Correlation & Regression (Edexcel IAL Maths)

PMCC

Spec point — 4.3

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 $-1\leq r\leq1$

    • Can be written as $\mid r \mid \le 1$

  • 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 _ x x end _ = capital sigma x ^2 - ( capital sigma x ) ^2 / n
  • $S_{yy} = Σy^{2} - \frac{( Σy )^{2}}{n}$
  • S_ x y end_=capital σ x y - ((capital σ x)(capital σ y))/(n)
  • These are given in the formula booklet

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

  • $S_{xx}$

    • capital sigma ( x - x -bar ) ^2
    • $Σx^{2}-n\overline{x}^{2}$
    • $nσ_{x}^{2}$

  • $S_{xy}$

    • $Σ(x-\overline{x})(y-\overline{y})$
    • $Σxy-n\overline{x}\overline{y}$

• 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

Spec point — 4.1

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 $( \bar{x} , \bar{y} )$ 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

Worked Example Ashika is a football coach to 20 children. She records how long it takes each of them to run a lap of the football pitch, $p$ seconds, and the distance that they can kick the football, $d$ metres.

Ashika calculates the following summary statistics:

$S_{pp}=687.2$ $\overline{p}=62.8$ $Σd=1566$ $Σd^{2}=124240$ $Σpd=99127$.

(a) Calculate $S_{pd}$.

(b) Calculate the product moment correlation coefficient between $p$ and $d$.

(c) Calculate the equation of the regression line of $d$ on $p$ giving your answer in the form $d=a+bp$

Answer:

Exam Hint

• Questions typically use different variables instead of x and y. It might help to label the independent variable as x and the dependent variable as y, this will help you when calculating the equation of the regression line.