Key takeaway: A residual is the vertical distance, with a sign, between an observed data value and the value a fitted line predicts for the same x, found by subtracting predicted from actual.
Grade 9 – Algebra 1S-ID.B.6bStatistics and Data
Definition of Residual
A residual measures how far one data point sits from a model. After you fit a line to a scatter plot, each point has an actual \(y\)-value and a predicted \(y\)-value read from the line at the same \(x\).
The leftover difference between them tells the story. When the point lies above the line, the model predicted too low and the difference is positive. When it lies below the line, the difference is negative. A value of zero means the line passes exactly through the point.
Formula and notation
Subtract the prediction from the observation:
\[ \text{residual} = y_{\text{actual}} – y_{\text{predicted}} \]
Some books write \(e = y – \hat{y}\), where \(\hat{y}\) (“y-hat”) is the value predicted by the equation of the line. The answer carries the same units as \(y\).
Key facts
- Points above the fitted line give positive values; points below it give negative values.
- For the least-squares line, these differences add up to zero.
- A plot of them with no pattern suggests a line is a good model.
- A curved pattern in that plot suggests a linear model is the wrong choice.
Residual example
A line of best fit for a sunflower is \(\hat{y} = 2.5x + 14\), where \(x\) is weeks since planting and \(y\) is height in inches. At week 6 the plant measures 33 inches. Find the residual and describe the point.
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Predicted height: \(\hat{y} = 2.5(6) + 14 = 15 + 14 = 29\) inches.
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Actual height: \(y = 33\) inches.
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Subtract: \(33 – 29 = 4\) inches.
The residual is +4 inches: the plant is 4 inches taller than the line predicts, so the point lies above the line.
Common mistake
Watch out
Leo takes a point with actual value 18 and predicted value 22, computes \(22 – 18 = 4\), and says the point is above the line.
Why it is wrong
He subtracted in the wrong order. Always take actual minus predicted: \(18 – 22 = -4\). The negative sign tells you the point is below the line, because the observed value is smaller than the prediction.
Related terms and lessons
- Linear Regression
- Line of Best Fit
- Scatter Plot
- Least-Squares Regression Line
- Correlation Coefficient
- Algebra 1 · Two-Way Tables, Scatter Plots and CorrelationLessonWorksheetQuiz
- AP Statistics · Regression AnalysisLessonWorksheetQuiz
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