Transcript Chapter 10: Re-expressing Data

Chapter 10:
Re-expressing Data
It’s easier than you think!
Goals of Re-expression

Goal 1:
• Make the distribution of a variable more
symmetric.
• Easier to summarize the center, using mean
and standard deviation.
• If distribution is unimodal, use the 68-95-99.7
Rule.
Goals of Re-expression

Goal 2:
• Make the spread of several groups mare
alike, even if their centers differ.

Goal 3:
• Make the form of the scatterplot more nearly
linear.

Goal 4:
• Make the scatter in the scatterplot spread out
evenly rather than following a fan shape.
The Ladder of Powers
Power
Name
2
y2
1
½
0
Comment
Unimodal, skewed left
Raw data Data that takes on +/- values
y1/2
Counted data
logarithm Measurements that cannot be -
-½
1/ y1/2
-1
-1/y
Preserves the direction of relationship
Ratios of two quantities
Attack of the Logarithms
Model Name
x-axis
Exponential
x
Logarithmic log(x)
Power
y-axis
Comment
log(y) Useful with values that
grow by % increase.
y
Useful with wide range of x
values or scatterplot
descending rapidly then
leveling off.
log(x) log(y) When one of the ladder’s
powers is too big and the
other is too small.
Let’s Try It! (Pg 192)
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Shutter speed and
f/stop of the lens
•
•
L1: shutter speed
L2: f/stop

Curved stat plot
•
•
•
Try logarithms
Take log of L1→L3
Take log of L2→L4
Let’s Try It!
Scatterplot #1:
Xlist→L3, Ylist→L2
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Scatterplot #2:
Xlist→L1, Ylist→L4
Let’s Try It!
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Use Scatterplot #3:
LinReg L3, L4
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LinReg L3, L4
Multiple Benefits
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A single re-expression may improve
each of our goals at the same time.
Re-expression certainly simplifies efforts
to analyze and understand relationships.
Simpler explanations and simpler
models tend to give a true picture of the
relationship. (Occam’s Razor)
TI Tips

Regressions that automatically and
appropriately re-express the data:
Equivalent Models
Type of
Re-expression
Model
Equation
Calculator’s
Curve
Command
Equation
Logarithmic
LnReg
Exponential
ExpReg
Power
PwrReg
What Can Go Wrong?!?

Beware of multiple modes.
• Re-expression cannot pull separate modes
together.
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Watch out for scatterplots that turn
around.
Watch out for negative data values.
• It is impossible to re-express negative values
by any power that is not positive.
What Can Go Wrong?!?

Watch for data far from one.
• Re-expressing data with a range from 1 to
1000 is far more effective than re-expressing
data with a range of 100,000 to 100,100.
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Don’t stray too far from the ladder.
• Stick to powers between -2 and 2.
• Stick to the simpler powers contained in the
“ladder.”