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Product Moment Correlation Coefficient (PMCC) (AQA Level 3 Mathematical Studies (Core Maths)): Revision Note

Exam code: 1350

Naomi C

Written by: Naomi C

Reviewed by: Dan Finlay

Updated on

Product Moment Correlation Coefficient (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

    • A positive value of r describes positive correlation

      • r equals 1 means perfect positive correlation

    • A negative value of r describes negative correlation

      • r equals negative 1 means perfect negative correlation

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

    • If r equals 0 there is no correlation

    • The closer the data points are to a straight line, the greater the strength of the correlation

  • The gradient does not change the value of size 16px r

A diagram of different scatter graphs showing different types of correlation. The first three graphs show perfect positive correlation, r=1, each with a different gradient. The second set of three graphs show perfect negative correlation, r=-1, each with a different gradient. Of the last set of three graphs, the first shows a positive correlation of r = 0.7, the second shows no correlation, r = 0, and the final graph shows negative correlation, r=-0.4.

Worked Example

Two variables have a pmcc of -0.827.

Describe the correlation between the variables.

The pmcc is negative so there is a negative correlation
It is also close to -1, so it is a strong correlation

The two variables have a strong, negative correlation

Calculating PMCC

How is the product moment correlation coefficient calculated?

  • You must learn how to use your calculator to calculate value of the pmcc, begin mathsize 16px style r end style, for the correlation between two variables

  • All calculators are different and you should make sure you can calculate the pmcc on your personal calculator

    • You may need to know how to put your calculator into the statistics mode

      • You will be given the option to turn the frequency on or off, choose off for most calculations of the pmcc

    • With the statistics mode switched on on your calculator, there will be a ‘statistics’ option, followed by a regression option in the form A + BX

      • Your calculator will give you two columns into which you can input the begin mathsize 16px style x end style and begin mathsize 16px style y end style data values

    • Once the data has been entered into your calculator, choose the ‘r’ value from the ‘STAT’ options

Worked Example

The table below shows data for a sample of 8 people comparing the maximum number of pull-ups they are able to complete, x, with the maximum number of press-ups, y.

Number of pull-ups (x)

5

10

8

3

6

8

1

4

Number of press-ups (y)

24

34

36

18

30

35

11

19

(a) Work out the pmcc for the data.

Enter the data into your calculator to find the pmcc

0.955914...

Round

pmcc = 0.956 (3 s.f.)

(b) Comment on the relationship between the two variables.

Comment on the strength and type of correlation
Relate this to the actual variables in context

There is a strong, positive correlation between the number of pull-ups that a person can do and the number of press-ups that they can do

As the number of press-ups that a person can do increases, the number of pull-ups that they can do also increases

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    Author
    Reviewer
    Naomi C

    Author: Naomi C

    Expertise: Maths Content Creator

    Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject 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.