Interpolation & Extrapolation using Linear Models (College Board AP® Statistics)

A linear model is any line of best fit drawn on a scatterplot

• There are many possible straight lines that could be drawn

The line of best fit shows the linear relationship between the explanatory variable, , and the response variable,

A linear model allows you to make predictions (estimates) of a -value, when given a specific -value

• The straight line is called a linear regression line ('regression line' for short)

For example, data for the price of a computer and the time it takes the computer to start up are shown below

• The regression line predicts that a computer worth £620

  • will start up in 3.4 seconds

Interpolation means using a regression line to predict a -value from a given -value

• where the value of lies within the interval of -values seen in the data

This is seen as a reliable prediction

For example, data for the price of the computer and the time it takes the computer to start up are shown below

• Predicting the start up time of a computer worth £620 is interpolation

  • because £620 lies within the interval of -values seen, from £220 to £900

Extrapolation means using a regression line to predict a -value from a given -value

• where the value of lies outside the interval of -values seen in the data

It can be thought of as extending the regression line on either side

• then using those line segments to predict values

  • e.g. from above, predicting the start up time for a computer costing over £900 would be extrapolation

This is far less reliable as you do not know how the variables relate outside of the range of data given

• The linear relationship might break down or change direction

The further you extrapolate, the less reliable the estimates become