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Introduction
Residuals and Model Accuracy helps you connect a mathematical rule to a result you can explain and check. Calculate observed minus predicted output. Use the sign and pattern of residuals to discuss model accuracy. This lesson belongs to linear functions and modeling and builds toward independent Algebra I problem solving.
Alignment uses 19 TAC §111.39(c). Review lessons sample their listed standards. Readiness and supporting labels describe STAAR emphasis, not whether a topic is required.
Learning objectives
- Identify the quantities and representations used in residuals and model accuracy.
- Apply the lesson's rule to solve a numerical or contextual problem.
- Explain each mathematical step using the correct vocabulary.
- Verify an answer and correct a likely error.
Prerequisite knowledge
Read coordinate pairs, calculate means, and interpret slope.
Previous lesson: Linear Regression
Key mathematical vocabulary
- variable
- A letter representing an unknown or changing quantity.
- correlation
- A measure of association, not proof of cause.
- residual
- Observed output minus predicted output.
- output
- The result of a rule at an allowed input.
- expression
- A mathematical phrase representing a value.
Formulas and rules
- a(b+c)=ab+ac
- Equivalent expressions agree for all allowed inputs.
- A solution satisfies every original condition.
Data, association, and linear models
Calculate observed minus predicted output. Use the sign and pattern of residuals to discuss model accuracy.
A scatterplot records paired quantitative observations. A fitted line describes a trend and usually does not pass through every point. Pearson correlation r measures the direction and strength of a linear association, with −1≤r≤1 when both variables have nonzero variation.
Technology can calculate m=Sxy/Sxx and b=ȳ−m x̄ for a least-squares line, and r=Sxy/√(Sxx Syy). A residual is observed y minus predicted y. A small linear correlation does not rule out a strong nonlinear pattern. Association alone cannot prove causation; confounding variables and study design matter.
A method you can explain
- Read the question and identify the known information, unknown quantities, and any domain restrictions. Select a suitable representation and tool.
- Apply the relationship explained above. Predicted value: ŷ=12(7)+22=106.
- Write each calculation or logical inference on a separate line. Keep units and signs visible. When two methods are possible, compare which one uses the given information most directly.
- Check the result in the ORIGINAL conditions, not merely the last simplified line. Prediction plus residual equals the observation.
- Interpret what the answer means. Decide whether its size, sign, and precision make sense; identify a restriction or assumption that would change the conclusion.
Five worked examples
Example 1 · Beginner
A linear model predicts ŷ=12x+22. At x=7, the observed y is 108. Find the residual.
| x | y |
|---|---|
| 1 | 34 |
| 2 | 46 |
| 3 | 58 |
| 4 | 70 |
- Predicted value: ŷ=12(7)+22=106.
- Residual means observed minus predicted: 108−106=2.
- A positive residual places the observed point 2 units ABOVE the fitted line. The residual is not the absolute error unless it is nonnegative.
Answer: 2
Verify: Prediction plus residual equals the observation.
Avoid this mistake: Interpreting correlation as proof of cause.
Example 2 · Basic
A linear model predicts ŷ=13x+22. At x=7, the observed y is 115. Find the residual.
| x | y |
|---|---|
| 1 | 35 |
| 2 | 48 |
| 3 | 61 |
| 4 | 74 |
- Predicted value: ŷ=13(7)+22=113.
- Residual means observed minus predicted: 115−113=2.
- A positive residual places the observed point 2 units ABOVE the fitted line. The residual is not the absolute error unless it is nonnegative.
Answer: 2
Verify: Prediction plus residual equals the observation.
Avoid this mistake: Reversing observed minus predicted when finding a residual.
Example 3 · Intermediate
A linear model predicts ŷ=14x+22. At x=7, the observed y is 122. Find the residual.
| x | y |
|---|---|
| 1 | 36 |
| 2 | 50 |
| 3 | 64 |
| 4 | 78 |
- Predicted value: ŷ=14(7)+22=120.
- Residual means observed minus predicted: 122−120=2.
- A positive residual places the observed point 2 units ABOVE the fitted line. The residual is not the absolute error unless it is nonnegative.
Answer: 2
Verify: Prediction plus residual equals the observation.
Avoid this mistake: Trusting distant extrapolation without checking the context.
Example 4 · Advanced
A linear model predicts ŷ=2x+23. At x=7, the observed y is 39. Find the residual.
| x | y |
|---|---|
| 1 | 25 |
| 2 | 27 |
| 3 | 29 |
| 4 | 31 |
- Predicted value: ŷ=2(7)+23=37.
- Residual means observed minus predicted: 39−37=2.
- A positive residual places the observed point 2 units ABOVE the fitted line. The residual is not the absolute error unless it is nonnegative.
Answer: 2
Verify: Prediction plus residual equals the observation.
Avoid this mistake: Interpreting correlation as proof of cause.
Example 5 · Challenge
A linear model predicts ŷ=3x+23. At x=7, the observed y is 46. Find the residual.
| x | y |
|---|---|
| 1 | 26 |
| 2 | 29 |
| 3 | 32 |
| 4 | 35 |
- Predicted value: ŷ=3(7)+23=44.
- Residual means observed minus predicted: 46−44=2.
- A positive residual places the observed point 2 units ABOVE the fitted line. The residual is not the absolute error unless it is nonnegative.
Answer: 2
Verify: Prediction plus residual equals the observation.
Avoid this mistake: Reversing observed minus predicted when finding a residual.
Technology lab: fit and inspect your data
Enter paired measurements to calculate correlation and fit linear, quadratic, and positive exponential models. Compare the rule with the actual context before choosing a model.
Common mistakes and corrections
- Interpreting correlation as proof of cause.
- Reversing observed minus predicted when finding a residual.
- Trusting distant extrapolation without checking the context.
Real-world applications
- Study the association between practice time and performance. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Predict costs from observed production data and inspect prediction errors. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Calculate observed minus predicted output. Use the sign and pattern of residuals to discuss model accuracy.
A scatterplot records paired quantitative observations. A fitted line describes a trend and usually does not pass through every point. Pearson correlation r measures the direction and strength of a linear association, with −1≤r≤1 when both variables have nonzero variation.
Frequently asked questions
What should I know before residuals and model accuracy?
Read coordinate pairs, calculate means, and interpret slope.
How can I check my work?
Prediction plus residual equals the observation. A numerical check can detect an error; when claiming an identity for all inputs, also explain the algebraic property that makes it true.
Which mistakes should I watch for?
Interpreting correlation as proof of cause. Reversing observed minus predicted when finding a residual. Trusting distant extrapolation without checking the context.
Can I use another valid method?
Yes. Explain your assumptions, show enough intermediate work to justify the result, and verify it in the original problem. Equivalent expressions and valid alternative methods receive credit.
Mathematical process practice · A.1A–A.1G
- A.1A: Study the association between practice time and performance.
- A.1B: Plan, solve, check, and interpret the written response.
- A.1C: Select paper, a calculator, a table, or a graph; explain the choice.
- A.1D: Connect symbols with a verbal explanation and another representation.
- A.1E: Organize the solution with a table, diagram, or labeled equations.
- A.1F: Explain how the representation and the algebraic rule express the same relationship.
- A.1G: Justify each operation and the reasonableness of the conclusion.
Quick revision notes
- Calculate observed minus predicted output. Use the sign and pattern of residuals to discuss model accuracy.
- Interpreting correlation as proof of cause.
- Reversing observed minus predicted when finding a residual.
- Trusting distant extrapolation without checking the context.
Practice and explain
Try all four multiple-choice questions, then show your reasoning in the written response. Objective scores and AI feedback assess different skills.
Continue learning
Original Cambridge Veritas practice. These are not official STAAR items. Cambridge Veritas is not affiliated with or endorsed by TEA.
Official Algebra I TEKS (§111.39) · TEA assessed curriculum · STAAR blueprint · TEA instructional materials