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Correlation & Regression (Edexcel International A Level Maths) : Revision Note

Dan Finlay

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PMCC

What is the product moment correlation coefficient?

  • The product moment correlation coefficient (PMCC) is a way of giving a numerical value to linear correlation of bivariate data

  • The PMCC of a sample is denoted by the letter r

    • r can take any value such that negative 1 less or equal than r less or equal than 1

      • Can be written as vertical line r vertical line less or equal than 1

    • A positive value of r describes positive correlation

    • A negative value of r  describes negative correlation

    • If r = 0 there is no correlation

    • r = 1 means perfect positive correlation and r = -1 means perfect negative correlation

    • The closer to 1 or -1, the stronger the correlation

  • The gradient of the regression line does not change the value of r

2-5-1-pmcc-diagram-1

How is the product moment correlation coefficient (PMCC) calculated?

  • For n pairs of bivariate data (x, y) we define the following statistics

    • begin mathsize 16px style S subscript x x end subscript equals straight capital sigma x squared minus open parentheses straight capital sigma x close parentheses squared over n end style

    • begin mathsize 16px style S subscript y y end subscript equals Σy squared minus open parentheses Σy close parentheses squared over n end style

    • begin mathsize 16px style S subscript x y end subscript equals straight capital sigma x y minus fraction numerator open parentheses straight capital sigma x close parentheses open parentheses straight capital sigma y close parentheses over denominator n end fraction end style 

    • These are given in the formula booklet

  • These are related to variance and can be written in several different ways:

    •  begin mathsize 16px style S subscript x x end subscript end style

      • begin mathsize 16px style straight capital sigma open parentheses x minus x with bar on top close parentheses squared end style

      • begin mathsize 16px style n sigma subscript x squared end style

    • size 16px S subscript size 16px x size 16px y end subscript

      • begin mathsize 16px style straight capital sigma x y minus n x with bar on top space y with bar on top end style

  • The product moment correlation coefficient (PMCC) is then calculated using the formula

    • begin mathsize 16px style r equals fraction numerator S subscript x y end subscript over denominator square root of S subscript x x end subscript S subscript y y end subscript end root end fraction end style

    • This is given in the formula booklet

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Calculating Regression Line

If the PMCC is close to 1 or -1 then this suggests the data follows a linear model. In this case a regression line of the form y = a + bx is appropriate.

How do I calculate the equation of the regression line of y on x?

  • The gradient b of the regression line is calculated using the formula

    • b equals S subscript x y end subscript over S subscript x x end subscript 

    • This is given in the formulae booklet

  • The y-intercept a of the regression line is calculated using the formula

    • a equals y with bar on top minus b x with bar on top 

    • This is given in the formulae booklet

    • This is found using the fact that the point open parentheses x with bar on top comma space y with bar on top close parentheses lies on the regression line

  • If you are asked to find the equation of the regression line of x on y

    • x = c + dy 

    • d equals S subscript x y end subscript over S subscript y y end subscript

    • c equals x with bar on top minus d y with bar on top

    • These are not given in the formulae booklet

Worked Example

Ashika is a football coach to 20 children. She records how long it takes each of them to run a lap of the football pitch, p seconds, and the distance that they can kick the football, d metres. 

Ashika calculates the following summary statistics:

 S subscript p p end subscript equals 687.2        p with bar on top space equals 62.8       straight capital sigma d equals 1566       straight capital sigma d squared equals 124240       straight capital sigma p d equals 99127.   

(a) Calculate S subscript p d end subscript.

 

(b) Calculate the product moment correlation coefficient between p and d.

 

(c) Calculate the equation of the regression line of d on p giving your answer in the form d equals a plus b p                

1-3-2-correlation-regression-we-solution-part-1
1-3-2-correlation-regression-we-solution-part-2
1-3-2-correlation-regression-we-solution-part-3

Examiner Tips and Tricks

  • Questions typically use different variables instead of x and y. It might help to label the independent variable as x and the dependent variable as y, this will help you when calculating the equation of the regression line.

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    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.