Pearson's Product Moment Correlation Coefficient (PMCC) (Edexcel GCSE Statistics)

Pearson's PMCC

What is Pearson’s product moment correlation coefficient (PMCC)?

• Pearson’s product moment correlation coefficient (PMCC) measures the strength of the linear correlation between two data sets

  • i.e., to what extent does one always go up when the other one goes up (or always go down when the other one goes up)

  • AND to what extent a scatter diagram of the data points lie along a straight line

    • This is what it means for the correlation to be linear

    • The line in question is the linear regression line for the data points

• The notation for the Pearson’s product moment correlation coefficient of a sample is usually  

• Pearson's PMCC is always a number between -1 and 1

  • i.e.   

    • This is the same as for Spearman's rank correlation coefficient

• The value of tells you about the type and strength of any correlation

  • A positive value () means there is positive correlation between the data sets

    • An value close to 1 means strong linear positive correlation

    • If , there is perfect linear positive correlation

      • i.e., the points lie exactly along a straight line that slopes 'up to the right' (positive gradient)

  • If is zero (), then there is no linear correlation

  • A negative value () means there is negative correlation between the data sets

    • An value close to -1 means strong linear negative correlation

    • If , there is perfect linear negative correlation

      • i.e., the points lie exactly along a straight line that slopes 'down to the right' (negative gradient)

  • In general, the closer to 1 or -1 that is, the stronger the linear correlation between the data sets

• Note that you will not be asked to calculate a Pearson's PMCC value on the exam

  • However you need to be able to interpret any values that are given

Spearman's Versus Pearson's Correlation Coefficients

What is the difference between Pearson's PMCC and Spearman's rank correlation coefficient?

• Pearson’s PMCC () tests for a linear relationship between two data sets

  • i.e. how close the data points are to lying along a straight line

  • It is most suitable for data that shows linear correlation

• Spearman’s rank correlation coefficient () tests for an 'always increasing' or 'always decreasing' relationship between two data sets

  • It will not tell whether a line or some other curve best fits the data

  • It is most suitable for data that shows non-linear correlation

How are Pearson’s and Spearman’s correlation coefficients connected?

• If there is strong linear correlation then there is also strong correlation in general

  • If is close (or equal) to 1 then will also be close (or equal) to 1

  • If is close (or equal) to -1 then will also be close (or equal) to -1

• However the opposite is not true

  • It is possible for to be equal to 1 (or -1) but for to be different

    • For example data that lies along an increasing curve

      • as the points are always increasing

      • as the points do not lie on a straight line

• Pearson's and Spearman's correlation coefficients will both be positive if the data has positive correlation (and will both be negative if the data has negative correlation)

  • However, if the data points do not lie close to a straight line then the Spearman's coefficient will be closer to 1 (or -1) than the Pearson's coefficient