MM207 Milestone 4

MM207 Milestone 4

Student Name

Purdue University Global

MM207 Statistics

Prof. Name:

Date

Understanding Scatterplots, Correlation, Regression, and Causation in Statistics

Statistics uses scatterplots, correlation, and regression analysis to examine relationships between variables. In a scatterplot, the explanatory variable is displayed on the horizontal (x) axis, while the response variable is shown on the vertical (y) axis. Correlation measures the strength and direction of a linear relationship between two variables, whereas regression helps predict outcomes. However, a strong correlation alone does not prove causation.

These concepts are fundamental in statistics, data science, economics, business analytics, healthcare, and scientific research because they help identify patterns, make predictions, and support data-driven decisions.

Understanding Explanatory and Response Variables

Scatterplots are commonly used to visualize the relationship between two quantitative variables.

The explanatory variable (also called the independent variable or predictor) is placed on the horizontal (x-axis). The response variable (dependent variable or outcome) appears on the vertical (y-axis).

For example, if a scatterplot displays voltage on the horizontal axis and motor rotation speed on the vertical axis, then:

• Explanatory variable: Voltage

• Response variable: Rotation speed

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

How to Identify Variables in a Scatterplot

When analyzing a scatterplot:

• Check the horizontal axis first.

• The horizontal axis always represents the explanatory variable.

• The vertical axis represents the response variable.

• Determine whether changes in the explanatory variable are associated with changes in the response variable.

Correctly identifying these variables is the first step in interpreting statistical relationships.

Understanding Correlation

Correlation measures how closely two variables move together.

The correlation coefficient (r) ranges from −1 to +1.

Values closer to either +1 or −1 indicate stronger linear relationships.

Positive and Negative Correlation

The direction of correlation explains whether variables increase together or move in opposite directions.

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

As one variable increases, the other generally increases as well.

Negative Correlation

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

For example:

• Miles driven and gallons of fuel remaining

• Product price and customer demand

• Exercise frequency and body fat percentage

As the number of miles driven increases, the amount of gasoline remaining typically decreases. Therefore, the correlation coefficient is expected to fall between 0 and −1.

Reading Scatterplots

Scatterplots provide a visual representation of relationships between variables.

When interpreting a scatterplot, evaluate:

• Form of the relationship

• Direction of association

• Strength of association

• Presence of unusual observations such as outliers

These characteristics help determine whether correlation or regression analysis is appropriate.

Understanding Best-Fit Lines

A best-fit line, also called a regression line, summarizes the overall trend in a scatterplot.

Instead of connecting every point, it estimates the average relationship between variables.

For example, if a basketball player’s shooting performance decreases with distance, the best-fit line can estimate how many baskets the player is expected to make from a specific distance.

If the regression line predicts approximately five baskets from a distance of 10 feet, that value becomes the expected outcome.

Regression lines are valuable because they simplify prediction using existing data.

Making Predictions Using Regression

Regression equations estimate response values for given explanatory values.

For example:

[
\hat{y}=0.375x+1.33
]

where:

• x = Hours of sleep

• ŷ = Predicted GPA

If a student sleeps 2.5 hours, substitute the value into the equation:

[
\hat{y}=0.375(2.5)+1.33
]

The predicted GPA is approximately 2.27.

Regression predictions are estimates rather than exact values because real-world data naturally contains variation.

Correlation Does Not Mean Causation

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

Two variables may appear closely related without one directly causing the other.

For example:

• Ice cream sales and drowning incidents both increase during summer.

• Neither variable causes the other.

• The lurking variable is higher summer temperatures.

Similarly, neighborhoods with more cars may also have higher rates of health insurance coverage.

Owning additional cars does not cause people to purchase insurance. Instead, household income influences both variables.

Researchers must distinguish association from true cause-and-effect relationships.

How to Establish Causality

Demonstrating causation requires more evidence than observing correlation.

Researchers generally rely on:

• Controlled experiments

• Randomized assignment

• Careful elimination of alternative explanations

• Evidence that the explanatory variable occurs before the response variable

Randomized controlled experiments provide the strongest evidence that one variable causes changes in another.

Understanding Slope

Slope measures the rate of change between two variables.

The slope formula is:

[
\text{Slope}=\frac{y_2-y_1}{x_2-x_1}
]

Suppose:

• Hamster: 0.5 pounds, $2 weekly feeding cost

• Labrador: 62.5 pounds, $10 weekly feeding cost

Using these values:

[
\frac{10-2}{62.5-0.5}
=\frac{8}{62}
\approx0.13
]

The slope is approximately $0.13 per pound.

This means each additional pound of animal weight increases the average weekly feeding cost by about 13 cents.

Interpreting Regression Slope

Consider the regression equation:

[
\hat{y}=17+0.8x
]

where:

• x = Baby’s age (months)

• y = Baby’s weight (pounds)

The slope (0.8) indicates that:

On average, a baby’s weight increases by 0.8 pounds for every additional month of age.

The slope always describes how much the response variable changes when the explanatory variable increases by one unit.

Key Takeaways

• Scatterplots visualize relationships between two quantitative variables.

• The explanatory variable belongs on the horizontal axis.

• The response variable belongs on the vertical axis.

• Correlation measures the strength and direction of a linear relationship.

• Positive correlations move in the same direction.

• Negative correlations move in opposite directions.

• Regression lines help predict outcomes.

• Correlation alone cannot establish causation.

• Randomized controlled experiments provide the strongest evidence of causal relationships.

• The slope measures the average rate of change between variables.

MM207 Milestone 4

Quick Reference

What is the explanatory variable?

The explanatory variable is the independent variable displayed on the horizontal (x) axis of a scatterplot. It is used to explain or predict changes in the response variable.

What is a response variable?

The response variable is the dependent variable displayed on the vertical (y) axis. It represents the outcome being measured.

Does correlation prove causation?

No. Correlation measures association but does not establish that one variable causes another. Controlled experiments are typically required to demonstrate causality.

What does a regression line do?

A regression line summarizes the trend in a scatterplot and predicts the expected value of the response variable based on the explanatory variable.

Understanding Regression Analysis, Residuals, R², Outliers, and Scatterplots

Regression analysis extends beyond identifying relationships between variables. It allows researchers to evaluate how well a model fits the data, make predictions, identify unusual observations, and interpret the strength of relationships. Key concepts include the coefficient of determination (R²), residuals, outliers, least-squares regression, and lurking variables.

Predicting Values Using Regression Equations

Regression equations estimate the expected value of a response variable based on an explanatory variable.

For example, consider the regression equation:

[
\hat{y} = 0.375x + 1.33
]

where:

• x = Hours of sleep

• ŷ = Predicted GPA

To predict the GPA of a student who sleeps 2.5 hours, substitute 2.5 for x:

[
\hat{y} = 0.375(2.5) + 1.33
]

The predicted GPA is 2.27.

Regression predictions estimate average outcomes and should not be interpreted as exact values.Recognizing Nonlinear Relationships

Correlation measures linear relationships only.

If a scatterplot forms a curve instead of a straight-line pattern, the relationship is nonlinear, and the correlation coefficient may underestimate or misrepresent the strength of the relationship.

Examples of nonlinear relationships include:

• Population growth

• Compound interest

• Projectile motion

• Seasonal temperature changes

Before calculating correlation, always examine the scatterplot for curvature.

Understanding the Coefficient of Determination (R²)

The coefficient of determination, represented by R², measures how much variation in the response variable is explained by the regression model.

For example:

[
R^2 = 0.9846
]

This means:

• Approximately 98.5% of the variation in yearly income is explained by its linear relationship with age.

• Only about 1.5% of the variation is due to other factors or random variation.

A higher R² generally indicates a better-fitting regression model.

Relationship Between R² and Correlation

The correlation coefficient (r) is related to R².

The relationship is:

[
R^2 = r^2
]

Therefore,

[
r=\sqrt{0.9846}\approx0.992
]

Because the relationship is positive, the correlation coefficient is:

r ≈ 0.992

This represents an extremely strong positive linear relationship.

Understanding Residuals

A residual measures the difference between the observed value and the predicted value from a regression equation.

The formula is:

[
\text{Residual} = \text{Actual Value} – \text{Predicted Value}
]

For example:

Regression equation:

[
\hat{y}=-210+5.6x
]

If a person’s height is 72 inches, the predicted weight is:

[
-210+5.6(72)=193.2
]

If the person’s actual weight is 182 pounds, then:

[
182-193.2=-11.2
]

The residual equals:

−11.2 pounds

A negative residual means the actual value is lower than the predicted value.

Why Residuals Matter

Residuals help evaluate regression models.

Small residuals indicate:

• Better prediction accuracy

• Stronger model fit

Large residuals may indicate:

• Outliers

• Missing variables

• Poor model fit

Residual plots are commonly used to determine whether a linear model is appropriate.

Understanding Outliers

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

Outliers can occur in:

• The x-direction

• The y-direction

• Both directions

A point that lies far outside the normal range for both variables may strongly influence the regression line.

Analysts should investigate outliers before removing them because they may represent:

• Data entry errors

• Measurement errors

• Rare but valid observations

Understanding Influential Points

Not every outlier significantly changes a regression model.

An influential point is an observation that noticeably changes the regression line if removed.

Influential points typically occur when:

• The x-value is far from the rest of the data.

• The observation has high leverage.

• The point substantially shifts the slope.

Identifying influential points helps improve model reliability.

Finding the Least-Squares Regression Line

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

The slope is calculated using:

[
b=r\left(\frac{s_y}{s_x}\right)
]

where:

• r = Correlation coefficient

• sᵧ = Standard deviation of y

• sₓ = Standard deviation of x

Once the slope is known, the intercept is:

[
a=\bar y-b\bar x
]

These formulas create the regression equation:

[
\hat y=a+bx
]

This equation is widely used in statistics, economics, finance, healthcare, engineering, and machine learning.

Describing Scatterplots

Scatterplots are commonly described using three characteristics.

Form

The form describes the overall shape.

Common forms include:

• Linear

• Curved

• Exponential

• Clustered

Direction

Direction indicates whether variables move together.

Possible directions are:

• Positive

• Negative

• No direction

Strength

Strength describes how closely points follow the pattern.

Strength is generally classified as:

• Weak

• Moderate

• Strong

A scatterplot showing a moderate positive linear trend indicates that the variables generally increase together but still contain noticeable variation.

What Correlation Can and Cannot Tell You

Correlation provides valuable information about relationships between variables.

It can describe:

• Direction of association

• Strength of association

• Linear relationship

Correlation cannot determine:

• Cause-and-effect relationships

• Exact predictions

• Whether a third variable influences both variables

Regression analysis is used for prediction, while experimental research is needed to establish causation.

Understanding Lurking Variables

A lurking variable is an unobserved variable that influences both the explanatory and response variables.

For example:

A study finds that neighborhoods with more cars also have higher rates of health insurance coverage.

Owning additional cars does not directly cause people to purchase insurance.

Instead, household income influences both:

• Ability to own more vehicles

• Ability to afford health insurance

Ignoring lurking variables can lead to incorrect conclusions.

Practical Applications of Correlation and Regression

Correlation and regression are widely used across many industries.

Common applications include:

• Predicting housing prices

• Forecasting sales revenue

• Measuring advertising effectiveness

• Evaluating medical treatments

• Monitoring climate trends

• Financial risk analysis

• Educational research

• Sports performance analysis

Understanding these statistical tools helps professionals make evidence-based decisions.

Frequently Asked Questions (FAQs)

What is the difference between correlation and regression?

Correlation measures the strength and direction of a linear relationship between two variables. Regression models that relationship and predicts values of the response variable.

What does a negative correlation mean?

A negative correlation indicates that as one variable increases, the other tends to decrease.

What is a good R² value?

A higher R² indicates that the regression model explains more of the variation in the response variable. However, what qualifies as “good” depends on the field of study and the complexity of the data.

Why are residuals important?

Residuals measure prediction errors and help determine how well a regression model fits the observed data.

Can a strong correlation prove causation?

No. A strong correlation does not establish a cause-and-effect relationship. Controlled experiments are typically required to demonstrate causality.

What is an outlier?

An outlier is an observation that differs significantly from the rest of the data and may influence statistical analyses.

What is a lurking variable?

A lurking variable is an unseen factor that affects both the explanatory and response variables, potentially creating a misleading correlation.

Key Statistical Facts

• The explanatory variable appears on the horizontal (x) axis.

• The response variable appears on the vertical (y) axis.

• Correlation coefficients range from −1 to +1.

• Positive values indicate variables move together.

• Negative values indicate variables move in opposite directions.

• Correlation measures linear association, not causation.

• Regression equations predict expected outcomes.

• Residuals represent prediction errors.

• R² measures the proportion of variation explained by the regression model.

• Outliers and influential points can significantly affect regression results.

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/

NIST/SEMATECH. (2023). e-Handbook of Statistical Methods. National Institute of Standards and Technology. https://www.itl.nist.gov/div898/handbook/

OpenStax. (2023). Introductory Statistics 2e. Rice University. https://openstax.org/details/books/introductory-statistics-2e

MM207 Milestone 4

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/