Scatter Diagram, Correlation and Regression Analysis

Learning Objectives

By the end of this lesson, the learner will be able to:

Explain the purpose of a scatter diagram in Six Sigma analysis.
Identify possible relationships between an input variable and an output variable.
Distinguish between correlation and causation.
Explain positive, negative, and weak/no correlation.
Interpret the correlation coefficient.
Understand the purpose and basic principles of regression analysis.
Distinguish between simple linear regression and multiple regression at a basic level.
Interpret a regression equation and its coefficients.
Understand the meaning of R-squared.
Use scatter diagrams, correlation, and regression appropriately in DMAIC Analyze.
Apply these tools to identify potential relationships between process variables.
Recognize common errors in interpreting correlation and regression results.
Understand how Minitab can be used to perform correlation and regression analysis.

1. Introduction

In Six Sigma, improvement decisions should be based on evidence rather than assumptions.

During the Analyze phase of DMAIC, the Green Belt often needs to determine whether changes in one process variable are associated with changes in another variable.

For example:

Does higher temperature increase defects?
Does longer processing time increase customer waiting time?
Does machine speed affect product thickness?
Does employee experience influence processing errors?
Does pressure affect product strength?

A useful starting point is the scatter diagram.

A scatter diagram provides a visual way to examine the relationship between two numerical variables. If a relationship appears to exist, correlation analysis can quantify the strength and direction of that relationship.

Regression analysis goes one step further by developing a mathematical model that can describe and, within appropriate limits, predict the response based on one or more input variables.

These tools are particularly valuable when a Six Sigma team is investigating potential X–Y relationships.

Y = Output / Response
X = Input / Potential Cause

The objective is not simply to find that two variables move together. The objective is to develop evidence that helps the team understand the process and identify potential critical inputs.

2. Concept / Theory

2.1 The X–Y Relationship

Six Sigma commonly uses the concept:

Y = f(X)

This means that the process output or response (Y) may be influenced by one or more process inputs (X).

For example:

Y = Customer waiting time

Potential X variables might include:

Number of customers waiting
Number of service employees
Transaction complexity
Processing time
Time of day

Another example:

Y = Product defect rate

Potential X variables might include:

Machine temperature
Pressure
Speed
Raw-material characteristics
Operator settings

Scatter diagrams, correlation, and regression can help investigate these relationships.

However, statistical association alone does not automatically establish that X causes Y.

2.2 Scatter Diagram

A scatter diagram, also called a scatter plot, displays paired observations for two numerical variables.

Typically:

The X variable is plotted on the horizontal axis.
The Y variable is plotted on the vertical axis.
Each point represents one observation.

For example, if a team records machine temperature and the corresponding defect count for several production runs, each production run becomes one point on the scatter diagram.

The pattern of points can provide an initial indication of whether a relationship exists.

2.3 Types of Scatter Patterns

A scatter diagram may show several types of patterns.

Positive Relationship

As X increases, Y tends to increase.

Example:

Training hours ↑ → Employee productivity ↑

The points generally move upward from left to right.

Negative Relationship

As X increases, Y tends to decrease.

Example:

Machine age ↑ → Equipment efficiency ↓

The points generally move downward from left to right.

Little or No Relationship

Changes in X do not show an obvious relationship with Y.

The points appear widely scattered without a clear pattern.

Curvilinear Relationship

The relationship may not follow a straight line.

For example, increasing temperature may initially improve a process but excessive temperature may increase defects.

In such situations, a simple straight-line model may not adequately represent the relationship.

2.4 Correlation

Correlation measures the strength and direction of a linear relationship between two numerical variables.

The commonly used Pearson correlation coefficient is represented by:

The value of r ranges from:

−1 to +1

Interpretation:

r close to +1 → strong positive linear relationship
r close to −1 → strong negative linear relationship
r close to 0 → weak or no linear relationship

For example:

Correlation	Possible Interpretation
+0.90	Strong positive linear relationship
+0.60	Moderate positive relationship
+0.10	Very weak positive relationship
0.00	No linear relationship
−0.55	Moderate negative relationship
−0.90	Strong negative linear relationship

The exact interpretation of “weak,” “moderate,” or “strong” should be treated as context-dependent rather than as an absolute rule.

2.5 Correlation Does Not Prove Causation

This is one of the most important concepts in Six Sigma analysis.

A strong correlation between X and Y does not automatically prove that X causes Y.

For example, suppose:

Ice cream sales increase.
Sunburn cases increase.

These two variables may show a positive relationship.

However, ice cream consumption does not necessarily cause sunburn.

A third variable, such as hot weather, may influence both.

Therefore:

Correlation indicates association, not necessarily causation.

The Six Sigma team must investigate the process, consider other variables, conduct appropriate analysis, and use process knowledge before concluding that an X is a root cause.

2.6 Regression Analysis

Regression analysis is used to model the relationship between a response variable and one or more predictor variables.

For simple linear regression:

Y = β₀ + β₁X + ε

Where:

Y = response variable
X = predictor/input variable
β₀ = intercept
β₁ = slope
ε = random error

The estimated regression equation may be written as:

Ŷ = b₀ + b₁X

The equation can be used to describe the observed relationship and, when the model is appropriate, estimate the expected response for values of X within the relevant data range.

2.7 Intercept

The intercept is the predicted value of Y when X equals zero.

For example:

Ŷ = 50 + 2X

The intercept is 50.

Mathematically, when X = 0:

Ŷ = 50

However, the intercept should not automatically be given a practical interpretation if X = 0 is outside the meaningful operating range of the process.

2.8 Slope

The slope indicates the expected change in Y for a one-unit increase in X, according to the fitted linear model.

For:

Ŷ = 50 + 2X

The slope is 2.

This means that a one-unit increase in X is associated with an estimated increase of 2 units in Y.

A negative slope indicates an inverse relationship.

For example:

Ŷ = 100 − 3X

A one-unit increase in X is associated with an estimated decrease of 3 units in Y.

2.9 R-Squared

R-squared (R²) describes the proportion of variation in the response that is explained by the fitted regression model, within the context of the data and model.

For example:

R² = 0.64

can be expressed as approximately 64% of the observed variation in Y being explained by the model.

R² is useful, but it should not be interpreted in isolation.

A high R² does not automatically prove:

Causation
Correct model specification
Statistical significance
Practical usefulness
Good future prediction outside the observed data range

The Green Belt should examine the complete statistical and process context.

2.10 Statistical Significance of Regression

Regression output commonly provides statistical tests for model terms.

For a slope, the hypotheses may be:

H₀: β₁ = 0

H₁: β₁ ≠ 0

If the p-value is sufficiently small relative to the chosen significance level, the team may conclude that the data provide statistical evidence of a non-zero linear relationship.

However, statistical significance should still be considered together with:

Effect size
Process knowledge
Practical significance
Model assumptions
Residual behavior
Measurement quality

2.11 Residuals

A residual is the difference between an observed value and the value predicted by the regression model.

Residual = Observed Y − Predicted Y

Residual analysis helps determine whether the regression model is appropriate.

Patterns in residuals may indicate:

Non-linearity
Unequal variation
Outliers
Missing variables
Model inadequacy

Therefore, regression analysis should not stop with looking at the regression equation or R².

3. Key Topics

3.1 Scatter Diagram

Key purposes:

Visualize relationships.
Identify possible trends.
Detect clusters.
Identify unusual observations.
Identify potential outliers.
Determine whether further statistical analysis is appropriate.

3.2 Correlation Coefficient

The Pearson correlation coefficient:

Measures linear association.
Indicates direction.
Indicates strength.
Ranges from −1 to +1.

The sign indicates direction:

Positive r → positive relationship

Negative r → negative relationship

The magnitude indicates the strength of the linear association.

3.3 Correlation and Outliers

Outliers can have a substantial effect on correlation.

For example, most observations may show little relationship, but one extreme observation may make the calculated correlation appear much stronger.

Therefore:

Always examine the scatter diagram before interpreting the correlation coefficient.

3.4 Simple Linear Regression

Simple linear regression involves:

One response variable Y.
One predictor X.

Example:

Y = Defect rate

X = Machine temperature

The analysis estimates the linear relationship between temperature and defect rate.

3.5 Multiple Regression

Multiple regression includes more than one predictor.

For example:

Y = Defect rate

Potential predictors:

Temperature
Pressure
Machine speed
Material moisture

A simplified model might be:

Ŷ = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + β₄X₄

Multiple regression can help the team study several potential contributors simultaneously.

3.6 Correlation vs Regression

Aspect	Correlation	Regression
Main purpose	Measure association	Model relationship
Variables	Two numerical variables	Response + predictor(s)
Direction	Yes	Yes, through coefficients
Prediction	Not its primary purpose	Can support prediction
Equation	No	Yes
R²	May be derived from simple linear regression relationship	Commonly reported
Causation	Does not prove causation	Does not prove causation

3.7 When to Use These Tools

Use a scatter diagram when:

You have paired numerical data.
You want to visually examine a possible relationship.
You need to identify unusual observations.

Use correlation when:

You want to quantify the strength and direction of a linear relationship.

Use regression when:

You want to model the relationship.
You want to quantify the effect of predictor variables.
You need estimates within an appropriate operating range.
You want to evaluate one or more predictors.

3.8 Minitab Application

In Minitab, a Green Belt can generally use:

Graph → Scatterplot

to create a scatter diagram.

For correlation, commonly use:

Stat → Basic Statistics → Correlation

For regression, commonly use:

Stat → Regression → Regression → Fit Regression Model

The exact menu wording may vary by Minitab version.

Typical workflow:

Prepare and verify the data.
Define the response variable.
Identify potential predictor variables.
Create a scatter diagram.
Examine the visual relationship.
Calculate correlation where appropriate.
Fit a regression model if justified.
Examine coefficients and p-values.
Examine R² and model fit.
Examine residuals.
Interpret results using process knowledge.
Confirm important findings through further investigation or appropriate experimentation.

4. Tools / Methodology

4.1 Step 1 — Define the Y

Clearly identify the response variable.

Examples:

Cycle time
Defect rate
Customer waiting time
Product strength
Delivery time

The operational definition should specify exactly how Y is measured.

4.2 Step 2 — Identify Potential X Variables

Develop a list of possible process inputs.

Sources may include:

Process maps
Cause-and-effect diagrams
Process knowledge
Historical data
Subject-matter expertise
Previous investigations

Avoid assuming that every available variable is a root cause.

4.3 Step 3 — Verify the Data

Before analysis, check:

Measurement definitions
Units
Missing values
Data-entry errors
Sampling method
Time period
Data consistency
Measurement-system concerns

Poor-quality data can produce misleading statistical conclusions.

4.4 Step 4 — Create the Scatter Diagram

Plot X against Y.

Look for:

Direction
Strength
Linearity
Clusters
Outliers
Changing variation
Possible curvature

Do not calculate correlation without first understanding the graphical pattern.

4.5 Step 5 — Calculate Correlation

Calculate the correlation coefficient.

Then consider:

Magnitude
Direction
Statistical significance, where appropriate
Sample size
Outliers
Process context

4.6 Step 6 — Fit Regression

If a meaningful relationship appears plausible, fit an appropriate regression model.

Review:

Regression equation
Coefficients
p-values
Confidence intervals
R²
Residual plots
Outliers and influential observations

4.7 Step 7 — Interpret the Model

Ask:

Is the relationship statistically supported?
Is the effect practically meaningful?
Does the model fit the process?
Are assumptions reasonably satisfied?
Are there important omitted variables?
Is the relationship likely to be causal, or merely associated?
Is the model being used only within an appropriate range?

4.8 Step 8 — Confirm the Potential Cause

If the analysis identifies a potentially important X, further confirmation may be necessary.

Possible approaches include:

Process observation
Stratification
Additional data collection
Controlled experiments
Designed experiments
Confirmation runs
Process trials

Statistical analysis should support the improvement investigation rather than replace process understanding.

5. Worked Example / Case Study

Case Study — Moulding Temperature and Defects

A manufacturing team notices that defect levels vary between production runs.

The team suspects that moulding temperature may influence the number of defects.

They collect data from several production runs.

Run	Temperature (°C)	Defects
1	180	12
2	182	11
3	184	10
4	186	9
5	188	8
6	190	7
7	192	6
8	194	5
9	196	4
10	198	4

Step 1 — Define X and Y

X = Moulding temperature

Y = Number of defects

Step 2 — Create a Scatter Diagram

The scatter diagram would show the observations with:

Temperature on the X-axis.
Defects on the Y-axis.

The points show a generally downward pattern.

This suggests a potential negative relationship.

Step 3 — Interpret the Pattern

As temperature increases, the number of defects generally decreases.

The team therefore has evidence of an association between temperature and defects in this dataset.

However, the team should not immediately conclude:

“Temperature is proven to be the root cause.”

Other variables could have changed during the production runs.

For example:

Raw-material batch
Machine condition
Operator
Production speed
Environmental conditions

could also influence the result.

Step 4 — Calculate Correlation

Suppose the analysis produces:

r = −0.98

This indicates a very strong negative linear association in the observed data.

The negative sign means that higher temperature values are associated with lower defect counts.

The magnitude indicates that the linear association is strong.

Step 5 — Regression Model

Suppose the fitted model is:

Ŷ = 120 − 0.58X

The negative slope indicates that, according to the fitted model, increasing temperature by one degree Celsius is associated with an estimated decrease of approximately 0.58 defects per production run.

The team should examine the statistical significance, confidence intervals, residuals, and operating range before using this equation for decisions.

Step 6 — Practical Interpretation

The analysis provides evidence that temperature is a potentially important X.

The Green Belt should now investigate whether the relationship is:

Consistent
Physically plausible
Reproducible
Practically important
Causal or confounded by other variables

The team may conduct controlled trials or an appropriate designed experiment to establish the effect of temperature more convincingly.

Important Learning Point

The correct Six Sigma sequence is:

Visualize → Quantify → Model → Interpret → Confirm

Not:

Calculate correlation → Declare root cause

6. Practical Application

Exercise 1 — Customer Waiting Time

A service organization records:

Number of customers waiting
Customer waiting time

Ask the learner to:

Identify X.
Identify Y.
Draw a scatter diagram.
Describe the pattern.
Calculate correlation.
Explain what the correlation means.
Identify additional variables that may influence waiting time.

Exercise 2 — Employee Experience

A company wants to determine whether employee experience is associated with processing errors.

Data collected:

Employee experience in months
Number of processing errors

Ask the learner to:

Create a scatter diagram.
Identify the direction of the relationship.
Calculate correlation.
Investigate possible outliers.
Discuss whether correlation establishes causation.

Exercise 3 — Regression Interpretation

Suppose a regression model is:

Ŷ = 25 + 1.8X

where:

X = processing time
Y = cost per transaction

Ask:

What is the intercept?
What is the slope?
What does the slope mean?
What would the model predict for X = 10?
Is the intercept necessarily meaningful in the real process?
What additional information should be reviewed before using the model?

For X = 10:

Ŷ = 25 + 1.8(10)

Ŷ = 43

Therefore, the model predicts a response of 43 units at X = 10, assuming the model is appropriate for that value.

Exercise 4 — Green Belt Investigation

Select a real or simulated process and identify:

One Y
At least three potential X variables

For each X:

Describe how it could influence Y.
Identify the required data.
Create a scatter diagram.
Calculate correlation where appropriate.
Determine whether regression analysis is appropriate.
Identify potential confounding variables.
Recommend how the relationship could be confirmed.

Practical Checklist

Before concluding that an X is important, ask:

Is X measured correctly?
Is Y measured correctly?
Are the data representative?
Does the scatter diagram show a meaningful pattern?
Is the relationship linear?
Are there outliers?
What is the correlation?
Is the relationship statistically significant?
What is the practical effect?
What does the regression model indicate?
Are the residuals acceptable?
Are there other variables involved?
Does process knowledge support the relationship?
Has the potential cause been confirmed?

7. Lesson Summary

In Six Sigma, scatter diagrams, correlation, and regression are important tools for investigating relationships between process variables.

A scatter diagram provides a visual representation of paired numerical data and helps identify possible relationships, patterns, clusters, and outliers.

Correlation quantifies the strength and direction of a linear relationship between two numerical variables.

Regression analysis develops a mathematical model relating a response variable to one or more predictor variables.

Important concepts include:

Positive and negative relationships
Correlation coefficient
Simple linear regression
Regression equation
Slope
Intercept
R²
p-values
Residuals
Model assumptions
Statistical versus practical significance

A critical principle is:

Correlation does not prove causation.

A strong statistical relationship should be followed by process investigation and, where appropriate, controlled experimentation or other confirmation methods.

The Green Belt should use these tools to move from observation to evidence-based understanding of potential X–Y relationships.

8. Lesson Learnt / Conclusion

The major lesson is that statistical association is a starting point for Six Sigma investigation, not automatically the final answer.

The Green Belt should:

Visualize the relationship.

Quantify the relationship.

Model the relationship when appropriate.

Evaluate statistical and practical significance.

Check the quality and assumptions of the analysis.

Investigate alternative explanations.

Confirm important potential causes.

Scatter diagrams help the team see relationships.

Correlation helps the team measure linear association.

Regression helps the team model the relationship and, where appropriate, estimate the response.

Used correctly, these tools help the Six Sigma team convert process data into meaningful evidence during the Analyze phase of DMAIC.

Good analysis identifies relationships.
Good Six Sigma practice investigates whether those relationships truly explain the process problem.