Correlation Coefficients (DP IB Applications & Interpretation (AI)): Revision Note

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 

• 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 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 

• 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 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 

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 can take any value such that 

  • 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

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

• is linked to the covariance

•  and  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