Regression Lines (AQA Level 3 Mathematical Studies (Core Maths))

Calculating & Using Regression Lines

How do I draw a line of best fit?

• If strong linear correlation exists on a scatter diagram, then a line of best fit can be drawn

• A line of best fit can be drawn by eye following the trend of the data points

  • For a linear correlation it is a straight line

  • It must go through the mean point

  • There should be an approximately equal number of points above and below the line

What is a regression line?

• A line of best fit drawn by eye can be inaccurate, however the equation of an accurate line of best fit can be calculated

  • This is called the regression 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

  • If the regression line is calculated by looking at the vertical distances it is called the regression line of y on x

  • If the regression line is calculated by looking at the horizontal distances it is called the regression line of x on y

    • The regression line of x on y is rarely used and is not in this course

• The regression line of y on x is written in the form

  • is the gradient of the line and represents the change in for each individual unit change in

  • is the -intercept and shows the value of  for which is zero

  • The point will lie on the regression line

• You may be expected to:

  • work out the equation of the regression line from raw data using your calculator

  • draw a line of regression onto a scatter diagram

  • interpret or use a regression line to predict values

  • define and in the context of the question

How do I find the line of regression?

• Most calculators will be able to calculate the equation for the line of regression when you input raw data

  • Select the statistics function on your calculator

  • Input the and values from the given data set

  • Use the option to generate the regression line or the line

• The calculated values for and are likely to be given to a number of decimal places

  • Round to an appropriate degree of accuracy when writing them into the equation, e.g. 13 s.f.

How do I draw the regression line from the equation?

• Drawing a regression line is done in the same way as drawing a straight line graph

  • Plot the point

  • Substitute values for into the equation to find values of

  • Connect two or more found points with a straight line

• The equation of the regression line can be used to decide what type of correlation there is if there is no scatter diagram

  • If a is positive then the data set has positive correlation

  • If a is negative then the data set has negative correlation

Interpolation & Extrapolation

• The equation of the regression line can also be used to predict the value of a dependent variable (y) from an independent variable (x)

  • The equation should only be used to make predictions for y

    • Using a y on x line to predict x is not always reliable

• Predictions should only be made for values of the dependent variable that are within the range of the given data

  • Making a prediction within the range of the given data is called interpolation

  • Making a prediction outside of the range of the given data is called extrapolation and is much less reliable

• The prediction will be more reliable if the number of data values in the original sample set is bigger