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

Revision Note

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Author

Dan

Last updated

10 October 2023

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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 $- 1 \leq r \leq 1$
  - Can be written as $| r | \leq 1$
- 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
- $S_{x x} = Σ x^{2} - \frac{{(Σ x)}^{2}}{n}$
- $S_{y y} = {Σy}^{2} - \frac{{(Σy)}^{2}}{n}$
- $S_{x y} = Σ x y - \frac{(Σ x) (Σ y)}{n}$
- These are given in the formula booklet
These are related to variance and can be written in several different ways:
- $S_{x x}$
  - $Σ {(x - \bar{x})}^{2}$
  - $Σ x^{2} - n {\bar{x}}^{2}$
  - $n {σ_{x}}^{2}$
- $S_{x y}$
  - $Σ (x - \bar{x}) (y - \bar{y})$
  - $Σ x y - n \bar{x} \bar{y}$
The product moment correlation coefficient (PMCC) is then calculated using the formula
- $r = \frac{S_{x y}}{\sqrt{S_{x x} S_{y y}}}$
- This is given in the formula booklet

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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
- $b = \frac{S_{x y}}{S_{x x}}$
- This is given in the formulae booklet
The y-intercept a of the regression line is calculated using the formula
- $a = \bar{y} - b \bar{x}$
- 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
- $d = \frac{S_{x y}}{S_{y y}}$
- $c = \bar{x} - d \bar{y}$
- 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:

$S_{p p} = 687.2$ $\bar{p} = 62.8$ $Σ d = 1566$ $Σ d^{2} = 124240$ $Σ p d = 99127$ .

(a)

Calculate

S_{p d}

(b)

Calculate the product moment correlation coefficient between

p

and

d

(c)

Calculate the equation of the regression line of

d

p

giving your answer in the form

d = a + b p

(a)

Calculate

S_{p d}

1-3-2-correlation-regression-we-solution-part-1

(b)

Calculate the product moment correlation coefficient between

p

and

d

1-3-2-correlation-regression-we-solution-part-2

(c)

Calculate the equation of the regression line of

d

p

giving your answer in the form

d = a + b p

1-3-2-correlation-regression-we-solution-part-3

Examiner Tip

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.

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