Linear Transformations
Worksheet 1
Weights, in pounds, of 6 football players: 250, 280, 290, 310, 330, 280.
Find the mean weight, the variance of the weights and the standard deviation of the weights. Include the unit of measurement.
A Canadian is interested in the mean and standard deviation. Canadians don't know "pounds," they know kilograms, which are metric units. To convert the data X
_{i}
to kilograms X
_{i}
^{*}
use X
_{i}
^{*}
= 0.455X
_{i}
. For example, since X
_{1}
= 250, X
_{i}
^{*}
= 0.455250 = 113.8 kg. List the remainder of the data
in kilograms
.
Compute the mean and standard deviation of the (transformed) data measured in kilograms.
A math whiz says "You don't have to change all the data to kilograms to find the new values of the mean and standard deviation. Just use the same conversion factor (multiply by 0.455) to get the new values (in kilograms). Multiply your mean and standard deviation from part 1 by 0.455. Compare your results to those of part 3. Is the math whiz correct?
Now, return to the original weights in pounds. These measurements are made with the players undressed. Full football gear adds 20 pounds to each player. To convert the data into full-gear weights use X
_{i}
^{*}
= X
_{i}
+ 20. List the full-gear weights of the players.
Using the full-gear data of part 5, compute the mean and standard deviation of the full-gear weights.
Compare your answers of part 6 to those of part 1. How does the mean change when 20 is added to each observation? How does the standard deviation change when 20 is added to each observation?
Suppose we measure the players' weights, in kg, in full gear. What are the mean and standard deviation?