MM207 Milestone 4

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Purdue University Globle

MM207 Statistics

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MM207 Milestone 4

What Are Scatterplots, Correlation, Regression, and Causation?

Scatterplots, correlation, regression, and causation are foundational concepts in statistics used to analyze relationships between quantitative variables.

Scatterplots visualize relationships between two variables.
Correlation measures the strength and direction of a linear relationship.
Regression predicts values and identifies trends.
Causation determines whether one variable directly influences another.

Together, these tools enable researchers and analysts to make data-driven decisions while avoiding misleading conclusions.

Understanding Explanatory and Response Variables

Every scatterplot contains two variables:

Explanatory Variable (Independent Variable): Plotted on the horizontal (x) axis.
Response Variable (Dependent Variable): Plotted on the vertical (y) axis.

The explanatory variable is used to explain or predict changes in the response variable.

Example

Suppose a scatterplot compares voltage and motor rotation speed:

Explanatory variable: Voltage
Response variable: Motor rotation speed

As voltage changes, researchers observe how motor speed responds.

How to Identify Variables in a Scatterplot

When interpreting a scatterplot, follow these steps:

Identify the horizontal (x) axis.
Determine the variable displayed on that axis.
Identify the vertical (y) axis.
Evaluate whether changes in the x-variable correspond to changes in the y-variable.

Correctly identifying variables is the first step toward understanding statistical relationships.

What Is Correlation?

Correlation measures how closely two variables move together. 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.

Positive Correlation

A positive correlation occurs when both variables increase together.

Examples include:

Study time and exam scores
Years of experience and salary
Advertising budget and sales revenue

Negative Correlation

A negative correlation occurs when one variable increases while the other decreases.

Examples include:

Miles driven and gallons of fuel remaining
Product price and customer demand
Exercise frequency and body fat percentage

For instance, as miles driven increase, the amount of fuel remaining typically decreases, producing a negative correlation.

How to Read a Scatterplot

Scatterplots provide a visual representation of data relationships. Analysts typically examine four characteristics:

Form of the relationship
Direction of association
Strength of association
Presence of outliers or unusual observations

Describing Scatterplots

Scatterplots are commonly described using:

Form

Linear
Curved
Exponential
Clustered

Direction

Positive
Negative
No apparent direction

Strength

Weak
Moderate
Strong

A scatterplot with a strong positive linear relationship will display points clustered closely around an upward trend.

Understanding Regression Analysis

Regression analysis estimates the expected value of a response variable using one or more explanatory variables.

A simple regression equation is written as:

ŷ = a + bx

Where:

ŷ = Predicted value
a = Intercept
b = Slope
x = Explanatory variable

Regression is widely used for:

Forecasting sales
Predicting housing prices
Evaluating healthcare outcomes
Measuring marketing effectiveness
Financial risk analysis

Example of Regression Prediction

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 data point, the line estimates the average relationship between variables. This makes it easier to predict future outcomes based on historical data.

For example, a regression line can estimate the number of basketball shots a player is likely to make from a specific distance or predict future sales based on advertising expenditures.

Understanding Slope in Regression

Slope measures the rate of change between variables.

The formula is:

Slope = (y₂ − y₁) / (x₂ − x₁)

Consider the following example:

Hamster: 0.5 pounds, $2 weekly feeding cost
Labrador: 62.5 pounds, $10 weekly feeding cost

Calculation:

(10 − 2) / (62.5 − 0.5) = 8 / 62 ≈ 0.13

The slope indicates that every 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 (0.8) means that, on average, a baby’s weight increases by 0.8 pounds for every additional month of age.

Correlation Does Not Equal Causation

One of the most important principles in statistics is that correlation does not imply causation.

Two variables may appear strongly related without one causing the other.

Example

Ice cream sales and drowning incidents often increase during the summer months. However, buying ice cream does not cause drowning.

The actual influencing factor is a lurking variable:

Higher temperatures
Increased outdoor activities
More people visiting pools and beaches

Similarly, neighborhoods with more vehicles may also have higher rates of health insurance coverage. Household income, not car ownership, likely explains both variables.

How Researchers Establish Causation

Demonstrating causation requires stronger evidence than simply observing a relationship.

Researchers typically rely on:

Controlled experiments
Randomized assignment
Temporal evidence (cause occurs before effect)
Elimination of alternative explanations

Randomized controlled trials (RCTs) remain the strongest method for establishing cause-and-effect relationships.

Understanding Nonlinear Relationships

Correlation only measures linear relationships. If data forms a curve rather than a straight line, the correlation coefficient may not accurately represent the relationship.

Examples of nonlinear relationships include:

Population growth
Compound interest
Projectile motion
Seasonal temperature changes

Always examine a scatterplot before calculating correlation coefficients.

What Is R² (Coefficient of Determination)?

The coefficient of determination (R²) measures how much variation in the response variable is explained by a 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.

Generally, higher R² values indicate better-fitting models, although acceptable values vary by discipline.

Relationship Between R² and Correlation

R² and correlation are mathematically related:

R² = r²

If:

R² = 0.9846

Then:

r ≈ 0.992

This indicates an extremely strong positive linear relationship.

Understanding Residuals

Residuals measure prediction errors in regression analysis.

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 indicates the actual value is lower than the predicted value.

Why Residuals Matter

Residuals help analysts determine:

Prediction accuracy
Model reliability
Presence of outliers
Suitability of a linear model

Smaller residuals generally indicate a better-fitting model.

Understanding Outliers and Influential Points

An outlier is a data point that differs significantly from the rest of the dataset.

Outliers may result from:

Data entry mistakes
Measurement errors
Rare but valid observations

Not all outliers significantly affect a regression model. Some become influential points—observations that substantially change the regression line when removed.

Influential points often:

Have extreme x-values
Possess high leverage
Alter the slope noticeably

Identifying these points improves the accuracy and reliability of statistical models.

Least-Squares Regression Line

The least-squares regression line minimizes the squared distance between observed and predicted values.

The slope is calculated using:

b = r(sᵧ/sₓ)

The intercept is:

a = ȳ − b x̄

The resulting equation becomes:

ŷ = a + bx

Least-squares regression is extensively used in:

Machine learning
Engineering
Economics
Healthcare
Finance
Business analytics

Understanding Lurking Variables

A lurking variable is an unseen factor that affects both the explanatory and response variables.

Example

Researchers may observe that communities with more vehicles tend to have greater health insurance coverage.

However, the true influencing factor is often:

Household income
Education level
Employment status

Ignoring lurking variables can lead to incorrect conclusions and flawed decision-making.

Practical Applications of Correlation and Regression

Correlation and regression have numerous real-world applications, including:

Predicting housing prices
Forecasting revenue
Evaluating medical treatments
Measuring advertising effectiveness
Monitoring climate patterns
Assessing investment risks
Analyzing sports performance
Conducting educational research

These techniques allow organizations to make evidence-based decisions across industries.

Frequently Asked Questions

What is the explanatory variable?

The explanatory variable is the independent variable shown on the horizontal (x) axis. It helps explain or predict changes in the response variable.

What is the response variable?

The response variable is the dependent variable shown on the vertical (y) axis and represents the measured outcome.

Does correlation prove causation?

No. Correlation identifies associations between variables but does not establish cause-and-effect relationships.

What does a regression line do?

A regression line summarizes trends in data and predicts expected values of the response variable.

What is a good R² value?

There is no universal benchmark. Higher values generally indicate stronger explanatory power, but acceptable values vary by field and dataset.

Why are residuals important?

Residuals measure prediction errors and help determine whether a regression model adequately fits the data.

What is an outlier?

An outlier is a data point that differs substantially from other observations and may influence statistical analyses.

What is a lurking variable?

A lurking variable is an unobserved factor that affects both variables under study, potentially creating misleading correlations.

Key Takeaways

Scatterplots visualize relationships between quantitative variables.
The explanatory variable appears on the x-axis.
The response variable appears on the y-axis.
Correlation coefficients range from -1 to +1.
Regression models predict expected outcomes.
Correlation measures association, not causation.
R² indicates how much variation is explained by a model.
Residuals reveal prediction errors.
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/