The Chi-Square Test for Independence is a statistical method used within the Lean Six Sigma framework to evaluate if two categorical variables are independent of each other across different populations. This test is particularly useful in identifying relationships between variables in a process or system that Lean Six Sigma projects aim to optimize.

Understanding the Chi-Square Test for Independence

The Chi-Square Test for Independence, represented by the symbol χ2, is a non-parametric test that does not assume a normal distribution of the data. It's used to determine whether there's a significant association between two categorical variables by comparing the observed frequencies in each category against the frequencies expected if the variables were independent.

How to Perform the Chi-Square Test for Independence

1. Data Collection: Collect data in a contingency table, which displays the frequency distribution of the variables.
   

2. Hypothesis Formulation:

   • Null Hypothesis (H0​): Assumes that there is no association between the two variables, meaning they are independent.

   • Alternative Hypothesis (H1​): Assumes that there is an association between the two variables, meaning they are not independent.
     

3. Calculate Expected Frequencies: For each cell in the contingency table, calculate the expected frequency based on the assumption that the variables are independent. The expected frequency for each cell is calculated as:

4. Compute Chi-Square Statistic: The chi-square statistic is calculated using the formula:

where Oij​  is the observed frequency in cell ij and Eij​ is the expected frequency for the same cell. The sum is taken over all cells in the table.

5. Determine Significance: Compare the calculated chi-square statistic to the critical value from the chi-square distribution table with the appropriate degrees of freedom (df). Degrees of freedom for this test are calculated as: df=(number of rows−1)∗(number of columns−1) (see below table); If the chi-square statistic is greater than the critical value, reject the null hypothesis.

6. Conclusion: Based on the comparison, if the null hypothesis is rejected, it is concluded that there is a significant association between the two variables, indicating that they are not independent.

Applications in Lean Six Sigma

In Lean Six Sigma projects, the Chi-Square Test for Independence can be used to:

• Identify relationships between different factors affecting a process, such as the relationship between machine settings and product defects.

• Evaluate the effectiveness of changes made to a process by analyzing before and after data.

• Determine if observed improvements in a process are statistically significant or due to random variation.

  
  
  
  

Conclusion

The Chi-Square Test for Independence is a powerful tool in the Lean Six Sigma toolkit, enabling practitioners to make data-driven decisions about the relationships between categorical variables. By rigorously testing for independence, Lean Six Sigma projects can more accurately identify areas for improvement and validate the impact of process changes, leading to more effective and efficient processes

Scenario:

The Chi-Square Test for Independence is a statistical hypothesis test used to determine whether there is a significant association between two categorical variables. Let's go through a real-life example step by step, using a small set of data to understand how to apply the Chi-Square Test for Independence.

A company wants to investigate if there is a relationship between gender (Male, Female) and preference for a new product (Like, Dislike). They surveyed 100 people, and here's the data collected:

Objective:

To determine if gender is independent of product preference.

Step 1: State the Hypotheses

• Null Hypothesis (H0): There is no association between gender and product preference. (They are independent.)

• Alternative Hypothesis (H1): There is an association between gender and product preference. (They are not independent.)
  

Step 2: Calculate Expected Frequencies

The expected frequency for each cell in a contingency table is calculated using the formula:

For the "Male & Like" cell:

Similarly, we calculate for other cells:

Step 3: Calculate Chi-Square Statistic

The Chi-Square statistic is calculated using the formula:

Where O is the observed frequency, and E is the expected frequency.

For "Male & Like":

Continuing this for all cells:

Summing these values gives the total Chi-Square statistic:

χ^2Total​=0.714+1.667+0.714+1.667=4.762

Step 4: Determine the Critical Value

The critical value for the Chi-Square statistic can be found in a Chi-Square distribution table. For our example, with 1 degree of freedom (df = (Rows - 1) (Columns - 1) = (2-1)(2-1) = 1) and a significance level of 0.05, the critical value is approximately 3.841 (see below table).

Step 5: Make the Decision

Since our calculated χ^2Total​=4.762 is greater than the critical value of 3.841, we reject the null hypothesis.

Conclusion:

There is sufficient evidence at the 0.05 significance level to conclude that there is an association between gender and preference for the new product. This means gender and product preference are not independent of each other in this sample.

This step-by-step process demonstrates how the Chi-Square Test for Independence can be applied to real-world scenarios to test associations between categorical variables.