MM207 Milestone 4:
Statistical tools such as scatterplots, correlation, regression, and causation help researchers understand relationships between variables, identify patterns, make predictions, and support evidence-based decision-making. Scatterplots visually display data, correlation measures the strength and direction of relationships, regression predicts outcomes, and causation determines whether one variable directly influences another. Learning these concepts is essential for accurately interpreting data and avoiding incorrect conclusions in academic, business, healthcare, and scientific research.
Understanding Scatterplots
A scatterplot is a graph that displays the relationship between two quantitative variables. Each point on the graph represents one observation, making it easier to identify patterns, trends, clusters, or unusual values.
Every scatterplot contains two variables:
Explanatory variable (Independent variable): Displayed on the horizontal (x-axis)
Response variable (Dependent variable): Displayed on the vertical (y-axis)
The explanatory variable helps explain or predict changes in the response variable.
Example of Explanatory and Response Variables
Suppose a study examines the relationship between voltage and motor rotation speed.
Explanatory Variable: Voltage
Response Variable: Motor Rotation Speed
As voltage changes, researchers observe whether motor speed increases, decreases, or remains unchanged.
How to Identify Variables in a Scatterplot
When interpreting a scatterplot, use the following process:
Identify the variable on the horizontal (x) axis.
Determine the variable on the vertical (y) axis.
Observe how changes in the x-variable affect the y-variable.
Look for patterns, trends, or unusual observations.
Correctly identifying the variables is the first step toward interpreting statistical relationships accurately.
What Is Correlation?
Correlation measures the strength and direction of a linear relationship between two quantitative variables. It is represented by the correlation coefficient (r), which ranges from −1 to +1.
| Correlation Coefficient (r) | Interpretation |
|---|---|
| +1 | Perfect positive correlation |
| 0 | No linear correlation |
| −1 | Perfect negative correlation |
Values closer to +1 or −1 indicate stronger linear relationships, while values closer to 0 indicate weaker relationships.
Positive Correlation
A positive correlation occurs when both variables increase together.
Common examples include:
Study time and exam scores
Years of work experience and salary
Advertising budget and sales revenue
As one variable increases, the other tends to increase as well.
Negative Correlation
A negative correlation occurs when one variable increases while the other decreases.
Examples include:
Miles driven and fuel remaining
Product price and customer demand
Exercise frequency and body fat percentage
For example, the more miles a vehicle travels, the less fuel typically remains in the tank.
How to Read a Scatterplot
Scatterplots provide valuable information about relationships between variables. When analyzing one, focus on four important characteristics:
Form of the relationship
Direction of the relationship
Strength of the relationship
Presence of outliers
Describing Scatterplots
Form
Relationships may appear as:
Linear
Curved
Exponential
Clustered
Direction
The association may be:
Positive
Negative
No clear direction
Strength
Relationships are generally classified as:
Weak
Moderate
Strong
A strong positive linear relationship appears as data points clustered closely around an upward-sloping trend.
Understanding Regression Analysis
Regression analysis estimates the expected value of a response variable based on one or more explanatory variables. Unlike correlation, regression is primarily used for prediction.
The simple linear regression equation is:
ŷ = a + bx
Where:
ŷ = Predicted value
a = Intercept
b = Slope
x = Explanatory variable
Regression analysis is widely used in:
Forecasting sales
Predicting housing prices
Evaluating healthcare outcomes
Measuring marketing performance
Assessing financial risk
Regression Prediction Example
Consider the equation:
ŷ = 0.375x + 1.33
Where:
x = Hours of sleep
ŷ = Predicted GPA
If a student sleeps 2.5 hours:
ŷ = 0.375(2.5) + 1.33 = 2.27
The model predicts a GPA of approximately 2.27.
What Is a Best-Fit Line?
A best-fit line, also called a regression line, summarizes the overall trend in a scatterplot. Instead of connecting every point, it represents the average relationship between variables.
Regression lines are commonly used to:
Predict future sales
Estimate athletic performance
Forecast business growth
Analyze healthcare outcomes
These predictions are based on historical data patterns.
Understanding the Slope of a Regression Line
The slope measures how much the response variable changes for each one-unit increase in the explanatory variable.
The slope formula is:
Slope = (y₂ − y₁) / (x₂ − x₁)
Example
Suppose:
Hamster weighs 0.5 pounds and costs $2 per week to feed.
Labrador weighs 62.5 pounds and costs $10 per week to feed.
Calculation:
(10 − 2) / (62.5 − 0.5) = 8 / 62 ≈ 0.13
This indicates that each additional pound increases the average weekly feeding cost by approximately $0.13.
Interpreting Regression Slope
Consider the regression equation:
ŷ = 17 + 0.8x
Where:
x = Baby’s age (months)
ŷ = Baby’s weight (pounds)
The slope of 0.8 means a baby’s weight is expected to increase by approximately 0.8 pounds for every additional month of age.
Correlation Does Not Mean Causation
One of the most important concepts in statistics is that correlation does not prove causation.
Two variables may be strongly related without one directly causing the other.
Example
Ice cream sales and drowning incidents both increase during summer.
However, purchasing ice cream does not cause drowning.
Instead, both are influenced by lurking factors such as:
Higher temperatures
Increased outdoor activities
More people swimming at beaches and pools
Likewise, neighborhoods with more vehicles may also have higher health insurance coverage because household income influences both variables.
How Researchers Establish Causation
To demonstrate causation, researchers need stronger evidence than a simple statistical relationship.
Common approaches include:
Controlled experiments
Randomized assignment
Demonstrating that the cause occurs before the effect
Eliminating alternative explanations
Randomized Controlled Trials (RCTs) remain the strongest method for establishing cause-and-effect relationships.
Understanding Nonlinear Relationships
Correlation measures only linear relationships.
If data follows a curved pattern, the correlation coefficient may not accurately describe the relationship.
Examples include:
Population growth
Compound interest
Projectile motion
Seasonal temperature changes
Always examine the scatterplot before interpreting a correlation coefficient.
What Is the Coefficient of Determination (R²)?
The coefficient of determination (R²) measures how much variation in the response variable is explained by the regression model.
For example:
R² = 0.9846
This means:
Approximately 98.5% of the variation is explained by the model.
Approximately 1.5% remains unexplained.
Higher R² values generally indicate a better-fitting model, although acceptable values vary by discipline.
Relationship Between R² and Correlation
The coefficient of determination and the correlation coefficient are mathematically related:
R² = r²
If:
R² = 0.9846
Then:
r ≈ 0.992
This represents an extremely strong positive linear relationship.
Understanding Residuals
Residuals measure the difference between observed and predicted values.
The formula is:
Residual = Actual Value − Predicted Value
Example
Regression equation:
ŷ = −210 + 5.6x
If:
Height = 72 inches
Predicted weight = 193.2 pounds
Actual weight = 182 pounds
Then:
Residual = 182 − 193.2 = −11.2
A negative residual means the actual value is lower than the predicted value.
Why Residuals Matter
Residual analysis helps evaluate:
Prediction accuracy
Model reliability
Presence of outliers
Whether a linear model is appropriate
Smaller residuals generally indicate a better-fitting regression model.
Understanding Outliers and Influential Points
An outlier is a data point that differs substantially from other observations.
Outliers may result from:
Data entry errors
Measurement errors
Rare but valid observations
Not every outlier significantly affects the regression model. Some become influential points, meaning they substantially change the regression line if removed.
Influential points often:
Have extreme x-values
Possess high leverage
Change the slope of the regression line
Identifying these observations improves the accuracy of statistical analyses.
Least-Squares Regression Line
The least-squares regression line minimizes the squared differences between observed and predicted values.
The slope is calculated using:
b = r(sᵧ / sₓ)
The intercept is:
a = ȳ − b x̄
The final regression equation becomes:
ŷ = a + bx
Least-squares regression is widely applied in:
Machine learning
Engineering
Healthcare
Economics
Finance
Business analytics
Understanding Lurking Variables
A lurking variable is an unseen factor that influences both the explanatory and response variables, creating a misleading relationship.
Example
Researchers may find that communities with more vehicles also have higher health insurance coverage.
However, this relationship is often explained by other variables such as:
Household income
Education level
Employment status
Ignoring lurking variables can lead to inaccurate conclusions and poor decision-making.
Practical Applications of Correlation and Regression
Correlation and regression are valuable tools across many industries.
Common applications include:
Predicting housing prices
Forecasting revenue
Evaluating medical treatments
Measuring advertising effectiveness
Monitoring climate patterns
Assessing investment risks
Analyzing sports performance
Conducting educational research
These methods support informed, data-driven decisions in real-world settings.
Frequently Asked Questions
What is the explanatory variable?
The explanatory variable, also called the independent variable, appears on the x-axis and is used to explain or predict changes in the response variable.
What is the response variable?
The response variable, also known as the dependent variable, appears on the y-axis and represents the measured outcome.
Does correlation prove causation?
No. Correlation identifies relationships between variables but does not establish that one variable causes changes in another.
What does a regression line do?
A regression line summarizes the overall trend in the data and predicts expected values for the response variable.
What is a good R² value?
There is no universal standard. Higher R² values generally indicate a stronger model fit, but acceptable values depend on the field of study and the data.
Why are residuals important?
Residuals measure prediction errors and help determine whether a regression model accurately represents the data.
What is an outlier?
An outlier is a data point that differs significantly from the rest of the dataset and may influence statistical analyses.
What is a lurking variable?
A lurking variable is an unobserved factor that affects both variables being studied, potentially creating misleading correlations.
Key Takeaways
Scatterplots visually display relationships between quantitative variables.
The explanatory variable is plotted on the x-axis.
The response variable is plotted on the y-axis.
Correlation coefficients range from −1 to +1.
Regression models estimate and predict outcomes.
Correlation indicates association but does not establish causation.
R² measures how much variation is explained by a regression model.
Residuals evaluate prediction accuracy.
Outliers and influential points can affect regression results.
Controlled experiments provide the strongest evidence for causation.
References
Freedman, D., Pisani, R., & Purves, R. (2007). Statistics (4th ed.). W. W. Norton & Company. https://wwnorton.com/
Moore, D. S., McCabe, G. P., & Craig, B. A. (2021). Introduction to the Practice of Statistics (10th ed.). W. H. Freeman. https://www.macmillanlearning.com/
National Institute of Standards and Technology. (2023). e-Handbook of Statistical Methods. https://www.itl.nist.gov/div898/handbook/
MM207 Milestone 4
OpenStax. (2023). Introductory Statistics 2e. Rice University. https://openstax.org/details/books/introductory-statistics-2e
Penn State Eberly College of Science. (2024). STAT 200: Elementary Statistics. https://online.stat.psu.edu/stat200/
UCLA Institute for Digital Research and Education. (2024). Regression with Stata, SPSS, R, and SAS. https://stats.oarc.ucla.edu/