Level 3MathsAQAMathematical Studies (Core Maths)Paper 2A: Statistical TechniquesRevision NotesCorrelation & RegressionCorrelation & RegressionProduct Moment Correlation Coefficient (PMCC)

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.

Answer:

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.

Answer:

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.

Answer:

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.