Side note: Pearson Coefficient is mostly use in Lean Six Sigma, so focus yourself on it, do not spend to much time on Spearman Coefficient. In the realm of Lean Six Sigma, understanding the relationship between variables is crucial for identifying and improving processes. Two statistical tools often used to measure the strength and direction of a relationship between two continuous or ordinal variables are the Pearson and Spearman correlation coefficients. This article delves into both, highlighting their significance, differences, and applications within Lean Six Sigma projects.

Pearson Correlation Coefficient

The Pearson correlation coefficient, denoted as r, measures the linear correlation between two variables, providing insight into the strength and direction of their linear relationship. Its value ranges from -1 to 1, where:

• 1 indicates a perfect positive linear relationship,

• -1 indicates a perfect negative linear relationship, and

• 0 indicates no linear relationship.

In Lean Six Sigma projects, the Pearson correlation coefficient is instrumental in quantifying the linear association between two continuous variables. For instance, it can be used to investigate the relationship between production speed and defect rates in a manufacturing process. A high positive correlation might indicate that as production speed increases, defect rates also increase, suggesting a potential area for process improvement.

Calculation of Pearson Correlation Coefficient

The Pearson correlation coefficient is calculated using the formula:

where Xi​ and Yi​ are the values of the two variables, and X̄ and Ȳ are the means of those variables, respectively.

Spearman Correlation Coefficient

The Spearman correlation coefficient, denoted as rs​, measures the strength and direction of the monotonic relationship between two ordinal or continuous variables. Unlike the Pearson coefficient, which assesses linear relationships, the Spearman coefficient is designed to identify any monotonic trend, whether linear or not. Its value also ranges from -1 to 1, with similar interpretations for these boundary values.

The Spearman correlation is particularly useful in Lean Six Sigma when data do not meet the assumptions required for Pearson's correlation, such as when data are not normally distributed, or when dealing with ordinal variables. For example, it could be used to evaluate the relationship between the ordinal rank of employee training levels and the number of errors reported in a process.

Calculation of Spearman Correlation Coefficient

The Spearman correlation coefficient is calculated by first ranking the values of both variables. Then, the difference between the ranks of each observation is squared and summed. The formula for rs​ is:

where d is the difference between the ranks of corresponding values, and n is the number of observations.

Application in Lean Six Sigma

Understanding both Pearson and Spearman correlation coefficients enables Lean Six Sigma practitioners to analyze the relationships between variables effectively. This analysis can lead to insights that drive data-driven improvements. For example, identifying a strong correlation between variables might suggest a causal relationship worth exploring through further analysis or experimentation, such as a designed experiment (DOE).

The choice between Pearson and Spearman correlation coefficients depends on the data's nature and the underlying assumptions. Pearson is preferred for continuous data that meet the assumptions of normality and linearity. In contrast, Spearman is ideal for ordinal data or when those assumptions are not met.

In conclusion, both Pearson and Spearman correlation coefficients are valuable tools in the Lean Six Sigma toolkit. They provide a foundation for understanding relationships between variables, crucial for identifying opportunities for process improvements and achieving operational excellence.

Scenario: Customer Satisfaction and Response Time

A company wants to analyze the relationship between customer satisfaction scores and the response time to customer inquiries. The hypothesis is that longer response times might be associated with lower customer satisfaction. The company collected data from 100 customer interactions. Here's a simplified version of the dataset for illustration purposes:

Pearson Correlation Coefficient Calculation

First, we calculate the mean of the satisfaction scores and response times:

Mean Satisfaction Score (X̄) = 8+7+6+9+55=758+7+6+9+5​=7

Mean Response Time (Ȳ) = 2+3+5+1+65=3.452+3+5+1+6​=3.4 Next, we calculate the numerator and denominator of the Pearson formula separately:

Therefore, the Pearson correlation coefficient r is:

Interpretation

The Pearson coefficient of −0.73 suggests a strong negative linear relationship between satisfaction scores and response time, corroborating the hypothesis that longer response times may lead to lower customer satisfaction.

Extra: Let's plot that:

The light red band, represents the 95% confidence interval for the regression estimate. This confidence interval provides a range of values which is likely to contain the true mean response for a given level of X (Satisfaction Score). Points within this band are consistent with the estimated linear relationship between the two variables at a 95% confidence level.

Spearman Correlation Coefficient Calculation

For the Spearman calculation, we first rank the satisfaction scores and response times:

Next, calculate the difference between ranks (d) and its square (d2) for each interaction:

Sum of d2 = 4+0+4+16+16=40

Using the formula for rs​, we have n=5:

Interpretation

The Spearman coefficient of −1−1 indicates a perfect negative monotonic relationship, further supporting this conclusion but from a rank-order perspective. These insights could guide the company in prioritizing efforts to reduce response times as a means to improve customer satisfaction.