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LESSON 2.2 • Transforming Data 101

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Error (m)
n − x s x Min Q 1 Med Q 3 Max IQR Range Lesson 2.2
Error (m) 44 3.02 7.14 –5 –2 2 4 27 6 32

Because the students are having some difficulty with the metric system, it may not be helpful to tell
them that their guesses tended to be about 2 meters too high. Let’s convert the error data to feet
before we report back to them.
There are roughly 3.28 feet in a meter. So, for the student whose error was 5meters− , that
translates to
3.28 feet
−5meters × = −16.4feet
1meter
To change the units of measurement from meters to feet, we multiply each of the error values by 3.28.

(a) What shape would the resulting distribution of error have?

(b) Find the median of the distribution of error in feet.
(c) Find the interquartile range of the distribution of error in feet.

SOLUTION:
(a) The same shape as the original distribution of guesses: Multiplying each data value by 3.28 doesn’t

skewed to the right with two distinct peaks. change the shape of the distribution.
(b) Median2 3.28 6.56 feet=× =
(c) IQR =×63.28 =19.68 feet
FOR PRACTICE TRY EXERCISE 9.

Figure 2.4 confirms the results of the example. TEACHING TIP:
Differentiate
Error (m) d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d FIGURE 2.4 Dotplots Here is a short algebraic justification that
d
and summary statistics
for the errors in Austra-
Error (ft) d d d d d d d d d d d d d d d d d d d d d d d d dd lian students’ guesses multiplying every value in a distribution
by a real number b multiplies the mean
of classroom width, in
by b. If you have students who want to

meters and in feet.
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–20 d d d (C) 2021 BFW Publishers -- for review purposes only.
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0
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n − x s x Min Q 1 Med Q 3 Max IQR Range use their algebraic skills in statistics, give
Error (m) 44 3.02 7.14 –5 –2 2 4 27 6 32 them the first line or two and see if they
Error (ft) 44 9.91 23.42 –16.4 –6.56 6.56 13.12 88.56 19.68 104.96 can complete it.
For a set of values x xx, 1 2 , 3 ,. .. , x n ,
Putting It All Together: Adding/Subtracting x + x + x + ⋅⋅⋅+ x n
1
3
2
and Multiplying/Dividing let x = n
What happens if we transform a data set by both adding or subtracting a constant and
multiplying or dividing by a constant? We just use the facts about transforming data Newmean =
⋅
that we’ve already established and the order of operations. xb) + xb) + xb) + ⋅⋅⋅+ xb)
⋅
(
⋅
⋅
= ( 1 ( 2 ( 3 n
n
= ⋅ bx ( 1 + x 2 + x 3 + ⋅⋅⋅+ x n )
n
03_StarnesSPA4e_24432_ch02_088_153.indd 101 07/09/20 1:54 PM x ( 1 + x 2 + x 3 + ⋅⋅⋅+ x n )
= ⋅ b
n
= ⋅bx
LESSON 2.2 • Transforming Data 101

03_TysonTEspa4e_25177_ch02_088_153_4pp.indd 101 10/11/20 7:43 PM