9.4: Interpreting Slope, Intercept, and Residuals
- Page ID
- 58934
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Once we’ve calculated the least-squares regression line, we want to understand what the slope and intercept really mean. And beyond that, how good our predictions are which is where residuals come in.
Let’s anchor these ideas in a story.
Story Example: Does Practice Help with Typing Speed?
Suppose a typing instructor is studying the relationship between number of practice sessions attended (x) and typing speed at the end of the month (y), measured in words per minute (wpm).
The instructor records data from 12 students, runs a least-squares regression analysis, and finds the equation:
Predicted typing speed: \( \hat{y} = 42 + 3.5x \)
Interpreting the Equation
- Slope (3.5): For every additional practice session, the model predicts an increase of 3.5 words per minute in typing speed.
- Intercept (42): If a student attends 0 practice sessions, the model predicts a typing speed of 42 wpm. (This may not be meaningful depending on context — we should always be cautious when interpreting an intercept.)
Definition: Residual
A residual is the difference between an observed value and its predicted value from the regression model:
\[ \text{Residual} = \text{Actual} - \text{Predicted} = y - \hat{y} \]
If the residual is positive, the prediction underestimated the actual value.
If the residual is negative, the prediction overestimated the actual value.
Example: Predictions and Residuals
A student attended 6 practice sessions and achieved a final speed of 62 wpm.
Prediction: \( \hat{y} = 42 + 3.5(6) = 63 \) wpm
Actual: 62 wpm
Residual: \( 62 - 63 = -1 \)
This tells us the model overestimated the student’s performance by 1 wpm.
Interactive Scatterplot: Explore the Line and Residual
Click in the box below to add a student data point. The regression line is fixed as: \(\hat{y} = 42 + 3.5x\)
The app will draw the prediction and calculate the residual for your clicked point.
What Should We Take Away?
- Slope tells us the change in predicted \( y \) for a one-unit increase in \( x \).
- Intercept tells us the predicted value when \( x = 0 \). Be cautious when interpreting these — sometimes they're outside the realistic data range.
- Residuals measure prediction error. Large residuals may indicate bad fits or outliers.
- If all the residuals are large, does that mean the model is bad?
- If the residuals are sometimes positive and sometimes negative, what does that suggest?
- Would this model be reliable for predicting the speed of someone who attends 15 practice sessions?