Pearson's Product-Moment Correlation Coefficient (DP IB Analysis & Approaches (AA): SL): Revision Note

PMCC

What is Pearson’s product-moment correlation coefficient?

• Pearson’s product-moment correlation coefficient (PMCC) is a way of giving a numerical value to the linear relationship of bivariate data

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

  • r can take any value in the interval $-1\le r\le 1$

  • A positive value of r describes positive correlation

  • A negative value of r describes negative correlation

  • r = 0 means there is no linear correlation

  • r = 1 means perfect positive linear correlation

  • r = -1 means perfect negative linear correlation

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

How do I calculate Pearson’s product-moment correlation coefficient (PMCC)?

• You will be expected to use the statistics mode on your GDC to calculate the PMCC

• The formula can be useful to deepen your understanding

$r=\frac{S_{xy}}{S_x{S}_y}$

• $S_{xy}=\sum _{i=1}^n{x}_i{y}_i-\frac{1}{n}\left(\sum _{i=1}^n{x}_i\right)\left(\sum _{i=1}^n{y}_i\right)$ is linked to a statistical measure known as covariance

• $S_x=\sqrt{\sum _{i=1}^n{x_i}^2-\frac{1}{n}{\left(\sum _{i=1}^n{x}_i\right)}^2}$ and $S_y=\sqrt{\sum _{i=1}^n{y_i}^2-\frac{1}{n}{\left(\sum _{i=1}^n{y}_i\right)}^2}$ are linked to the variances

• You do not need to learn this as using your GDC will be expected

When does the PMCC suggest there is a linear relationship?

• Critical values of r indicate when the PMCC would suggest there is a linear relationship

  • In your exam you will be given critical values where appropriate

  • Critical values will depend on the size of the sample

• If the absolute value of the PMCC is bigger than the critical value then this suggests a linear model is appropriate