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AP®StatisticsCollege BoardStudy GuidesUnit 2: Exploring Two-Variable DataScatterplots & RegressionResiduals

Residuals (College Board AP® Statistics) : Study Guide

Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

Updated on

Residuals

What are residuals?

  • A residual of a data point on a scatterplot is its vertical distance above the regression line

    • A positive residual means the point lies above the regression line

    • A negative residual means the point lies below the regression line

  • When a residual is positive, the regression line underestimates the y-value of that data point

    • whereas when a residual is negative, the regression line overestimates it

What is the formula for calculating a residual?

  • The formula for calculating a residual is

    • residual = y value of the data point - y value of regression line

      • i.e. residual = actual y-value - predicted y-value

Worked Example

A scatterplot and regression line are shown below. Calculate the residual for each of the five data points.

A scatterplot with points shown and a dashed regression line on a grid.

Answer:

The residuals are the numbers shown in brackets on the diagram below

The residuals +2, -3, 0, +3, -2 shown between data points on a scatterplot and the regression line.

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    Author
    Reviewer
    Mark Curtis

    Author: Mark Curtis

    Expertise: Maths Content Creator

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.