Statistics and Probability · 35 minutes
Suggested grades 10–12 · Unit 1: Distributions and paired-data models · Lesson 2 of 2
A Paired-Data Model and Its Residuals
Check a proposed line against observations instead of equating fit with truth.
Mr. S Teaches · Original starter lesson · HS-STAT-02 · Grade range is planning guidance.Watch, notice, discuss
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Introduction to residuals and least squares regression
By Khan Academy
Connect observed minus predicted values to the lesson’s paired-data model.
Open on YouTube: Introduction to residuals and least squares regression (opens a new tab) ↗Residual plots | Exploring bivariate numerical data | AP Statistics | Khan Academy
By Khan Academy
Look for patterns in residuals and discuss when a straight-line model may be inadequate.
Open on YouTube: Residual plots | Exploring bivariate numerical data | AP Statistics | Khan Academy (opens a new tab) ↗What does the sign and size of a residual tell you about the prediction?
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Learning objective
Calculate residuals, assess a simple model and interpret a contextual prediction.
Before you begin
- Linear equations
- Scatter plots
Gather your materials
- Paper and pencil
- Graph paper
- Calculator for checking results
Language to know
- residual
- Observed output minus model-predicted output.
- association
- A pattern relating two measured variables.
- extrapolation
- Prediction outside the observed input range.
Understand the concept
Fictional workshop data pair practice sessions with completed sample tasks:(1,3),(2,5),(3,6),(4,9). A trial model y=2x+1 is close to these points but does not explain the cause of the pattern. Residuals show vertical prediction errors. A pattern in residuals can reveal a missing feature even when the scatter plot looks roughly linear.
Teach it step by step
- Plot the paired values and sketch the trial line y=2x+1.
- Make a table of observed output, prediction and residual for every point.
- Interpret the slope as modeled tasks per additional practice session and distinguish association from causation.
- Predict at x=5 and discuss why a prediction at x=100 is a much stronger extrapolation.
Make the thinking visible
Worked example or model response
Predictions 3,5,7,9 give residuals 0,0,−1,0. At three sessions the model overpredicts by one task. At five sessions it predicts 11; the small data set does not establish that this rate continues indefinitely.
Guided practice
For points(1,4),(2,5),(3,8), test y=2x+2 and list residuals.
Independent practice
1. A model predicts 18 while an observation is 21. Find the residual. 2. Explain why a zero residual at one point does not validate the entire model.
Quick assessment
What does a negative residual say about the model’s prediction at that input?
Answer key & assessment criteria
Answers and teaching guidance
- Guided: Predictions 4,6,8; residuals 0,−1,0.
- Independent 1:+3. 2: A model can match one observation and miss others or fail outside the observed range.
- Quick check: The observed output is below the prediction; the model overpredicts by the residual’s magnitude.
Look for: Require the learner to explain the interpretation and check, not only state a result. Distinguish an exact value from a rounded approximation and attend to the stated domain.
Make it work for your learner
More support
Use three labeled columns and compute one residual at a time.
Extension challenge
Propose a second line and compare residual patterns and total absolute error.
Verified references · Selected contributions
Standards connections
Subject-content standards
These records identify the part of a subject standard practiced here. Course placement is instructional guidance; the source’s category or band remains authoritative.
Contributes to Common Core standard
HSS.ID.B.6.a
Common Core · High school · Statistics and Probability
Regular high school mathematics standard
Use a fitted contextual function to interpret and predict paired data.
Read the official standard ↗This lesson contributes selected skills; broader applications and full standard coverage require further instruction.
Verified 2026-11-04. Selected contribution, not full attainment.
Contributes to Common Core standard
HSS.ID.B.6.b
Common Core · High school · Statistics and Probability
Regular high school mathematics standard
Assess fit by calculating and inspecting residuals.
Read the official standard ↗This lesson contributes selected skills; broader applications and full standard coverage require further instruction.
Verified 2026-11-04. Selected contribution, not full attainment.