The document discusses different types of data transformations that can be used when a scatterplot shows a non-linear regression line. It describes squared, log, and reciprocal transformations and explains how each transformation affects data values. The best transformation is the one that results in the highest r-squared value and a randomly scattered residual plot, indicating a linear relationship between the variables after transformation. The document provides examples of applying each type of transformation.

1. Further Maths
Data Transformation
K McMullen 2012

2. Data Transformations
When a scatterplot is non-linear we can linearise the
regression line by transforming the data using one of
the following transformations:
Squared transformation
Log transformation
Reciprocal transformation
Always use the table of transformations to help you
choose the right transformations
Once you have chosen your transformations, the one
with the highest r2 value is the best transformation
given that it’s residual plot is randomly scattered
(which should be the case is r2 is high)
K McMullen 2012

3. Data Transformations
Squared transformations
Has the effect of decreasing values less than 1
and increasing values greater than 1
The effect of the squared transformation is to
stretch the values
Example of x2 transformation:
Example of y2 transformation:
K McMullen 2012

4. Data Transformations
Log transformation: reduces all values, and
values between 0 and 1 become negative
Large values are reduced much more than small
values
The effect of the log transformation is to
compress the values
Example of a log (x) transformation:
Example of a log (y) transformation:
K McMullen 2012

5. Data Transformations
Reciprocal transformations: reduces all values
greater than 1
Large values are reduced much more than small
values
The effect of the reciprocal transformation is to
compress the large values to and even greater
extent than the log transformation
Example of a 1/x transformation:
Example of a 1/y transformation:
K McMullen 2012

6. Data Transformations
To do a transformation:
Draw your scatterplot
Look at the table of transformations to decide which
transformations to try
For each transformation:
Transform your x or y values depending on the
transformation
Calculate the r2 value for each transformation
Draw the residual graph against your x values (if you
transformed x then you use the transformed values) for each
transformation and comment (remember that residual values
are always on your y-axis)
Compare each transformation and decide which is best
When writing your equation make sure you use the
transformed variable in your answer and calculations
K McMullen 2012