The Effect of Linear Transformations on Measures of Center & Spread

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  • 0:02 Effects of Linear…
  • 1:42 Transforming Data
  • 3:54 Example 2
  • 5:16 Lesson Summary
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Lesson Transcript
Instructor: Cathryn Jackson

Cat has taught a variety of subjects, including communications, mathematics, and technology. Cat has a master's degree in education and is currently working on her Ph.D.

Linear transformations can be a great way to manipulate and analyze data. This lesson will show you how those transformations affect the center and spread of data.

The Effects of Linear Transformations

Professor Shannon is a science professor at a local university. Recently, Professor Shannon administered a test to one of her science classes. These are the percentage grades each of the students received on their tests: 31, 17, 27, 35, 22, 35, 15, 17, 21, 27, 17.

Uh oh. It looks like the students didn't do too well on this test. The mean of this data set is 24, which means that the students scored on average 24% on their tests. This is an example of measure of center in the data set. The standard deviation of this data set is approximately 7.4. This is an example of a measure of spread in the data set. Since 7.4 is a rather small number for standard deviation, this means that the numbers in the data set are pretty close together. Since the numbers are close together and don't spread out very far from the mean, we know that the students all failed this test and have roughly a similar understanding of the concept.

If you are unfamiliar with these concepts, pause the video here and check out some of our other lessons on summarizing data.

Professor Shannon decides to help the students out by grading this test on a curve. A curve is one example of a linear transformation, which is when a variable is multiplied by a constant and then added to a constant. When using linear transformations on a data set, all variables in the data set are transformed. Therefore, Professor Shannon wants to change all of the students' grades to a higher value in the data set.

Transforming Data

We can transform the data in this data set by using the following formula for linear transformations: a + bx. In this case, x = the number in the data set, a = the constant being added to the variable and b = the constant being multiplied to the variable.

The best way to figure out how to transform this data is to look at the mean, 24. Professor Shannon feels that it would be fair for the average student to score a 75% on this test. If the average score is 24, then we can use the formula for linear transformations to change the average score of 24 to an average score of 75 like this: 27 + 2(24) = 75.

Now that we have our formula, we can transform all of the values in this data set. Take a look at this chart below to see the new values for the data set.

New data after transformation
chart showing linear transformation

You may notice that the highest score changed from a 35 to a 97, and the lowest score changed from a 15 to a 57. Therefore, with the new scores, only one student failed the science test. Our new mean for this data set is 75, which is the mean we used to figure out our average score. The standard deviation for this new data set is 14.14 - almost exactly double of our original standard deviation. Take a look at this graph to see the differences between the first data set and the second data set:

Chart comparing first and second data sets
chart comparing old and new data sets

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