Lines of Best Fit & Regression Lines (Edexcel GCSE Statistics)

Line of Best Fit Basics

What is a line of best fit?

• If a scatter graph suggests that there is a positive or negative correlation

  • a line of best fit can be drawn on the scatter graph

    • This can then be used to make predictions

How do I draw a line of best fit?

• A line of best fit can often be drawn by eye

  • It is a straight line (use a ruler!)

  • It must extend across the full data set

  • There should be roughly as many points on either side of the line (along its whole length)

  • The spaces between the points and the line should roughly be the same on either side

• If there is one extreme value (outlier) that does not fit the general pattern

  • then ignore this point when drawing a line of best fit

What is the double mean point?

• A question may talk about the double mean point

  • This is the point

    • is the mean of the data values that are plotted along the x-axis

    • is the mean of the data values that are plotted along the y-axis

  • The question may give you the values of and

    • Or you may need to calculate the means from the data

• If a question mentions the double mean point, then the line of best fit must go through the double mean point

  • It should still follow all the other rules for drawing a line of best fit (roughly same number of points on each side, etc.)

• If a question doesn't mention the double mean point

  • then you don't need to calculate it or worry about drawing the line through it

How do I use a line of best fit?

• The line of best fit can be used to predict the value of one variable from the other variable

  • See the Worked Example

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

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

    • This will normally give a reliable result

  • Making a prediction outside of the range of the given data is called extrapolation 

    • This is much less reliable

What about the gradient and y-intercept of a line of best fit?

• You need to be able to interpret the meaning of the gradient and y-intercept of a line of best fit

• The gradient is the slope or 'steepness' of the line

  • A question may tell you the gradient of the line of best fit

  • If you need to find it you can calculate it using 'rise over run'

    • Pick two points on the line with coordinates and

    • 

    • Be careful – the plotted data points will usually not be points on the line!

• The gradient of the line of best fit tells you the rate of change of the y-axis variable with respect to the x-axis variable

  • This needs to be interpreted in context

    • For example if the x-axis variable is distance travelled in a taxi (in miles) and the y-axis variable is the cost of the taxi ride (in pounds £)

    • then the gradient of the line of best fit (£ per mile) is the cost in pounds for increasing the distance travelled by 1 mile

• The y-intercept is the value of the y-coordinate at the point where the line crosses the y-axis

  • This can be read off the graph

• The y-intercept of the line of best fit tells you the value of the y-axis variable when the x-axis variable is equal to zero

  • This needs to be interpreted in context

    • For example if the x-axis variable is distance travelled in a taxi (in miles) and the y-axis variable is the cost of the taxi ride (in pounds £)

    • then the y-intercept of the line of best fit tells you the 'flat fee' that is added onto every taxi ride

Regression Lines

What is a regression line?

• Statistical software can calculate the equation for an 'ideal' line of best fit

  • This 'ideal' line of best fit is known as a regression line

    • It is more accurate than a line of best fit drawn by eye

  • You do not need to calculate the equation for a regression line

    • It will be given to you in the question

    • You need to be able to use and interpret it

• The equation of a regression line will be given in the following form

  • 

    • is the y-intercept of the regression line

    • is the gradient of the regression line

    • Both of those have the same meaning that they do for any line of best fit

• You may be asked to draw a regression line onto a scatter diagram

  • You need to know two points on the line

    • Choose two values (they don't need to correspond to any data values!)

    • Substitute into the equation of the regression line to find the corresponding values

  • Plot those two points on the scatter diagram and draw a straight line through them

    • Use a ruler!

• A regression line drawn from its equation will always go through the double mean point for the data set

  • You may be required to use this fact in an exam question