Effects of Linear Transformations
This section covers the effects of linear transformations on measures of central tendency and variability.
Example
Table below shows the temperatures of 5 cities to see the Linear transformation: temperatures of cities.
| City | Degrees Fahrenheit | Degrees Centigrade |
|---|---|---|
| Houston | 54 | 12.22 |
| Chicago | 37 | 2.78 |
| Minneapolis | 31 | -0.56 |
| Miami | 78 | 25.56 |
| Phoenix | 70 | 21.11 |
| Mean | 54.000 | 12.220 |
| Median | 54.000 | 12.220 |
| Variance | 330.00 | 101.852 |
| SD | 18.166 | 10.092 |
To transform the degrees Fahrenheit to degrees Centigrade, we use the formula
C = 0.556F - 17.778
To get the mean in Centigrade, you multiply the mean temperature in Fahrenheit by 0.556 and then subtract 17.778 .
(0.556)(54) - 17.778 = 12.22.
- The same is true for the median.
- This relationship holds even if the mean and median are not identical as they are in the table above.
- The formula for the standard deviation is just as simple: the standard deviation in degrees Centigrade is equal to the standard deviation in degrees Fahrenheit times 0.556.
- Since the variance is the standard deviation squared, the variance in degrees Centigrade is equal to 0.5562 times the variance in degrees Fahrenheit.
If a variable X has a mean of μ, a standard deviation of σ, and a variance of σ2, then a new variable Y created using the linear transformation
Y = bX + A will have a mean of bμ+A, a standard deviation of bσ, and a variance of b2σ2.
Quiz
