Home


  1. Introduction
    1. Contents
      Standard
    2. What are Statistics?
      Standard         Video
    3. Importance of Statistics
      Standard         Video
    4. Descriptive Statistics
      Standard         Video
    5. Inferential Statistics
      Standard         Video
    6. Sampling Demonstration
      Standard
    7. Variables
      Standard         Video
    8. Percentiles
      Standard         Video
    9. Levels of Measurement
      Standard         Video
    10. Measurement Demonstration
      Standard
    11. Distributions
      Standard         Video
    12. Summation Notation
      Standard         Video
    13. Linear Transformations
      Standard  
    14. Logarithms
      Standard         Video
    15. Statistical Literacy
      Standard
    16. Exercises
      Standard

  2. Graphing Distributions
  3. Summarizing Distributions
  4. Describing Bivariate Data
  5. Probability
  6. Research Design
  7. Normal Distribution
  8. Advanced Graphs
  9. Sampling Distributions
  10. Estimation
  11. Logic of Hypothesis Testing
  12. Tests of Means
  13. Power
  14. Regression
  15. Analysis of Variance
  16. Transformations
  17. Chi Square
  18. Distribution Free Tests
  19. Effect Size
  20. Case Studies
  21. Calculators
  22. Glossary
 

Linear Transformations

Author(s)

David M. Lane

Prerequisites

None

Learning Objectives
  1. Give the formula for a linear transformation
  2. Determine whether a transformation is linear
  3. Describe what is linear about a linear transformation

Often it is necessary to transform data from one measurement scale to another. For example, you might want to convert height measured in feet to height measured in inches. Table 1 shows the heights of four people measured in both feet and inches. To transform feet to inches, you simply multiply by 12. Similarly, to transform inches to feet, you divide by 12.

Table 1. Converting between feet and inches.
Feet Inches
5.00
6.25
5.50
5.75 		60
75
66
69

Some conversions require that you multiply by a number and then add a second number. A good example of this is the transformation between degrees Centigrade and degrees Fahrenheit. Table 2 shows the temperatures of 5 US cities in the early afternoon of November 16, 2002.

Table 2. Temperatures in 5 cities on 11/16/2002.
City Degrees Fahrenheit Degrees Centigrade
Houston
Chicago
Minneapolis
Miami
Phoenix		 54
37
31
78
70
 12.22
2.78
-0.56
25.56
21.11

The formula to transform Centigrade to Fahrenheit is:

F = 1.8C + 32

The formula for converting from Fahrenheit to Centigrade is

C = 0.5556F - 17.778

The transformation consists of multiplying by a constant and then adding a second constant. For the conversion from Centigrade to Fahrenheit, the first constant is 1.8 and the second is 32.

Figure 1 shows a plot of degrees Centigrade as a function of degrees Fahrenheit. Notice that the points form a straight line. This will always be the case if the transformation from one scale to another consists of multiplying by one constant and then adding a second constant. Such transformations are therefore called linear transformations.

Figure 1. Degrees Centigrade as a function of degrees Fahrenheit.

Many transformations are not linear. With nonlinear transformations, the points in a plot of the transformed variable against the original variable would not fall on a straight line. Examples of nonlinear transformations are: square root, raising to a power, logarithm, and any of the trigonometric functions.

Please answer the questions:
correct feedback