Stat 101L: Lecture 15
Re-expressing Data
 Chapter
6 – Normal Model
–What if data do not follow a
Normal model?
 Chapters
8 & 9 – Linear Model
–What if a relationship between
two variables is not linear?
1
Re-expressing Data
 Re-expression
is another name
for changing the scale of
(transforming) the data.
 Usually we re-express the
response variable, Y.
2
Goals of Re-expression
 Goal
1 – Make the distribution
of the re-expressed data more
symmetric.
 Goal 2 – Make the spread of
the re-expressed data more
similar across groups.
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1
Stat 101L: Lecture 15
Goals of Re-expression
 Goal
3 – Make the form of a
scatter plot more linear.
 Goal 4 – Make the scatter in
the scatter plot more even
across all values of the
explanatory variable.
4
Ladder of Powers
 Power:
2
2
 Re-expression: y
 Comment: Use on left skewed
data.
5
Ladder of Powers
 Power:
1
 Re-expression: y
 Comment: No re-expression.
Do not re-express the data if
they are already well behaved.
6
2
Stat 101L: Lecture 15
Ladder of Powers
 Power:
½
y
 Re-expression:
 Comment: Use on count data
or when scatter in a scatter plot
tends to increase as the
explanatory variable increases.
7
Ladder of Powers
 Power:
“0”
 Re-expression: log  y 
 Comments: Not really the “0”
power. Use on right skewed
data. Measurements cannot be
negative or zero.
8
Ladder of Powers
 Power:
–½, –1 1
1

,
 Re-expression:
y
y
 Comments: Use on right
skewed data. Measurements
cannot be negative or zero.
Use on ratios.
9
3
Stat 101L: Lecture 15
Goal 1 - Symmetry
 Data
are obtained on the time
between nerve pulses along a
nerve fiber.
 Time is rounded to the nearest
half unit where a unit is 1 50 of a
second.
th
– 30.5 represents 30.5 50  0.61 sec
3
.99
2
.95
.90
.75
.50
1
0
.25
.10
.05
.01
Normal Quantile Plot
10
-1
-2
-3
40
Count
60
20
0
10
20
30
40 th
50
60
70
Time ( 1 50 sec)
11
Time – Nerve Pulses
 Distribution
is skewed right.
 Sample mean (12.305) is much
larger than the sample median
(7.5).
 Many potential outliers.
 Data not from a Normal model.
12
4
3
.99
2
.95
.90
1
.75
0
.50
.25
Normal Quantile Plot
Stat 101L: Lecture 15
-1
.10
.05
-2
.01
-3
40
20
Count
30
10
0
1
2
3
4
5
6
7
8
9
13
3
.99
2
.95
.90
1
.75
0
.50
.25
Normal Quantile Plot
Sqrt(Time)
-1
.10
.05
-2
.01
-3
20
Count
30
10
-1
0
1
2
3
4
5
14
Log(Time)
Summary
 Time
– Highly skewed to the
right.
 Sqrt(Time) – Still skewed right.
 Log(Time) –Fairly symmetric and
mounded in the middle.
– Could have come from a Normal
model.
15
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