4.2.2 Correlation & Regression

Linear Regression

What is linear regression?

If strong linear correlation exists on a scatter diagram then the data can be modelled by a linear model
- Drawing lines of best fit by eye is not the best method as it can be difficult to judge the best position for the line
The least squares regression line is the line of best fit that minimises the sum of the squares of the gap between the line and each data value
It can be calculated by either looking at:
- vertical distances between the line and the data values
  - This is the regression line of y on x
- horizontal distances between the line and the data values
  - This is the regression line of x on y

How do I find the regression line of y on x?

The regression line of y on x is written in the form $y = a x + b$
a is the gradient of the line
- It represents the change in y for each individual unit change in x
  - If a is positive this means y increases by a for a unit increase in x
  - If a is negative this means y decreases by |a| for a unit increase in x
b is the y – intercept
- It shows the value of y when x is zero
You are expected to use your GDC to find the equation of the regression line
- Enter the bivariate data and choose the model “ax + b”
- Remember the mean point $(\bar{x}, \bar{y})$ will lie on the regression line

How do I find the regression line of x on y?

The regression line of x on y is written in the form $x = c y + d$
c is the gradient of the line
- It represents the change in x for each individual unit change in y
  - If c is positive this means x increases by c for a unit increase in y
  - If c is negative this means x decreases by |c| for a unit increase in y
d is the x – intercept
- It shows the value of x when y is zero
You are expected to use your GDC to find the equation of the regression line
- It is found the same way as the regression line of y on x but with the two data sets switched around
- Remember the mean point $(\bar{x}, \bar{y})$ will lie on the regression line

How do I use a regression line?

The regression line can be used to decide what type of correlation there is if there is no scatter diagram
- If the gradient is positive then the data set has positive correlation
- If the gradient is negative then the data set has negative correlation
The regression line can also be used to predict the value of a dependent variable from an independent variable
- The equation for the y on x line should only be used to make predictions for y
  - Using a y on x line to predict x is not always reliable
- The equation for the x on y line should only be used to make predictions for x
  - Using an x on y line to predict y is not always reliable
- Making a prediction within the range of the given data is called interpolation
  - This is usually reliable
  - The stronger the correlation the more reliable the prediction
- Making a prediction outside of the range of the given data is called extrapolation
  - This is much less reliable
- The prediction will be more reliable if the number of data values in the original sample set is bigger
The y on x and x on y regression lines intersect at the mean point $(\bar{x}, \bar{y})$

Exam Tip

Once you calculate the values of a and b store then in your GDC
- This means you can use the full display values rather than the rounded values when using the linear regression equation to predict values
- This avoids rounding errors

Worked example

The table below shows the scores of eight students for a maths test and an English test.

Maths ( $x$ )	7	18	37	52	61	68	75	82
English ( $y$ )	5	3	9	12	17	41	49	97

Write down the value of Pearson’s product-moment correlation coefficient,

r

4-2-2-ib-aa-sl-linear-reg-a-we-solution

Write down the equation of the regression line of

y

x

, giving your answer in the form

y = a x + b

where

a

and

b

are constants to be found.

4-2-2-ib-aa-sl-linear-reg-b-we-solution

Write down the equation of the regression line of

x

y

, giving your answer in the form

x = c y + d

where

c

and

d

are constants to be found.

4-2-2-ib-aa-sl-linear-reg-c-we-solution

Use the appropriate regression line to predict the score on the maths test of a student who got a score of 63 on the English test.

4-2-2-ib-aa-sl-linear-reg-d-we-solution

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 a linear relationship of bivariate data
The PMCC of a sample is denoted by the letter $r$
- r can take any value such that $- 1 \leq r \leq 1$
- 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

2-5-1-pmcc-diagram-1

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

$r = \frac{S_{x y}}{S_{x} S_{y}}$

- - $S_{x y} = \sum_{i = 1}^{n} x_{i} y_{i} - \frac{1}{n} (\sum_{i = 1}^{n} x_{i}) (\sum_{i = 1}^{n} y_{i})$ is linked to the covariance
  - $S_{x} = \sqrt{\sum_{i = 1}^{n} {x_{i}}^{2} - \frac{1}{n} {(\sum_{i = 1}^{n} x_{i})}^{2}}$ and $S_{y} = \sqrt{\sum_{i = 1}^{n} {y_{i}}^{2} - \frac{1}{n} {(\sum_{i = 1}^{n} y_{i})}^{2}}$ 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

DP IB Maths: AA HL

Revision Notes