The Wolf STAT Company
We have a company with 14 employees that earn the following monthly
salaries:
3500
1300
1200
1500
1900
1700
1400
2300
2100
1200
 = _______________
1800
1400
1200
1300
 = _______________
Business has been good, so every employee receives a raise of $500. What will happen to the mean and
standard deviation?
 = _______________
 = _______________
Business has not been good. Suppose that we have to cut everyone’s pay by $500. What will happen to the
mean and standard deviation?
 = _______________
 = _______________
Construct parallel boxplots of the original salaries and the increased & decreased salaries.

What happens to the mean and standard deviation if a number is added to each data value?
Suppose that we give everyone a 30% raise. What will happen to the mean and standard deviation?
 = _______________
 = _______________
Construct parallel boxplots of the original salaries and the increased salaries.

What happens to the mean and standard deviation if a number is multiplied to each data value?
Rule:
Linear Transformation of Random Variables Activity
Fritz
The Fritzi STAT Company
Teacher Page
We do this activity together, giving students time to work each part. Class
discusses each part as we proceed.
1.
Students should find the mean and standard deviation of the salaries.
 = $1657.14
= $634.39
Discuss notation –
Q: Why do we use mu () and lower case sigma () for the mean and standard deviation of salaries?
A: This is a population of workers’ salaries.
2. Students should add $500 to all the salaries. Then calculate the mean and standard deviation of these new
salaries.
 = $2157.14
= $634.39
Note:



3.
If using a TI calculator, adding $500 to each value is easily done.
Data is in L1 (list 1).
Move cursor to top of L2 (list 2).
L2 = L1(2nd 1 key) + 500
(This adds 500 to each value in L1.)
Discuss what happens to the mean and the standard deviation when you add $500 to each salary.
Q: Why does the standard deviation not change?
A: We just shifted the distribution of salaries to the right by 500.
Graph parallel boxplots of the original salaries and of the salaries + $500. This helps students visualize that
the distribution is just shifted.
Making connections:
Remind students of transformations in Algebra 2 or Precalculus – when you add a value to a function – it shifts
that function. The same thing happens in Statistics. If a value is added to every observation, then the
distribution shifts. This does not change the spread of the distribution, thus not changing the standard
deviation.
4. Students should subtract $500 to all the salaries. Then calculate the mean and standard deviation of
these new salaries.
 = $1157.14
= $634.39
5.
Discuss what happens to the mean and standard deviation when you subtract $500 from each salary.
Graph parallel boxplots of the original salaries, of the salaries + $500, and of the salaries -$500.
Linear Transformation of Random Variables Activity
Fritz
The Fritzi STAT Company
Teacher Page Continued
6. Students should multiply all the salaries by 1.3. Then calculate the mean and standard deviation of these
new salaries.
 = $2154.29
= $824.71
Discuss why an increase of 30% is the same as multiplying by 1.3.
7.
Discuss what happens to the mean and standard deviation when you multiply each salary by 1.3.
Q: Why does the standard deviation change?
A: When you multiply each observation by a value, you stretch the distribution, thus changing the standard
deviation.
Graph parallel boxplots of the original salaries, of the salaries + $500, and of the salaries  1.3.
Making Connections:
Once more, remind students of transformations in Algebra 2 or Precalculus – when you multiply a function by a
value – it stretches (or shrinks) that function. The same thing happens in Statistics. If every observation is
multiplied by a value, then the distribution stretches (or shrinks). This does change the spread of the
distribution, thus changing the standard deviation.
Rule:
 a bx  a  b x
a bx  b  x
Notes:
 Discuss the symbolic notation above – most students do not readily understand it.
 This rule applies when transforming ONE random variable into another random variable by using
, , ,  .
Q: Why do we use the absolute value of b in calculating the standard deviation?
A: Standard deviation is always positive!
Q: Do I need to memorize this rule?
A: YES!!!
Linear Transformation of Random Variables Activity
Fritz