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Learning Objectives:
Calculate a residual using the difference between an actual response and a predicted response.
Interpret residuals in context and use their signs to describe underprediction and overprediction.
Construct and read residual plots.
Describe what residual plots reveal about whether a linear model is appropriate.
Use apparent randomness or visible patterns in residual plots to judge model fit.
Essential Knowledge:
The residual is the difference between the actual value and the predicted value: residual = y - ŷ.
A positive residual means the actual response is greater than predicted, so the model underestimated.
A negative residual means the actual response is less than predicted, so the model overestimated.
A residual plot graphs residuals against explanatory-variable values or predicted response values.
Apparent randomness in a residual plot is evidence that a linear model is appropriate.
Visible patterns in a residual plot suggest the selected model may not fit well.
The difference between an actual response value and the model's predicted response value.
Actual Value
The observed y-value from the data set.
Predicted Value
The value ŷ produced by the regression model.
Residual Plot
A graph of residuals against explanatory-variable values or predicted response values.
Positive Residual
A residual above zero, showing the actual value is greater than predicted.
Negative Residual
A residual below zero, showing the actual value is less than predicted.
Underestimate
When a model predicts too low, which gives a positive residual.
Overestimate
When a model predicts too high, which gives a negative residual.
Model Fit
How well a selected model describes the relationship in the data.
Apparent Randomness
Scatter with no clear pattern, often centered around zero in a residual plot.
Pattern
A visible structure, such as curvature, that suggests the model missed something.
Centered at Zero
Residuals are balanced around 0 rather than mostly above or below it.
Video 1: Calculating and Interpreting Residuals
[0:00 – 5:42]
1.[0:00]
Topic 2.7 explores calculating and interpreting as well as constructing .
2.[0:14]
The lesson goals are to calculate a , interpret a , and construct .
Healthy Food Access Context[0:22 – 1:47]
Linda Saucedo studied supermarket locations in San Antonio, Texas. She compared average household income in a store's zip code with the number of organic items offered, then fit a least-squares regression line to see whether neighborhood income helps predict access to healthy foods.
3.[1:35]
Linda's scatterplot showed a trend: as average household went up, the number of organic items tended to go up too.
4.[1:47]
A residual is the difference between the response value and the model's response value, found by minus .
Residual Formula and Meaning[1:47 – 3:36]
residual = y - ŷ
y is the actual observed response.
ŷ is the predicted response from the model.
A positive residual means the model underestimated the actual value.
A negative residual means the model overestimated the actual value.
5.[1:58]
In the example for zip code 78240, the average household income used for x was $ and the actual store offered organic items.
6.[2:29]
Substituting x = 66,730 into the regression model gave a predicted response of organic items.
7.[3:06]
The residual was calculated as 84 minus 51.4, which equals .
8.[3:06]
Because the residual was , the actual number of organic items was greater than predicted.
9.[3:36]
A positive residual means the model the actual response value.
10.[3:36]
A negative residual means the model the actual response value.
11.[3:54]
In a residual plot, the y-axis shows instead of the original .
12.[4:58]
Residual plots help us get a better look at how well the model the data.
13.
Reflection: What does a residual tell you about a single data point, and how does the sign of the residual help you interpret the model's prediction?
Video 2: Using Residual Plots to Judge Fit
[0:00 – 4:31]
Assessing Model Fit[0:05 – 1:18]
The second video uses residual plots to decide whether a linear model is appropriate. Instead of focusing on the original y-values, it focuses on the leftover residuals so patterns in the fit become easier to see.
14.[0:05]
This video uses to determine whether linear models are for different data sets.
15.[1:47]
In a good-fitting residual plot, we see apparent with scatter centered at and no clear .
16.[2:04]
Random residual suggests the model captured the linear in the data.
Reading Residual Plots[1:47 – 4:13]
Apparent randomness centered around zero is a good sign for linear model fit.
A visible pattern, such as curvature, is a warning sign.
Patterns suggest the residuals are not just random noise.
Residual plots help us investigate whether a selected model is appropriate.
17.[2:46]
A bad residual plot can show a definite , such as up, down, up, which suggests .
18.[3:01]
When residuals show curvature, a simple model may not be the best .
19.[3:07]
If residuals show patterns, we might consider models or other modeling procedures.
20.[3:31]
In the original San Antonio example, the residual plot showed random scatter centered at , but there were still hints of a .
21.[4:06]
Apparent randomness is a sign for linear model fit, while patterns among residuals are a sign.
22.[4:13]
When a residual plot shows a clear pattern, we should question whether the selected model is .
23.
Reflection: How can a residual plot tell you whether a linear model is appropriate, and what would make you question the model?
Exit Ticket
A linear model predicts the number of organic items in a grocery store from the average household income in the store's zip code. For one store, the model predicts 51.4 organic items when x = 66,730, but the actual store offers 84 organic items. A residual plot for the full data set shows points scattered randomly around 0 with no clear pattern.
(a) Calculate the residual for this store.
(b) Interpret the residual in context, and state whether the model underpredicted or overpredicted.
(c) Explain what the residual plot suggests about whether a linear model is appropriate for the full data set.
(d) Describe what kind of residual-plot pattern would make you doubt that a linear model is a good fit.
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