Chapter 6 — Front Matter
Purpose and Overview
So far in this course, you have studied how to display and summarize a single variable at a time. You have described the shape, center, spread, and outliers of one data set. But many of the most important questions in statistics involve two variables at once: Does studying more lead to higher exam scores? Does a city's average income predict its average life expectancy? Does a car's weight affect its fuel efficiency?
Chapter 6 introduces the tools for exploring relationships between two quantitative variables. You will learn to display bivariate data on a scatter plot, describe the direction, form, and strength of the relationship, measure the linear relationship with correlation, model the relationship with a line of best fit, use the model to make predictions, and measure how well the model performs using residuals.
This chapter connects directly to what you already know from Algebra I. A line of best fit is still a linear equation. Its slope still measures rate of change. Its intercept still describes a starting value. The new skill is learning how to interpret those familiar algebraic objects in a statistical context — where data are noisy, patterns are approximate, and conclusions must be stated with appropriate caution.
Opening Question
A school district wants to know whether attendance rate is related to graduation rate. They collect data from 20 high schools, recording the percentage of days students attend and the percentage of students who graduate. Is there a relationship? If so, how strong is it? Can they predict the graduation rate of a school from its attendance rate? How accurate would that prediction be? And if a school improves attendance, does that guarantee a higher graduation rate? These are the kinds of questions Chapter 6 is designed to answer.
14Sections
10Objectives
16Key Terms
14+Worked Examples
Learning Objectives
- 1Identify the explanatory variable and the response variable in a bivariate context.
- 2Construct and interpret scatter plots for two quantitative variables.
- 3Describe the direction, form, and strength of an association from a scatter plot.
- 4Explain what correlation measures and what its value indicates.
- 5State what correlation does not prove and explain the distinction between association and causation.
- 6Identify a line of best fit and interpret its slope and intercept in context.
- 7Use a linear model to make predictions.
- 8Calculate and interpret residuals.
- 9Distinguish between interpolation and extrapolation and explain why extrapolation is risky.
- 10Identify common mistakes in interpreting correlation and linear models.
Sections 6.1 – 6.5
Displaying and Describing Bivariate Data
Two Variables · Scatter Plots · Direction, Form, Strength · Outliers
Section 6.1
Two Quantitative Variables
Until now, every chapter in Part II asked you to organize, display, or summarize one variable. That work is called univariate analysis — analysis of a single variable. You described a distribution's shape, found its center, measured its spread, and identified any outliers. Chapter 6 moves into bivariate analysis — the study of two quantitative variables measured simultaneously on the same individuals. The goal is no longer simply to describe one variable. It is to understand how two variables relate to each other.
Explanatory and Response Variables
When two variables are related, statisticians often think of one variable as helping to explain or predict the other. The variable that does the explaining is called the explanatory variable, and the variable being explained is called the response variable.
| Context | Explanatory Variable (x) | Response Variable (y) |
|---|
| Education research | Hours studied per week | Score on an exam |
| Health science | Cigarettes smoked per day | Risk of lung disease |
| Environmental science | Average daily temperature | Electricity consumption |
| Sports analytics | Games played | Points scored |
| Finance | Advertising budget | Revenue |
Labeling Convention
The labels "explanatory" and "response" do not always mean that one variable causes the other. They simply reflect the direction of the question being asked. In a scatter plot, the explanatory variable is always placed on the horizontal axis (x) and the response variable on the vertical axis (y) — the same convention you used for input and output in Algebra I.
Section 6.2
Scatter Plots
A scatter plot is the standard graphical tool for displaying the relationship between two quantitative variables. Each individual in the data set is represented by a single point. The point's horizontal position reflects its value on the explanatory variable, and its vertical position reflects its value on the response variable.
How to Construct a Scatter Plot
1. Place the explanatory variable on the horizontal axis. Label it with the variable name and units.
2. Place the response variable on the vertical axis. Label it with the variable name and units.
3. Choose a scale for both axes that gives a clear picture of the data. The scale does not need to start at zero.
4. Plot each data point as a dot at coordinates (x, y) for that individual.
5. Give the scatter plot a title describing the context.
Example 6.2.1
Constructing and Reading a Scatter Plot
- ProblemA teacher records hours studied and test scores for eight students. Construct a scatter plot and describe the pattern.
Data: (1, 55), (2, 60), (3, 65), (4, 70), (5, 80), (6, 85), (7, 88), (8, 90)
- ThinkHours studied is the explanatory variable (x). Test score is the response variable (y). Place hours on the horizontal axis and scores on the vertical axis. Plot each pair as a point.
- Display
Scatter Plot: Test Score vs. Hours Studied
Horizontal axis: Hours Studied (0 to 9, increments of 1)
Vertical axis: Test Score (50 to 100, increments of 10)
Points: (1,55) (2,60) (3,65) (4,70)
(5,80) (6,85) (7,88) (8,90)
Pattern: Points rise steadily from lower-left to upper-right.
- InterpretStudents who studied more tended to score higher. The pattern is roughly linear with no obvious curves or extreme outliers.
Answer: The scatter plot shows a positive, roughly linear association between hours studied and test score, with no obvious outliers.
Section 6.3
Direction, Form, and Strength
| Feature | What It Describes | Possible Values |
|---|
| Direction | Whether y tends to increase or decrease as x increases | Positive · Negative · None |
| Form | The shape of the pattern — straight line or curve | Linear · Curved · No form |
| Strength | How closely the points follow the overall pattern | Strong · Moderate · Weak |
Describing an Association — Always Use All Three
When asked to describe an association from a scatter plot, name the direction, form, and strength together and include context. For example: "The scatter plot shows a strong, negative, linear association between car weight (lbs) and fuel efficiency (MPG)." Saying only "positive" or "negative" is incomplete.
Example 6.3.1
Describing an Association — Car Weight and Fuel Efficiency
- ProblemTen vehicles are measured for weight (lbs) and fuel efficiency (MPG). The data range from 2400 lbs at 36 MPG to 4800 lbs at 18 MPG, with values decreasing steadily in between. Describe the association.
- ThinkWeight is the explanatory variable. MPG is the response. As weight increases, check whether MPG increases, decreases, or shows no trend. Then judge how closely the points follow that pattern.
- AnalyzeDirection: As weight increases, MPG decreases — negative direction. Form: The points fall along a roughly straight path — linear. Strength: The points cluster fairly tightly around this downward line — moderately strong to strong.
- InterpretHeavier cars tend to have lower fuel efficiency, and this trend is quite consistent across the ten vehicles.
Answer: Strong, negative, linear association between car weight and fuel efficiency. Heavier cars tend to get fewer miles per gallon.
Section 6.4
Positive and Negative Association
| Positive Associations | Negative Associations |
|---|
| Height and weight in adults | Age of a used car and its resale value |
| Income level and years of education | Altitude and air temperature |
| Hours of practice and performance score | Hours of sleep and frequency of errors |
| Temperature and ice cream sales | Price of a product and quantity demanded |
Weak Association ≠ No Association
Be careful not to confuse a weak association with no association. A weak positive association still exists even if the scatter is large. "No association" means there is truly no directional trend at all — the points are completely random. "Weak" means there is a slight trend that is hard to see through the scatter.
Section 6.5
Outliers in Scatter Plots
An outlier in a scatter plot is a point that does not follow the overall pattern of the other points. Outliers here are more subtle than outliers in a single-variable distribution. A point can be unusual because it is extreme in x alone, or because it falls far away from the line that fits the rest of the data — even when neither its x nor y value appears extreme individually. The second type is more important in regression because it can strongly pull the line of best fit away from the true trend.
Example 6.5.1
Identifying an Outlier in a Scatter Plot
- ProblemA class records absences and final grades for ten students. Nine students follow a clear downward pattern: more absences, lower grade. Student 10 has 4 absences but a 96 final grade. Identify the outlier and explain why it is unusual.
- ThinkPlot all ten points. A student with 4 absences would typically score somewhere around the mid-70s based on the downward trend of the other nine. Student 10 sits far above that expected value.
- InterpretStudent 10's point (4 absences, 96 final grade) falls well above the downward trend. It is an outlier because it does not fit the pattern, not because the values themselves are extreme. A possible explanation: extenuating circumstances for the absences while the student maintained strong performance.
Answer: Student 10 (4 absences, 96 grade) is an outlier — far above the downward trend of the other nine students.
Data Ethics
Removing outliers without a valid reason — or only reporting the pattern that fits a preferred conclusion — is a form of data manipulation. Real data sometimes contain genuine unusual cases. Responsible analysis identifies outliers, investigates them, and reports them honestly rather than discarding inconvenient points.
Sections 6.6 – 6.8
Measuring and Interpreting Association
Correlation · What Correlation Measures · What Correlation Does Not Prove
Section 6.6
Correlation
Visual descriptions are useful, but statisticians prefer numerical summaries. For two quantitative variables, the correlation coefficient — written r — measures both the direction and strength of the linear relationship between them.
The Correlation Coefficient r
The correlation r is always between −1 and 1: −1 ≤ r ≤ 1
r = 1 → Perfect positive linear relationship; every point on a line with positive slope.
r = −1 → Perfect negative linear relationship; every point on a line with negative slope.
r = 0 → No linear relationship; no consistent directional trend.
| |r| Value | Interpretation |
|---|
| 0.00 – 0.30 | Weak linear association |
| 0.30 – 0.60 | Moderate linear association |
| 0.60 – 0.80 | Moderately strong linear association |
| 0.80 – 1.00 | Strong linear association |
The Formula for Correlation
Correlation Coefficient
r = (1 / n−1) · Σ [ (x_i − x̄) / s_x ] · [ (y_i − ȳ) / s_y ]
Technology Note
You do not need to memorize the formula for most high school purposes. A graphing calculator or spreadsheet will compute r for you. But understand what the formula means: it standardizes both variables using z-scores and then measures how much they tend to move together in the same direction. When both variables are above their means at the same time, the product of their z-scores is positive, increasing r.
Properties of Correlation
Correlation has no units — it does not depend on the units of measurement for either variable. Correlation is symmetric: the correlation between x and y is the same as the correlation between y and x. Correlation only measures linear association; a strong curved relationship can produce a low r. And correlation is sensitive to outliers — a single unusual point can substantially change the value of r.
Example 6.6.1
Interpreting a Correlation Coefficient
- ProblemA researcher studies average daily temperature (°F) and hot drinks sold at a café over 30 days. She reports r = −0.82. Interpret this value.
- ThinkThe sign tells us direction. The magnitude tells us strength. Both must be interpreted in context.
- InterpretThe negative sign means that on days with higher temperatures, fewer hot drinks were sold — a negative direction. The magnitude of 0.82 falls in the "strong" range, meaning the trend is quite consistent across the 30 days. Knowing the temperature provides substantial predictive information about hot drink sales.
Answer: There is a strong, negative linear association between temperature and hot drink sales. On warmer days, customers buy significantly fewer hot drinks, and this pattern holds consistently across the 30-day data set.
Section 6.7
What Correlation Measures
Correlation measures the strength and direction of the linear relationship between two quantitative variables. It reflects how well the data follow a straight-line pattern. It does not measure whether the relationship is causal, whether it is curved, or how much absolute variation exists.
Two data sets can have the same correlation but very different levels of scatter. And a strong curved relationship can produce an r close to zero — not because the variables are unrelated, but because their relationship is not linear.
Student Warning — r ≈ 0 Does Not Mean No Relationship
Never conclude "no relationship" simply because r ≈ 0. A low r means no linear relationship. Always look at the scatter plot first. If the points form a clear curve, r fails to capture it. The scatter plot is irreplaceable.
Example 6.7.1
When Low Correlation Does Not Mean No Relationship
- ProblemA scientist measures outdoor temperature (x) and bird sightings at a feeder (y). Bird activity is high at mild temperatures and low at both extremes (very cold and very hot). She reports r = −0.05. What does this tell us?
- ThinkA near-zero r suggests weak linear association. But the description of the data suggests a curved relationship — bird activity peaks in the middle of the temperature range. This is not linear, so r fails to capture it.
- InterpretThe near-zero correlation does not mean temperature and bird activity are unrelated. It means they do not have a linear relationship. The actual relationship is likely bell-shaped or quadratic. A scatter plot would reveal the true pattern that r hides.
Answer: r = −0.05 indicates no linear association, but the data likely show a nonlinear curved pattern. Correlation alone does not capture this. The scatter plot must be examined directly.
Section 6.8
What Correlation Does Not Prove
This section addresses one of the most important ideas in all of statistics: correlation does not prove causation. When two variables have a strong correlation, it is tempting to conclude that one causes the other. But this conclusion requires more than a statistical association. Consider: ice cream sales and drowning deaths are positively correlated — but eating ice cream does not cause drowning. Both increase during summer when the weather is warm. The real explanation involves a third variable: warm weather causing both.
Confounding Variables
A confounding variable (also called a lurking variable) is a variable related to both the explanatory and response variables whose presence can make it appear that one causes the other when the true relationship is more complex. Confounding variables are the reason correlation alone cannot establish causation. The only reliable way to establish causation is through a well-designed experiment with random assignment, as discussed in Chapter 2.
When data are observational — without random assignment — any correlation could reflect direct causation, reverse causation, a confounding variable causing both, or coincidence. This does not make observational data worthless; scientists use it constantly to build strong cases for causal claims. But the statistical association alone is not sufficient evidence.
Data Ethics
Claiming causation from correlation is one of the most common forms of statistical misuse. It appears in advertising ("People who use our product earn more money"), political arguments, and health claims. As a statistically literate reader, always ask: Could a confounding variable explain this? Was this an experiment or an observational study? Has the association been replicated consistently?
Example 6.8.1
Identifying a Possible Confounding Variable
- ProblemA data analyst finds that counties with more fast-food restaurants per capita have higher obesity rates (r = 0.72). Is it fair to conclude that fast food causes obesity?
- ThinkThe correlation is strong and positive, but correlation does not prove causation. Ask: could a third variable explain both the density of fast-food restaurants and the obesity rate?
- AnalyzeOne plausible confounding variable is income level. Lower-income areas may have both more fast-food restaurants (affordable, accessible food options) and higher obesity rates (limited access to fresh food, time constraints, other socioeconomic pressures). Income, food environment, and health outcomes are all interrelated.
- InterpretThe strong correlation does not prove causation. A confounding variable such as socioeconomic status may explain much of the observed association. More rigorous study designs would be needed to isolate the causal effect of fast-food density.
Answer: Correlation alone is insufficient to conclude causation. Socioeconomic status or food environment may serve as confounding variables linking fast-food density and obesity rates.
Sections 6.9 – 6.11
Lines of Best Fit and Prediction
Least-Squares Regression Line · Slope and Intercept in Context · Prediction
Section 6.9
Lines of Best Fit
When a scatter plot shows a roughly linear pattern, it makes sense to model the data with a line. The line of best fit — also called the least-squares regression line or regression line — is the line that minimizes the sum of the squared vertical distances between each data point and the line. Each distance is a residual; the least-squares line makes the sum of squared residuals as small as possible.
The Regression Equation
ŷ = a + bx
ŷ (read "y-hat") is the predicted value of the response variable.
a is the y-intercept of the line.
b is the slope of the line.
x is the value of the explanatory variable.
The hat on ŷ is essential — it signals that this is a predicted value, not an observed one.
Formulas for Slope and Intercept
Slope of the Least-Squares Line
b = r · (s_y / s_x)
Intercept of the Least-Squares Line
a = ȳ − b · x̄
Key Property
The least-squares line always passes through the point (x̄, ȳ) — the intersection of the two sample means. This is a useful check when verifying technology output: substitute x̄ into the regression equation and confirm that ŷ = ȳ.
Example 6.9.1
Finding the Line of Best Fit
- ProblemUse the hours-studied data from Example 6.2.1. Technology gives these summary statistics:
x̄ = 4.5 ȳ = 74.1 s_x = 2.45 s_y = 12.8 r = 0.993
Find the equation of the least-squares regression line.
- ThinkBegin with slope (which requires r, s_y, and s_x), then use the means to find the intercept.
- Calculate
b = r · (s_y / s_x)
= 0.993 · (12.8 / 2.45)
= 0.993 · 5.22
≈ 5.18
a = ȳ − b · x̄
= 74.1 − (5.18)(4.5)
= 74.1 − 23.31
= 50.79
- CheckDoes the line pass through (x̄, ȳ) = (4.5, 74.1)?
ŷ = 50.79 + 5.18(4.5) = 50.79 + 23.31 = 74.10 ✓
- InterpretThe equation ŷ = 50.79 + 5.18x models the relationship between hours studied (x) and predicted test score (ŷ).
Answer: The line of best fit is ŷ = 50.79 + 5.18x.
Section 6.10
Slope and Intercept in Context
A regression equation becomes meaningful only when you connect its parts to the real context of the problem. The slope and intercept are not abstract numbers — they tell a story about the relationship between the variables.
Interpretation Templates
Slope: "For each additional [one unit of x], the predicted [response variable] [increases / decreases] by [|b|] [units of y]."
Intercept: "When [x = 0], the predicted [response variable] is [a] [units of y]." — Only interpret literally if x = 0 is meaningful in context.
Example 6.10.1
Interpreting Slope and Intercept — Car Weight and MPG
- ProblemA regression line for car weight (x, in pounds) and fuel efficiency (y, in MPG) is ŷ = 47.8 − 0.007x. Interpret the slope and intercept.
- Slopeb = −0.007. For each additional pound of weight, the predicted fuel efficiency decreases by 0.007 MPG. Equivalently, for every 1,000 additional pounds, predicted MPG decreases by about 7 miles per gallon.
- Intercepta = 47.8. A car with a weight of zero pounds would be predicted to get 47.8 MPG. This is not a meaningful real-world value — no car weighs zero pounds. The intercept should not be interpreted literally; it is simply the mathematical starting point of the line.
Answer: The slope means heavier cars get lower predicted MPG at about 7 MPG per additional 1,000 pounds. The intercept of 47.8 MPG has no meaningful real-world interpretation.
Student Warning — When the Intercept Is Not Meaningful
If x = 0 is outside the range of the data or physically impossible (a car with zero weight, a person with zero height), interpret the intercept cautiously. Acknowledge that it is a mathematical artifact of extending the line, not a real prediction. Focus interpretation on the slope.
Section 6.11
Prediction with Linear Models
One of the primary uses of a regression line is prediction: using the equation to estimate the response variable for a given value of the explanatory variable. To make a prediction, substitute the given x-value into the regression equation and compute ŷ. Remember that ŷ is a prediction — an estimate based on the model — not a guarantee.
Example 6.11.1
Predicting a Test Score
- ProblemUsing ŷ = 50.79 + 5.18x, predict the test score for a student who studies 6 hours.
- Calculate
ŷ = 50.79 + 5.18(6)
= 50.79 + 31.08
= 81.87
- CheckFrom the original data, the student who studied 6 hours scored 85. Our prediction of ≈ 81.9 is close but not exact — which is expected of any model.
- InterpretA student who studies 6 hours is predicted to score approximately 81.9 points. The actual score may differ by the amount of the residual.
Answer: The predicted test score for a student studying 6 hours is approximately 81.9 points.
Example 6.11.2
Predicting Fuel Efficiency
- ProblemUsing ŷ = 47.8 − 0.007x, predict the fuel efficiency of a car weighing 3,500 pounds.
- Calculate
ŷ = 47.8 − 0.007(3500)
= 47.8 − 24.5
= 23.3 MPG
- InterpretA 3,500-pound car is predicted to get approximately 23.3 miles per gallon.
Answer: Predicted fuel efficiency for a 3,500-pound car is approximately 23.3 MPG.
Sections 6.12 – 6.14
Residuals, Model Fit, and Common Mistakes
Residuals · Interpolation and Extrapolation · Common Mistakes
Section 6.12
Residuals
Every prediction made by the regression line has some error. The residual for a data point is the difference between the actual observed y-value and the predicted ŷ-value:
Residual
Residual = y − ŷ (Actual − Predicted)
Interpreting Residuals
Positive residual: Actual > predicted. The model underestimated the response. The point sits above the line.
Negative residual: Actual < predicted. The model overestimated the response. The point sits below the line.
Residual = 0: The actual value falls exactly on the regression line.
For any least-squares line, the sum of all residuals always equals zero. Positive and negative residuals cancel exactly.
Example 6.12.1
Calculating Residuals
- ProblemUsing ŷ = 50.79 + 5.18x, calculate the residuals for the student who studied 2 hours (scored 60) and the student who studied 6 hours (scored 85).
- Calculate
Student A: x = 2, y = 60
ŷ = 50.79 + 5.18(2) = 50.79 + 10.36 = 61.15
residual = 60 − 61.15 = −1.15
Student B: x = 6, y = 85
ŷ = 50.79 + 5.18(6) = 50.79 + 31.08 = 81.87
residual = 85 − 81.87 = +3.13
- InterpretStudent A (2 hours) scored 1.15 points less than the model predicted — their point falls slightly below the line. Student B (6 hours) scored 3.13 points more than predicted — their point falls slightly above the line.
Answer: Residuals are −1.15 (for 2 hours) and +3.13 (for 6 hours).
Residual Plots
A residual plot displays the residuals on the vertical axis and the explanatory variable (or predicted values) on the horizontal axis. A well-fitting linear model produces residuals that are randomly scattered around zero, with no systematic pattern and roughly uniform spread across all x-values. If the residual plot shows a curved pattern, the linear model is not appropriate and a nonlinear model should be considered.
Residual Plot — What to Look For
Good sign: Residuals scattered randomly around zero, no visible curve, no dramatic widening or narrowing across x-values. The linear model is appropriate.
Warning sign: A curved or wave-shaped pattern in the residuals indicates that the linear model is missing a nonlinear trend. Try a quadratic or exponential model (Chapter 7).
Example 6.12.2
Interpreting a Residual Plot
- ProblemAfter fitting a linear model, a researcher's residual plot shows a clear U-shaped curve: small residuals at the left, large positive residuals in the middle, small residuals again at the right. What does this suggest?
- ThinkA U-shaped pattern in the residuals is systematic, not random. The model consistently underestimates values in the middle of the x-range, suggesting the true relationship is curved.
- InterpretThe U-shaped residual plot indicates the linear model is inadequate. The relationship between x and y appears nonlinear — likely quadratic. The researcher should consider a curved model.
Answer: A curved residual plot signals the linear model is inappropriate. A nonlinear model would likely fit the data better.
Section 6.13
Interpolation and Extrapolation
| Type | Definition | Reliability |
|---|
| Interpolation | Prediction for an x-value within the observed data range | Generally reliable; model was built on data in this range |
| Extrapolation | Prediction for an x-value outside the observed data range | Unreliable; no data to confirm the pattern continues |
Student Warning — Extrapolation Is Risky
Relationships that are linear within the observed data often stop being linear outside that range. Physical limits, biological ceilings, and practical constraints can cause a trend to level off, accelerate, or reverse entirely. For the test score data, predicting a score for a student who studies 20 hours gives ŷ = 50.79 + 5.18(20) = 154.4 — impossible on a 100-point test. The model has been taken far beyond where it makes sense. Always note whether a prediction requires extrapolation.
Example 6.13.1
Distinguishing Interpolation and Extrapolation
- ProblemA real estate model predicts home sale price (ŷ, in dollars) from living area (x, in sq ft): ŷ = 45,000 + 150x. Data covered homes from 800 to 3,500 sq ft. Classify each prediction:
(a) 2,000 sq ft (b) 500 sq ft (c) 4,000 sq ft
- ThinkThe data range is 800 to 3,500 sq ft. Predictions inside that range are interpolation; outside are extrapolation.
- Classify
(a) 2,000 sq ft → within range (800–3,500) → INTERPOLATION
(b) 500 sq ft → below the range → EXTRAPOLATION
(c) 4,000 sq ft → above the range → EXTRAPOLATION
- InterpretThe prediction for 2,000 sq ft (ŷ = $345,000) is based on data in that range and is reasonably reliable. Predictions for 500 sq ft and 4,000 sq ft are extrapolations — the linear pattern may not hold outside the observed range, so these should be used with caution or not at all.
Answer: (a) Interpolation — reliable. (b) Extrapolation — unreliable. (c) Extrapolation — unreliable.
Section 6.14
Common Correlation and Linear Model Mistakes
Mistake 1 — Confusing Correlation with Causation
"r = 0.85 proves that x causes y."
Correction: Correlation measures association, not causation. A confounding variable may explain the relationship. Only a well-designed experiment with random assignment can establish causation. Always ask: "Could something else explain this?"
Mistake 2 — Thinking r = 0 Means No Relationship
"r ≈ 0 means the variables are completely unrelated."
Correction: r ≈ 0 means no linear relationship. A strong nonlinear relationship can produce r ≈ 0. Always look at the scatter plot before interpreting r.
Mistake 3 — Ignoring Outliers
Reporting r and the regression line without checking the scatter plot for unusual points.
Correction: A single influential point can dramatically change r and the regression equation. Always examine the scatter plot for unusual points before interpreting any model.
Mistake 4 — Extrapolating Beyond the Data
Using the regression equation to predict at x = 20 when the data only covered x = 1 to 8.
Correction: The linear pattern observed in the data may not continue beyond the data's range. Always check whether a prediction requires extrapolation. Extrapolated predictions must be reported with a clear warning.
Mistake 5 — Reporting r Without Examining the Scatter Plot
Computing r = 0.78 and concluding the relationship is linear without drawing the scatter plot.
Correction: Correlation summarizes one feature of the data. The scatter plot reveals whether the relationship is linear, whether outliers are present, and whether the linear model is appropriate. Always draw or examine the scatter plot first.
Mistake 6 — Forgetting That ŷ Is a Prediction, Not a Guarantee
"The model says exactly 81.9, so the score must be 81.9."
Correction: The regression line describes the average trend. Individual data points scatter around the line. ŷ is the best estimate based on the model, but individual cases will typically differ from ŷ by the amount of the residual.
Mistake 7 — Reversing Explanatory and Response Variables
Building a model of x on y instead of y on x, or confusing which variable is being predicted.
Correction: The regression of y on x is not the same as the regression of x on y. Think carefully about which variable you are trying to predict from which before building the model.
Mistake 8 — Interpreting the Intercept When x = 0 Is Meaningless
"A car with zero pounds of weight gets 47.8 MPG."
Correction: The intercept is a mathematical requirement of the line, not always a practical prediction. If x = 0 is unrealistic, say so clearly. Focus interpretation on the slope.
Chapter 6 — Back Matter
Chapter Summary
- Sec 6.1Bivariate analysis studies two quantitative variables simultaneously. The explanatory variable (x) explains or predicts the response variable (y). Labeling does not imply causation; it reflects the direction of the question.
- Sec 6.2Scatter plots display bivariate quantitative data. The explanatory variable goes on the horizontal axis, the response variable on the vertical axis. Each individual is represented by one point at coordinates (x, y).
- Sec 6.3An association is described by its direction (positive, negative, or none), form (linear or curved), and strength (strong, moderate, or weak). Always use all three and reference the context and variable names.
- Sec 6.4Positive associations show y increasing as x increases. Negative associations show y decreasing as x increases. A weak association is not the same as no association; no association means there is truly no directional trend.
- Sec 6.5Outliers in scatter plots are points that do not follow the overall pattern. They may be extreme in x, extreme in y, or simply far from the fitted line. Outliers can strongly influence both r and the regression line.
- Sec 6.6The correlation coefficient r measures direction and strength of the linear relationship between two quantitative variables. It ranges from −1 to 1. Sign indicates direction; magnitude indicates strength. r has no units.
- Sec 6.7Correlation measures only linear association. A strong curved relationship can produce r ≈ 0. Never conclude "no relationship" from a low r without examining the scatter plot directly.
- Sec 6.8Correlation does not prove causation. A strong association could be explained by a confounding variable. Only a controlled experiment with random assignment can establish causation from observational data alone.
- Sec 6.9The least-squares regression line ŷ = a + bx minimizes the sum of squared residuals. The slope is b = r · (s_y / s_x) and the intercept is a = ȳ − bx̄. The line always passes through (x̄, ȳ).
- Sec 6.10The slope describes the predicted change in y per one-unit increase in x. The intercept describes the predicted y when x = 0. Always include direction, magnitude, units, and the word "predicted" when interpreting both.
- Sec 6.11Predictions are made by substituting a specific x into ŷ = a + bx. The result is a predicted value, not a guarantee. Individual predictions will differ from ŷ by the amount of the residual.
- Sec 6.12Residuals measure prediction error: residual = y − ŷ. Positive residuals are above the line; negative residuals are below. The sum of all residuals equals zero. A residual plot with random scatter confirms that a linear model is appropriate.
- Sec 6.13Interpolation is prediction within the data range — generally reliable. Extrapolation is prediction outside the data range — unreliable, because the linear trend may not continue. Always identify which type a prediction requires.
- Sec 6.14Eight common mistakes: confusing correlation with causation; misreading r ≈ 0; ignoring outliers; extrapolating; skipping the scatter plot; treating ŷ as exact; reversing x and y; and interpreting the intercept when x = 0 is meaningless.
Key Terms Review
Bivariate data
Data that record two variables on each individual in the study.
Explanatory variable
The variable used to explain or predict the other; placed on the x-axis.
Response variable
The variable being explained or predicted; placed on the y-axis.
Scatter plot
A graph displaying pairs of (x, y) data as points on a coordinate plane.
Direction
Whether the association between two variables is positive, negative, or neither.
Form
Whether the pattern in a scatter plot appears linear, curved, or random.
Strength
How closely the data points follow the overall pattern — strong, moderate, or weak.
Correlation (r)
A number between −1 and 1 measuring the direction and strength of the linear relationship between two quantitative variables.
Line of best fit
The least-squares regression line ŷ = a + bx that minimizes the sum of squared residuals.
Slope (b)
The predicted change in y per one-unit increase in x; direction indicates positive or negative association.
Intercept (a)
The predicted value of y when x = 0; not always meaningful in context.
Predicted value (ŷ)
The value of the response variable estimated by the linear model for a given x.
Residual
The difference between an observed value and its predicted value: y − ŷ.
Residual plot
A scatter plot of residuals versus x (or ŷ) used to assess whether the linear model is appropriate.
Interpolation
Using a model to predict within the range of the observed data — generally reliable.
Extrapolation
Using a model to predict outside the range of the observed data — generally unreliable.
Confounding variable
A third variable related to both x and y that can create a spurious apparent association between them.
Skill Practice
- What is the difference between the explanatory variable and the response variable? Give an example of each in a context of your own choosing.
- A scatter plot shows that as x increases, y decreases. Is the correlation positive or negative? How does the scatter plot tell you this?
- A researcher computes r = 0.12 for a data set. What does this value suggest about the relationship?
- Explain why r = 0 does not always mean that two variables are completely unrelated.
- True or false: If r = 0.90, then increasing the explanatory variable causes the response variable to increase. Explain your answer.
- What does a residual of +8 mean in context? What does a residual of −3 mean?
- What is the difference between interpolation and extrapolation? Which is more reliable and why?
- A student reads a headline: "Cities with more green spaces have lower rates of depression." They conclude that planting more trees would reduce depression. What is wrong with this conclusion?
- The following data record daily practice time (minutes) and performance exam scores for eight students: (10, 52), (20, 60), (30, 67), (40, 73), (50, 80), (60, 85), (70, 89), (80, 94). (a) Identify explanatory and response variables. (b) Describe the display you would construct. (c) Describe the direction, form, and strength of the association.
- An environmental researcher records weekly rainfall (inches) and fire incidents across 10 weeks. As rainfall increases from 0.1 to 3.5 inches, incidents fall steadily from 18 to 2. (a) Which variable is explanatory? (b) Describe direction, form, and strength. (c) Does a linear model seem appropriate? Explain.
- Match each correlation to the best description: r = 0.95, r = −0.41, r = 0.03, r = −0.88. Descriptions: strong negative; weak or none; strong positive; moderate negative.
- A researcher studies machine run-time per day (x) and defective parts produced (y), reporting r = 0.68. (a) Describe direction and strength. (b) Does this prove running machines longer causes more defects? Name one possible confounding variable.
- For each situation, predict whether r is likely positive, negative, or near zero: (a) sunlight hours and crop yield; (b) age of a cell phone battery and battery life per charge; (c) shoe size and IQ score; (d) daily caloric intake and body weight.
- Summary statistics for study time (x, hours) and exam score (y): x̄ = 5, ȳ = 78, s_x = 2.0, s_y = 10.0, r = 0.85. (a) Find slope b. (b) Find intercept a. (c) Write the equation. (d) Interpret the slope in context. (e) Predict the score for a student who studies 7 hours.
- A linear model for resting heart rate (beats per minute) from age (years) is ŷ = 95 − 0.4x. (a) Interpret the slope. (b) Interpret the intercept — is it meaningful? (c) Predict the heart rate for a 30-year-old. (d) Is predicting for a 120-year-old interpolation or extrapolation?
- Using ŷ = 50.79 + 5.18x from Example 6.9.1, calculate residuals for: Student A (1 hour, scored 55), Student D (4 hours, scored 70), Student H (8 hours, scored 90). Show your work for each.
- A residual plot for a data set shows that residuals are mostly positive for small and large x-values, and mostly negative for middle x-values. What does this pattern suggest about the linear model?
- A student says: "My regression model has a residual sum of zero, so all my predictions must be correct." Explain the error in this reasoning.
Real-World Applications
For each problem: identify the explanatory and response variables, interpret any model given, and address the specific questions with full sentences.
- Public Health: A county health department collects data from 12 clinics on health educators on staff (x) and vaccination rate (y, percent). The regression equation is ŷ = 58 + 3.2x. (a) Interpret the slope in context. (b) A county official says: "We should hire more health educators because this model proves they cause higher vaccination rates." Is this claim valid? Why or why not? (c) What information would you want before drawing causal conclusions?
- Finance: A financial analyst builds a model predicting quarterly profit (ŷ, millions) from advertising spending (x, millions): ŷ = 5 + 1.8x. Data covered advertising budgets from $1M to $8M. (a) Predict profit if advertising spending is $4M. (b) Predict profit if spending is $20M. Is this reliable? What risk does the analyst face in presenting this extrapolated prediction?
- Environmental Science: A researcher reports that average global CO₂ concentration (x, ppm) and average global surface temperature (y, °C) have r = 0.96 over 150 years. (a) Describe the association. (b) A skeptic says: "Correlation doesn't prove causation, so this data tells us nothing about whether CO₂ causes warming." Evaluate the logic and conclusion of this argument. (c) What would strengthen the causal case beyond the correlation?
- Sports Analytics: A coach tracks minutes of practice per week (x) and free-throw percentage (y) for 15 players. She finds r = 0.72 and the line ŷ = 45 + 0.3x. (a) Interpret the slope. (b) Predict the free-throw percentage for a player who practices 100 minutes per week. (c) The data ranged from 20 to 120 minutes per week. Is your prediction interpolation or extrapolation?
- Real Estate: A model predicts home sale price (ŷ, in $1,000s) from square footage (x): ŷ = 80 + 0.12x. The data covered homes from 1,000 to 4,000 sq ft. (a) Interpret the slope in dollars. (b) Predict the sale price for a 2,500 sq ft home. (c) A 500 sq ft studio apartment is on the market. Can this model reliably predict its price? Explain.
Error Analysis
Each problem shows a student's work or statement. Identify the error and write the correct interpretation or calculation.
Problem 1
"r = 0.81 proves that caloric intake directly causes weight gain."
Identify two problems with this statement and write corrected interpretations.
Problem 2
A student uses a model built from readers of 10–90 minutes/day to predict that a 200 min/day reader scores 142 out of 100.
Identify the error and explain what went wrong.
Problem 3
"This scatter plot of child age (x) and height (y) has r = 0.93, so growing taller causes children to get older."
What fundamental logical error has the student made?
Problem 4
"My residual plot has both positive and negative residuals, so the model is fine."
The residual plot actually shows a clear wave-shaped pattern. Explain why the student is wrong and what the pattern means.
Problem 5
A student computes r = 0.04 and concludes "temperature and bird activity are completely unrelated."
Describe a situation where this conclusion would be wrong and explain what r = 0.04 actually means.
Chapter 6 Checkpoint
Chapter 6 Checkpoint
Complete without assistance — assesses all key skills and concepts from Chapter 6
Section 1 — Vocabulary and Concepts
- Define explanatory variable and response variable. Give one example of each from health science.
- What do direction, form, and strength describe in a scatter plot? Why is it important to report all three?
- Why does correlation not prove causation? Give an example of two variables that are correlated but where causation cannot be assumed.
Section 2 — Calculation Skills
- Given summary statistics: x̄ = 12, ȳ = 45, s_x = 3.0, s_y = 9.0, r = −0.70. (a) Compute slope b. (b) Compute intercept a. (c) Write the regression equation.
- Using your equation from Question 4, predict ŷ when x = 15.
- The actual value of y when x = 15 is 24. Calculate the residual. Is the model overestimating or underestimating here?
Section 3 — Interpretation
- A model predicts annual salary (ŷ, in $1,000s) from years of experience (x): ŷ = 38 + 2.5x. Data covered employees with 0–25 years of experience. (a) Interpret the slope. (b) Interpret the intercept — is it meaningful here? (c) Predict salary for 10 years of experience. (d) Predict salary for 50 years of experience. Is this interpolation or extrapolation? Is it reliable?
- After fitting a linear model, a residual plot shows residuals randomly scattered around zero with no systematic pattern. What does this tell you about the model? If the residual plot had shown a curved pattern instead, what would you recommend?
Section 4 — Error Analysis
- A student simplifies a regression result as follows: "The correlation is 0.78, which proves that advertising causes sales to increase." Identify all errors in this statement and write a corrected interpretation.
- A model for predicting plant height (cm) from days grown is ŷ = 2 + 0.8x, built from data covering days 5 through 60. A student predicts the height on day 200. What is wrong, and what should the student have noted?
Bridge to College Statistics and Beyond
Chapter 7: Nonlinear Models and Transformations
The ideas in Chapter 6 are not just high school tools. They form the foundation of regression analysis — one of the most widely used techniques in statistics, data science, economics, psychology, medicine, and scientific research.
In college statistics, you will study regression more formally: testing whether the slope differs significantly from zero, building confidence intervals for predicted values, and checking the conditions required for regression inference (linearity, constant variance, independence, and normality of residuals). You will also encounter multiple regression, which uses two or more explanatory variables simultaneously.
In data science, regression is a central tool in predictive modeling. The steps you have learned — building a model, computing residuals, checking model fit — are exactly the steps used in machine learning pipelines, from financial forecasting to medical diagnosis.
Chapter 7 will extend your modeling skills beyond straight lines into quadratic, exponential, logarithmic, and power models. You will learn to recognize when a linear model is inappropriate and to choose a more suitable model based on the shape of the data and the behavior of the residuals.
MGU Library Connection
- Appendix G: Correlation and Regression Reference — formula sheets and notation summary
- Appendix F: Summary Statistics Formula Sheet — means, standard deviations, z-scores
- Algebra I, Chapter 5: Linear Equations and Slope — connection to the regression slope concept
- Chapter 7: Nonlinear Models and Transformations — when a line is not the right model
- Chapter 12: Sampling Distributions and Estimation — inference about regression slopes
- MGU Dictionary entries: correlation, residual, least-squares regression, confounding variable, extrapolation