Correlation & Regression (Edexcel International AS Maths: Statistics 1): Revision Note

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 $-1\le r\le 1$

    • Can be written as $\left|r\right|\le 1$

  • 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

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

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

  • $S_{xx}=\mathrm{\Sigma }{x}^2-\frac{{\left(\mathrm{\Sigma }x\right)}^2}{n}$

  • $S_{yy}={\mathrm{\Sigma y}}^2-\frac{{\left(\mathrm{\Sigma y}\right)}^2}{n}$

  • $S_{xy}=\mathrm{\Sigma }xy-\frac{\left(\mathrm{\Sigma }x\right)\left(\mathrm{\Sigma }y\right)}{n}$ 

  • These are given in the formula booklet

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

  •  $S_{xx}$

    • $\mathrm{\Sigma }{\left(x-\overline{x}\right)}^2$

    • $\mathrm{\Sigma }{x}^2-n{\overline{x}}^2$

    • $n{{\sigma }_x}^2$

  • $S_{xy}$

    • $\mathrm{\Sigma }(x-\overline{x})\left(y-\overline{y}\right)$

    • $\mathrm{\Sigma }xy-n\overline{x}\overline{y}$

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

  • $r=\frac{S_{xy}}{\sqrt{S_{xx}{S}_{yy}}}$

  • This is given in the formula booklet
    

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

  • $b=\frac{S_{xy}}{S_{xx}}$ 

  • This is given in the formulae booklet

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

  • $a=\overline{y}-b\overline{x}$ 

  • This is given in the formulae booklet

  • This is found using the fact that the point $\left(\overline{x},\overline{y}\right)$ lies on the regression line

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

  • x = c + dy 

  • $d=\frac{S_{xy}}{S_{yy}}$

  • $c=\overline{x}-d\overline{y}$

  • These are not given in the formulae booklet