# 1.3.2 Correlation & Regression

## 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
• Can be written as
• A positive value of r describes positive correlation
• A negative value of 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
•
• These are given in the formula booklet
• These are related to variance and can be written in several different ways:
•
• The product moment correlation coefficient (PMCC) is then calculated using the formula
• 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
•
• This is given in the formulae booklet
• The y-intercept a of the regression line is calculated using the formula
•
• This is given in the formulae booklet
• This is found using the fact that the point  lies on the regression line
• If you are asked to find the equation of the regression line of x on y
• x = c + dy
• 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,  seconds, and the distance that they can kick the football,  metres.

Ashika calculates the following summary statistics:

.

(a)
Calculate .

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

(c)
Calculate the equation of the regression line of  on giving your answer in the form
(a)
Calculate .

(b)
Calculate the product moment correlation coefficient between  and .
(c)
Calculate the equation of the regression line of  on giving your answer in the form

#### Exam 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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