction factor.

   
   

3. Determine Significance: Lastly, we determine if the calculated U value is significant by comparing it against critical values from a Mann-Whitney U table or calculating a p-value.

Step 1: Combine and Rank the Data

Combining both groups, we get:

• Combined Scores: 4, 5, 6, 7, 6, 8, 9, 7, 8, 9

  
  

Ranked:

• Ranks: 1, 2, 3.5, 3.5, 5.5, 5.5, 8, 9.5, 9.5 (Note: 6 and 7 are repeated, so they receive the average rank of their positions.)

  
  

Let's assign these ranks to each group and calculate the sum of ranks for both groups.

Step 2: Calculate U for Each Group

After assigning ranks:

• Before Protocol (Group A) Ranks: 1, 2, 3.5, 5.5, 5.5

• After Protocol (Group B) Ranks: 8, 9.5, 9.5, 8, 9.5

  
  

Sum of ranks:

• Sum of Ranks for Group A (R_A): 1 + 2 + 3.5 + 5.5 + 5.5 = 17.5

• Sum of Ranks for Group B (R_B): 8 + 9.5 + 9.5 + 8 + 9.5 = 44.5

  
  

U statistic is calculated using the formula:

Where R is the sum of ranks for the group, and n is the number of observations in the group.

Calculating U for both groups:

Let's calculate these U value:

U for Group A (Before Protocol): 2.5

U for Group B (After Protocol): 29.5

Then the U value is the min(UA;UB): U value = min(2.5;29.5)

U value= 2.5

Step 3: Compare with critical value

Finding the Critical Value in a Mann-Whitney U Table

1. Identify the Sample Sizes: For our example, both groups have a sample size of n=5. In a Mann-Whitney U table, you will look for the section that corresponds to the sample sizes of the two groups you are comparing. If the table is organized by sample size, find the column and row for 5n1=5 and n2​=5.

   
   

2. Select the Significance Level: Decide on your significance level (α). A common choice is α=0.05 for a 95% confidence level. This means you're accepting a 5% chance of incorrectly rejecting the null hypothesis (Type I error).

   
   

3. Look Up the Critical Value: With your sample sizes and chosen α, find the critical value in the table. This value is the threshold at which you would reject the null hypothesis. For a two-tailed test with n1=5 & n2​=5 & α=0.05.

The critical value for the Mann-Whitney U test for sample sizes of n1​=5 and n2​=5 at the α=0.05 significance level is 2. Therefore, with a calculated U value of 2.5 for Group A and a critical value of 2 from the table, the calculated U value exceeds the critical value, which would mean we fail to reject the null hypothesis based on this test at the 0.05 significance level. This suggest that there is not a statistically significant difference between the two groups.