Correlation & Regression (DP IB Maths: AA HL)

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 

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