Study GuidesExam QuestionsPast Papers
AP®StatisticsCollege BoardStudy GuidesUnit 2: Exploring Two-Variable DataScatterplots & RegressionAssociation & Correlation Coefficients

Association & Correlation Coefficients (College Board AP® Statistics)

Study Guide

Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

Association

What is an association?

  • An association between two variables means that the variables are related to each other in some particular way

    • A change in one variable corresponds to a change in the other

  • It is possible for two variables to have no association

    • A change in one variable does not correspond to a change in the other

What is the direction of an association?

  • If an association exists, it can have different directions:

    • Positive association is when one variable increases, the other tends to increase

      • For example, as temperature increases, sales of cold drinks tend to increase

    • Negative association is when one variable increases, the other tends to decrease

      • For example, increasing the age of a car tends to decrease its value

What is the form of an association?

  • Having a positive or negative association does not mean the association is linear (a straight line)

  • There are many different forms an association could take, for example:

    • Linear forms follow straight lines

    • Non-linear forms follow curved lines, including:

      • quadratics and cubics

      • reciprocals, e.g. y equals 1 over x

      • exponentials, e.g. y equals 2 to the power of x

Nine graphs show functions y = x, y = x², y = x³, y = 1/x, y = kˣ, y = -x, y = -x², y = -x³, and y = 1/x .
Linear and non-linear forms of association

What is the strength of an association?

  • The strength of an association is how well the data points on a scatterplot follow the form of the association

  • Strengths are described as either strong, moderate or weak

    • The stronger the strength, the more closely data points follow the form

      • e.g. data points may show a 'weak quadratic' association

What are unusual features of a scatterplot?

  • Unusual features of a scatterplot include

    • clusters

      • where data points appear to be in groups (clouds)

    • outliers

      • data points that do not appear to fit the general pattern shown

Examiner Tips and Tricks

In the exam, if asked to describe the relationship shown on a scatterplot, you should comment in context on the direction (positive, negative, none), form (linear, non-linear) and strength (strong, moderate, weak) of an association, as well as any unusual features (clusters, outliers).

Worked Example

Describe the relationship shown between the hours spent on a phone per day and the hours spent on a computer per day for nine students in a class, shown on the scatterplot below.

Scatterplot showing hours spent on a computer per day (y-axis) versus hours spent on a phone per day (x-axis).

Answer:

You must comment on the strength, direction and form of the association seen

You must also comment on unusual features, in particular outliers and clusters

Remember to give your answer in context

The scatterplot reveals a strong, negative, roughly linear association between the hours spent on a phone per day and the hours spent on a computer per day for the nine students in the class

There are no significant outliers, though there is a slight clustering of points into two clusters (top left, between 1 and 3 hours on a phone per day, and bottom right, between 5 and 9 hours on a phone per day)

Correlation coefficients

What is correlation?

  • Correlation is a numerical measure of the direction and strength of a linear association between two variables

Five scatter plots illustrating different correlations: Strong Positive, Weak Positive, No Correlation, Weak Negative, and Strong Negative.

What is the correlation coefficient?

  • The correlation coefficient, r, is a value between -1 and 1 where

    • r equals 1 means a perfect positive linear association

      • All points lie along the same straight line with a positive slope

    • r equals 0 means no linear association

    • r equals negative 1 means a perfect negative linear association

      • All points lie along the same straight line with a negative slope

  • Values in between can be described as weak, moderate or strong

    • e.g. r equals 0.9 is a 'strong positive linear' association

      • Points appear to roughly follow a straight line with a positive slope

Nine scatter plots with correlation coefficients: six perfect (r = 1 or -1) showing diagonal lines, two moderate (r ≈ 0.7 and -0.4), and one (r = 0) showing no correlation.

What is the formula for the correlation coefficient?

  • For n data points with coordinates open parentheses x subscript i comma space y subscript i close parentheses, the formula for the correlation coefficient is

    • r equals fraction numerator 1 over denominator n minus 1 end fraction sum from blank to blank of open parentheses fraction numerator x subscript i minus x with bar on top over denominator s subscript x end fraction close parentheses open parentheses fraction numerator y subscript i minus y with bar on top over denominator s subscript y end fraction close parentheses space

    • where s subscript x equals square root of fraction numerator 1 over denominator n minus 1 end fraction sum from blank to blank of open parentheses x subscript i minus x with bar on top space close parentheses squared space end root equals square root of fraction numerator sum from blank to blank of open parentheses x subscript i minus x with bar on top space close parentheses squared space over denominator n minus 1 end fraction end root is the sample standard deviation of the x-values

      • recall that x with bar on top equals 1 over n sum from blank to blank of x subscript i equals fraction numerator sum from blank to blank of x subscript i over denominator n end fraction

    • and where s subscript y equals square root of fraction numerator 1 over denominator n minus 1 end fraction sum from blank to blank of open parentheses y subscript i minus y with bar on top space close parentheses squared space end root equals square root of fraction numerator sum from blank to blank of open parentheses y subscript i minus y with bar on top space close parentheses squared space over denominator n minus 1 end fraction end root is the sample standard deviation of the y-values

      • and y with bar on top equals 1 over n sum from blank to blank of y subscript i equals fraction numerator sum from blank to blank of y subscript i over denominator n end fraction

  • However, in practice, the correlation coefficient is found using technology

    • e.g. using a calculator

Examiner Tips and Tricks

The formulas for r, s subscript x and x with bar on top are given in the exam, but the formulas for s subscript y and y with bar on top are not (though they can easily be formed by looking at s subscript x and x with bar on top).

What else do I need to know about correlation coefficients?

  • You need to know that correlation coefficients, r, are

    • always in the range negative 1 less or equal than r less or equal than 1

    • only measure strengths of linear relationships

      • so r equals 0 has no linear association, but may have a non-linear (curved) association

    • independent of units

      • changing the units of the x and y variables does not affect r

    • affected by outliers

    • not affected by swapping the axes

      • i.e. plotting y values on the x axis and vice versa

What does the phrase "correlation does not imply causation" mean?

  • If two variables appear to correlate, it does not mean that one variable causes changes in the other variable

  • For example, each day you record the height of a sunflower and the weight of a puppy

    • As the height of the sunflower increases, the weight of the puppy increases

      • This shows a positive correlation

    • But you cannot claim that:

      • 'increasing the heights of sunflowers causes puppies to weigh more'

      • or 'heavier puppies lead to taller sunflowers'!

    • Both variables are actually increasing separately due to a third variable

      • In this case, time

Last updated:

You've read 0 of your 5 free study guides this week

Sign up now. It’s free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

    Test yourself

    Did this page help you?

    Previous:ScatterplotsNext:Interpolation & Extrapolation using Linear Models
    Author
    Reviewer
    Mark Curtis

    Author: Mark Curtis

    Expertise: Maths

    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

    Author: 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.