### (continued) Linear Transformations

```2.4 (cont.)
Changing Units of
Measurement
How shifting and rescaling
data affect data summaries
Shifting and rescaling: linear
transformations
Original data x1, x2, . . . xn
Linear transformation:
x* = a + bx, (intercept a, slope b)
Shifts data
by a
Changes
scale
x*
a
0
x
Linear Transformations
2.54
32
12
40
100
00
0a+
9/5 x
x* = 150
b
Examples: Changing
1. from feet (x) to inches (x*): x*=12x
2. from dollars (x) to cents (x*): x*=100x
3. from degrees celsius (x) to degrees
fahrenheit (x*): x* = 32 + (9/5)x
4. from ACT (x) to SAT (x*): x*=150+40x
5. from inches (x) to centimeters (x*):
x* = 2.54x
Shifting data only: b = 1
x* = a + x
 Adding the same value a to each value in
the data set:
 changes the mean, median, Q1 and Q3 by a
 The standard deviation, IQR and variance are
NOT CHANGED.
Everything shifts together.
Spread of the items does not change.
Shifting data only: b = 1
x* = a + x (cont.)
 weights of 80 men age 19 to 24
of average height (5'8" to 5'10")
x = 82.36 kg
 NIH recommends maximum healthy
weight of 74 kg. To compare their
weights to the recommended
maximum, subtract 74 kg from each
weight; x* = x – 74 (a=-74, b=1)
 x* = x – 74 = 8.36 kg
1.
No change in
shape
2.
No change in
3.
Shift by 74
Shifting and Rescaling data:
x* = a + bx, b > 0
Original x data:
x1, x2, x3, . . ., xn
Summary statistics:
mean x
median m
1st quartile Q1
3rd quartile Q3
stand dev s
variance s2
IQR
x* data: x* = a + bx
x1*, x2*, x3*, . . ., xn*
Summary statistics:
new mean x* = a + bx
new median m* = a+bm
new 1st quart Q1*= a+bQ1
new 3rd quart Q3* = a+bQ3
new stand dev s* = b  s
new variance s*2 = b2  s2
new IQR* = b  IQR
Rescaling data:
x* = a + bx, b > 0 (cont.)
 weights of 80 men age 19 to 24,
of average height (5'8" to 5'10")
 x = 82.36 kg
 min=54.30 kg
 max=161.50 kg
 range=107.20 kg
 s = 18.35 kg
 Change from kilograms to pounds:
x* = 2.2x (a = 0, b = 2.2)
 x* = 2.2(82.36)=181.19 pounds
 min* = 2.2(54.30)=119.46 pounds
 max* = 2.2(161.50)=355.3 pounds
 range*= 2.2(107.20)=235.84 pounds
 s* = 18.35 * 2.2 = 40.37 pounds
Example of x* = a + bx
4 student heights in inches
(x data)
not
62, 64, 74, 72
necessary!
UNC
x = 68 inches
method
s = 5.89 inches
Suppose we want
Go directly to
x* = 2.54x
this. NCSU
(a = 0, b = 2.54) method
4 student heights in centimeters:
157.48 = 2.54(62)
162.56 = 2.54(64)
187.96 = 2.54(74)
182.88 = 2.54(72)
x* = 172.72 centimeters
s* = 14.9606 centimeters
Note that
x* = 2.54x = 2.54(68)=172.2
s* = 2.54s = 2.54(5.89)=14.9606
Example of x* = a + bx
x data:
Percent returns from 4
investments during
2003:
5%, 4%, 3%, 6%
not
x = 4.5%
necessary!
s = 1.29%
Inflation during 2003:
2%
x* data:
Go directly to
this
x* = x – 2%
(a=-2, b=1)
x* data:
3% = 5% - 2%
2% = 4% - 2%
1% = 3% - 2%
4% = 6% - 2%
x* = 10%/4 = 2.5%
s* = s = 1.29%
x* = x – 2% = 4.5% –2%
s* = s = 1.29% (note! that
s* ≠ s – 2%) !!
Example
 Original data x: Jim Bob’s jumbo watermelons from his
garden have the following weights (lbs):
23, 34, 38, 44, 48, 55, 55, 68, 72, 75
s = 17.12; Q1=37, Q3 =69; IQR = 69 – 37 = 32
Melons over 50 lbs are priced differently; the
amount each melon is over (or under) 50 lbs is:
x* = x  50 (x* = a + bx, a=-50, b=1)
-27, -16, -12, -6, -2, 5, 5, 18, 22, 25
s* = 17.12; Q*1 = 37 - 50 =-13, Q*3 = 69 - 50 = 19
IQR* = 19 – (-13) = 32
NOTE: s* = s, IQR*= IQR
Z-scores: a special linear
transformation a + bx
z
xx
s

x
s

1
s
x  a  bx where a  
x
s
,b 
1
s
Example. At a community college, if a student takes x credit
hours the tuition is x* = \$250 + \$35x. The credit hours taken by
students in an Intro Stats class have mean x = 15.7 hrs and
standard deviation s = 2.7 hrs.
Question 1. A student’s tuition charge is \$941.25. What is the z-score of this
tuition?
x* = \$250+\$35(15.7) = \$799.50; s* = \$35(2.7) = \$94.50
z
941.25  799.50 141.75

 1.5
94.50
94.50
Z-scores: a special linear
transformation a + bx (cont.)
Example. At a community college, if a student takes x credit hours
the tuition is x* = \$250 + \$35x. The credit hours taken by students
in an Intro Stats class have mean x = 15.7 hrs and standard
deviation s = 2.7 hrs.
Question 2. Roger is a student in the Intro Stats class who has a
course load of x = 13 credit hours. The z-score is
z = (13 – 15.7)/2.7 = -2.7/2.7 = -1.
What is the z-score of Roger’s tuition?
Roger’s tuition is x* = \$250 + \$35(13) = \$705
Since x* = \$250+\$35(15.7) = \$799.50; s* = \$35(2.7) = \$94.50
The z-score does not depend
705 - 799.50 -94.50
on the unit of measurement.
z=
=
=-1
94.50
94.50
This is why z-scores are so
useful!!
SUMMARY: Linear
Transformations x* = a + bx
Assembly Time (seconds)
Assembly Time (minutes)
30
20
15
10
5
0
Frequency
Frequency
25
30
20
10
0
Linear transformations do not affect the shape
of the distribution of the data
-for example, if the original data is rightskewed, the transformed data is right-skewed
SUMMARY: Shifting and
Rescaling data, x* = a + bx, b > 0
original data x1 , x2 , x3 ,... transformed data x1* , x2* , x3* ,...
summary statistics
mean x    
median m
  
summary statistics
new mean x *  a  bx
new median m*  a  bm
1st Q1
   
new Q1*  a  bQ1
3rd Q3
   
new Q3*  a  bQ3
st dev s    
new st dev s*  bs
var. s 2
   
new var. s *2  b 2 s 2
IQR
   
new IQR*  bIQR
```