Linear transformations occur when our data is transformed from one set of units/measurements to
another. Common examples of linear transformations include:
Currency exchanges (i.e. Canadian to US dollars)
Temperature conversions (i.e. degrees Celsius to degrees Fahrenheit)
Weight changes (i.e. pounds to kilograms)
Calculating a Z-score!
We can transform our data by adding a constant to each observation, and/or multiplying each
observation by a constant.
Adding a Constant
Consider the following data set: 2, 4, 6, 8, 10.
The mean, median, variance, and standard deviation of this data set is:
Consider what happens when we add a constant (say, 3) to each observation, such that our new data
set is: 5, 7, 9, 11, 13. The mean, median, variance, and standard deviation of the new data set is:
In general, adding a constant to every observation in the data set will change the Measures of Centre,
but not the Measures of Variability.
Multiplying by a Constant
Consider our first data set again: 2, 4, 6, 8, 10
Now, consider what happens when we multiply each observation in the data set by a constant (say, 2),
such that our new data set is: 4, 8, 12, 16, 20.
The mean, median, variance, and standard deviation of the new data set is: