NUM8ERS study notes · Unit 5

Regression Analysis

Explore how two quantitative variables move together. Read scatterplots, describe correlation, predict with a linear model and check the residuals. Finish by explaining how least squares chooses a line and what its fit measure means.

2026–27 curriculum5 topic lessons3 learning stages6 self-check questions

Start with Topic 5.1 →Score Calculator Prediction Paper 2027

Before you start: Review quantitative variables, graph axes, means and standard deviations from Unit 1. Be comfortable substituting into a linear equation and interpreting units.

01 · Unit 5

Explore all 5 topics

Choose a topic to open its study notes, worked examples, visuals and practice. Follow the cards in order, or jump to the skill you want to review.

Explore the relationship

Use a scatterplot and correlation to describe association before modeling.

Topic 5.1
Graphical Representations Between Two Quantitative Variables
Build a scatterplot for paired quantitative values. Describe direction, form, strength and unusual points using both variables.
Study Topic 5.1
Topic 5.2
Correlation
Interpret the sign and magnitude of correlation. Recognize its unitless scale, sensitivity to unusual points and focus on linear association.
Study Topic 5.2

Predict and check the misses

Make predictions, calculate residuals and look for patterns the line leaves behind.

Topic 5.3
Linear Regression Models
Predict a response from an explanatory value. Interpret the slope and intercept, and distinguish interpolation from extrapolation.
Study Topic 5.3
Topic 5.4
Residuals
Calculate actual minus predicted, explain the sign and units, and use residual plots to assess a straight-line model.
Study Topic 5.4

Fit and interpret the line

Understand least squares, read technology output and explain coefficients and model fit.

Topic 5.5
Least-Squares Regression
Find the fitted line using technology. Explain squared errors, the mean point, coefficient meanings and the proportion of response variation explained.
Study Topic 5.5

02 · Unit 5

The big picture

A delivery team records each trip’s distance and travel time. A scatterplot shows whether longer trips tend to take more time. Correlation describes the linear association; a fitted line predicts time from distance. Residuals show the prediction misses, helping the team judge whether a straight line is a reasonable description.

A strong association is useful evidence about a relationship, but it does not automatically establish cause or justify predictions outside the observed range. Read the graph and inspect the model’s misses before relying on a fitted equation.

By the end of this unit, you should be able to:

Construct and describe a scatterplot using the variables, form, direction, strength and unusual features.
Interpret correlation as a unitless measure of linear association.
Use a linear model for predictions and explain interpolation versus extrapolation.
Calculate and interpret residuals and assess their patterns.
Explain the least-squares criterion and interpret slope, intercept and explained variation.

03 · Unit 5

Your study roadmap

Use these stages as a learning order. During revision, choose the stage that matches the skill you need. Each stage links back to its topic cards above.

Stage 01 · Topics 5.1–5.2
Explore the relationship
Use a scatterplot and correlation to describe association before modeling.
Stage 02 · Topics 5.3–5.4
Predict and check the misses
Make predictions, calculate residuals and look for patterns the line leaves behind.
Stage 03 · Topic 5.5
Fit and interpret the line
Understand least squares, read technology output and explain coefficients and model fit.

For each lesson: read the key idea, follow a worked example, attempt practice before revealing solutions, then explain the result in your own words.

04 · Unit 5

Revise with a purpose

Choose the route that fits your goal. A correct calculation needs an explanation of the method, meaning and limitations.

Learning for the first time

Follow all three stages using one paired data set. Describe the scatterplot before calculating correlation, then connect each prediction to its residual. Open the starting lesson →

Revising predictions and interpretation

Use Topics 5.3–5.5. Write which variable is the input, keep the response units and check whether the input lies inside the observed range. Open the starting lesson →

Revising model evaluation

Use Topics 5.2, 5.4 and 5.5 together. Compare correlation, the residual pattern and explained variation; do not let one summary replace the graph. Open the starting lesson →

Choose and explain your method

Describe before calculating

Name both variables, then describe form, direction, strength and unusual points. A scatterplot can show curvature that a linear correlation summary misses.

Review Topic 5.1

Correlation is not slope

Correlation $r$ is unitless and describes linear association. A fitted slope carries response units per explanatory unit and tells you the change in a predicted response.

Review Topic 5.2

Prediction is not an observation

A model supplies $\hat{y}$ at a given $x$ . An observed response is $y$ . Their residual is $e=y-\hat{y}$ , in response units. New cases have no residual until the actual response is known.

Review Topic 5.4

Fit measures need context

The coefficient $r^2$ describes explained response variation in simple least-squares regression with an intercept. It is not the percentage of predictions that are correct, and a high value does not rule out curvature.

Review Topic 5.5

Common mistakes to catch

Claiming causation from correlation
Association alone does not establish a causal relationship. Consider the study design, possible confounding and the population represented. Review Topic 5.2
Reversing the residual subtraction
Calculate $e=y-\hat{y}$ . Positive means the model underpredicts the actual response; negative means it overpredicts. Review Topic 5.4
Interpreting an unsupported intercept
The intercept predicts the response at $x=0$ . Check whether that input is sensible and supported by the observed range before assigning physical meaning. Review Topic 5.5
Trusting a line because its fit measure is high
Inspect the scatterplot and residual plot. Systematic curvature and changing spread can remain even when the fit summary appears strong. Review Topic 5.4

05 · Unit 5

Check your understanding

Try each original question before opening its answer. These short teaching situations help you identify the step to revisit; follow the review link for more practice.

Question 1

A scatterplot shows a clear U-shaped relationship but correlation $r$ is near $0$ . Does this imply no relationship?

Check answer 1

No. Correlation measures linear association. A strong curved relationship can have $r$ near $0$ . Describe the curve rather than concluding there is no association.

Review Topic 5.2

Question 2

A delivery model is $\hat{y}=12+1.8x$ , with distance $x$ in km and time $y$ in minutes. Interpret the slope.

Check answer 2

For each additional kilometre, the model predicts an increase of $1.8$ minutes in delivery time, on average. It does not say every delivery takes exactly that much longer or establish a causal effect by itself.

Review Topic 5.3

Question 3

At $x=10$ km, that model predicts $30$ minutes. The actual time is $29$ minutes. Find and interpret the residual.

Check answer 3

$e=29-30=-1$ minute. The actual time is $1$ minute below the prediction, so the model overpredicts by $1$ minute. On a residual plot against distance, the point is $(10,-1)$ .

Review Topic 5.4

Question 4

A simple least-squares model has correlation $r=-0.7$ . Find the coefficient of determination and explain whether it shows direction.

Check answer 4

$r^2=(-0.7)^2=0.49$ . About $49\%$ of response variation is explained by its linear relationship with the explanatory variable. Squaring removes the sign, so this coefficient does not show direction; the correlation and slope do.

Review Topic 5.5

Question 5

For the same observations, line A has residuals $(-2,1,1)$ and line B has residuals $(-1,0,1)$ . Which has the smaller squared-error total?

Check answer 5

$\mathrm{SSE}_A=4+1+1=6$ and $\mathrm{SSE}_B=1+0+1=2$ . Line B is better by this criterion. Both signed totals are $0$ , showing why signed residuals alone are not a useful fitting objective. Comparing two lines does not establish the global least-squares minimum.

Review Topic 5.5

Question 6

The observed distances range from $5$ to $25$ km. Is using the fitted line at $x=40$ km interpolation or extrapolation? Would a high fit measure make it safe?

Check answer 6

It is extrapolation because $40$ is outside the observed range. A strong fit within the recorded distances does not establish that the relationship stays linear there. Seek supporting data or a justified model before relying on that prediction.

Review Topic 5.3

06 · Unit 5

Are you ready to move on?

Explain each item aloud or on paper without looking at the notes. The boxes are study-session reminders, not an assessment score.

I can describe a scatterplot with both variables and relevant context.
I can interpret correlation without confusing it with slope or causation.
I can make a prediction, check the input range and explain coefficients with units.
I can calculate a residual and assess patterns in a residual plot.
I can explain least squares and interpret explained variation correctly.

If a skill is not comfortable yet, use the roadmap to revisit the relevant topic and retry its practice. Then work on mixed questions where the method is not named for you.

07 · Unit 5

Continue learning

Complete your AP Statistics review

Complete your course review by mixing data description, probability, inference and regression questions. Choose methods from the question and design instead of relying on a topic label.

Unit 1Exploring One-Variable Data and Collecting Data Unit 2Probability, Random Variables, and Probability Distributions Unit 3Inference for Categorical Data: Proportions Unit 4Inference for Quantitative Data: Means

Return to the topic cards · Back to the unit overview

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