Reflection: Grappling with Complexity Linear Regression and Residuals - Section 2: Direct Instruction

 

To many students, a residual plot looks exactly like a scatter plot, but yet they are told that a pattern on a scatter plot is a good thing, while a pattern on a residual plot is a bad thing. Especially for lower level students, this can be extremely frustrating. At best, they might memorize the rules so that they can pass the assessments, never fully understanding the value in the distinctions and similarities.


A compare and contrast discourse can be helpful in making the distinction. Showing several examples of scatter plots (with regression equations superimposed) side-by-side with their residual plots can help students generalize the following:

  1. A residual plot is an interpretation of how much residual error exists between the data and the model on the scatter plot (i.e. “Is this the best model here?”)

  2. A Scatter Plot and its Residual Plot will always have the same horizontal label. (i.e. if the scatter plot is showing height in terms of time, then the residual plot will show residual error (for height) in terms of time)

  3. A pattern on a residual plot says that I should have been able to make a better model. Some examples are useful here:

    1. a pattern that shows residuals that are consistently low (or high), means I should have used a model that gave me lower (or higher) values.

    2. residual patterns that are symmetric or curved means that I could get a better model by making those predictable changes.

  4. The absence of a pattern on a residual plot does not necessarily mean that the model is a ‘good’ model, only that there is nothing better.


For example, show a scatter plot that has a linear regression imposed on a pattern that is clearly (or even subtley) exponential. Display the residual plot and discuss its implications. Change the model to an exponential regression and view the resulting residual plot. Also, show a scatter plot with a linear regression for data that clearly has no association. View the residual plot, noting that the lack of pattern doesn’t mean our linear model is any good… only that nothing else will be better!

  Scatter Plot vs. Residual Plot: What's the Difference?
  Grappling with Complexity: Scatter Plot vs. Residual Plot: What's the Difference?
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Linear Regression and Residuals

Unit 7: Statistics: Two Variables
Lesson 4 of 4

Objective: SWBAT create residual plots and use them to determine if a linear model is an appropriate for a given two-variable data set.

Big Idea: Examining the size and distribution of errors made by a model can help us determine if the model is appropriate.

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