Residual
In short: A residual is the actual value of the response variable minus the value predicted by a model, so it measures how far a data point sits from the fitted line.
Algebra 1StatisticsS-ID.B.6b
What is Residual?
A residual is the leftover error at one data point. After a line is fitted to a scatter plot, each point is either above the line, below it, or right on it, and the vertical gap is that point’s error.
Formally, the gap equals \(y – \hat{y}\), where \(y\) is the observed value and \(\hat{y}\) is the value the line predicts at the same \(x\). A positive result means the model predicted too low; a negative result means it predicted too high.
How to spot it
- A problem gives a table of data and a fitted equation, then asks how well the equation matches one particular row.
- You see the phrase observed minus predicted, or the notation y minus y-hat.
- A plot of errors is shown, with dots scattered above and below a horizontal line at zero.
- The question asks whether a linear model is appropriate or which point the line misses by the most.
The method
- Substitute the given \(x\) into the model equation to get the predicted value \(\hat{y}\).
- Read the observed value \(y\) for that same \(x\) from the table or the graph.
- Subtract: observed minus predicted, \(y – \hat{y}\), keeping the sign.
- Interpret the sign: positive means the point lies above the line, negative means below.
Residual: solved example
Problem
A robotics club in Ohio fits the model \(\hat{y} = 4x + 7\) to predict the number of parts \(y\) a team assembles after \(x\) hours. At \(x = 6\) hours the team actually assembled 38 parts. Find the residual.
Solution
Predicted value: \(\hat{y} = 4(6) + 7 = 31\).
Observed minus predicted: \(38 – 31 = 7\). The team did better than the line expected, by 7 parts, so the point sits above the line and the answer is +7.
Try it
Question 1
A model predicts \(\hat{y} = 52\) at \(x = 9\), but the data table shows \(y = 47\). Find the error and say where the point lies.
Show the answer
\(47 – 52 = -5\). The point is 5 units below the line.
Question 2
For \(\hat{y} = 2.5x – 4\), a student measured \(y = 16\) at \(x = 8\). Compute the residual.
Show the answer
\(\hat{y} = 2.5(8) – 4 = 16\), so the result is \(16 – 16 = 0\): the line passes exactly through that point.
Residual vs. Line of Best Fit
The line of best fit is the model drawn through the whole cloud of points. A residual is a single number attached to a single point, the vertical distance between that point and the line. Choosing the best line means making these gaps as small as possible overall; each one only grades one observation.
Keep going
Lessons
Worksheets and quizzes
- Scatter Plots, Lines of Fit and Two-Way Tables – Worksheet
- Scatter Plots, Lines of Fit and Two-Way Tables – Quiz
- Least-Squares Regression and Residuals – Worksheet
- Least-Squares Regression and Residuals – Quiz
Related words
- Linear Regression
- Line of Best Fit
- Scatter Plot
- Correlation Coefficient
- Least-Squares Regression Line
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