Linear Transformations


Overview

Linear transformations can refer to either (1) adding a constant to each term in a dataset or (2) multiplying the dataset by a constant. These two types of transformations affect measures of center and measures of spread in different ways. Understanding how transformations will affect each type of measure is useful in many real-life situations. For example, if a certain dataset in centimeters needed to be converted to inches, one would have to use a linear transformation (multiplying by a constant) in order to make the desired change. 

 

Add Constant (x + c) 

Measures of center (mean and median) shifted by c.

Measures of spread (standard deviation and IQR) NOT impacted. 

Example: Let the data set {1, 2, 3, 4} equal x, and let the y be a new dataset where y = x + 5. Thus, y = {6, 7, 8, 9}. Compare the mean, median, standard deviation, and IQR of dataset x to dataset y. 

  mean median

standard deviation

IQR
x 2.5 2.5 1.29 2
y 7.5 7.5 1.29 2

 

Observe how the mean and median (measures of center) went from 2.5 to 7.5; they increased by the constant 5. This proves that the measures of center are shifted by the same amount as the constant. However, the standard deviation and IQR (measures of spread) are NOT affected at all.  

Adding a constant only impacts measures of center. The new measures of center are shifted by the amount of the constant.

 

Multiply by Constant (x • c)

Measures of center (mean and median) impacted.

Measures of spread (standard deviation and IQR) impacted. 

Example: Let the data set {1, 2, 3, 4} equal x, and let the z be a new dataset where z = 2x. Thus, z = {2, 4, 6, 8}.  Compare the mean, median, standard deviation, and IQR of dataset x to dataset z. 

  mean median

standard deviation

IQR
x 2.5 2.5 1.29 2
z 5 5 2.58 4

 

Observe how the mean, median, standard deviation, and IQR were all doubled. This proves that the measures of center and spread increase by the same factor as the constant. 

Multiplying by a constant impacts BOTH measures of center and spread. The new measures change by the factor of the constant.