AP Statistics
Chapter 4
Chapter 4: More on Two Variable Statistics
Section 4.1: Transforming Functions
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Transform a set of data using nonlinear transformations
• Report information in the appropriate format.
Transformations to linearize nonlinear data
Linearized data
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Allows for the use of Chapter 3 statistics tools on the transformed data
o Correlation Coefficient
o Coefficient of Determination.
o Least – Squares Regression
Transforming Data
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Re-expressing data – changing the scale of measurement that was used when data was
collected.
Can transform:
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The explanatory variable
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The response variable
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Both
The transformation done depends on the data given and the relationship of the variables.
Linear Transformations cannot straighten a curved relationship between two variables.
Transformations are done with nonlinear functions.
Exponential Growth Model
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Variable is multiplied by a fixed number (ratio) in each time period.
Verify the above is ACTUALLY occurring so an exponential growth model is the correct
conclusion rather than a different model.
Increase appears slow for a long period of time and then shoots up rapidly.
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Exponential Growth Model =>
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y = ab x where a and b are constants
Power Law Model
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p
Power Law Model =>
y = a⋅ x
Power “p” in the power law becomes the slope of the straight line that links the
transformed data. €
Slope is only an estimate of the p in an underlying power model.
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To Linearize the Data – use logarithmic properties
Product Property
Quotient Property
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Power Property €
log a MN = log a M + log a N
Ex:
M
log a
= log a M − log a N
N
€ Ex:
log2x = log2 + log x
log 3
6
= log 3 6 − log 3 y
y
log a M p = plog a M
Ex:
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€
€
log 4 x 5 = 5log 4 x
Determining Linear Growth or Exponential Growth
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Visually = not a reliable strategy
Check relationship = is it mirroring the following (approximately)
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Linear – when a fixed increment is added at equal time periods over
time.
Exponential – when a fixed increment is multiplied at equal time
periods over time.
Determining Exponential Growth or Power Law
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Check that a fixed increment is multiplied at equal time periods over time.
Transform the data (use the correct transformation based on the type of model you
believe it to be)
Look at it on a scatterplot to see if it has been linearized from performing the
transformations.
Once the Transformations have been completed,
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Calculate the least – squares regression, correlation, and coefficient of determination.
These calculations can only be used for linear data.
Residual Plot
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Ideally, the residual plot displays a random scatter above and below the line with no
distinct pattern.
Curved pattern shows the linear model is not appropriate for the data.
ALWAYS look at a residual plot to determine if the LSRL is a good model.
A residual plot will sometimes exaggerate or display a pattern a regular scatterplot did
not.
Knowing the background of the topic and thinking of the purpose of the work being
done will help understand the possible pattern emerging.
Look for trends, especially those that support the purpose of the study or experiment.
Making Predictions
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The regression line must make predictions using the original data (not the transformed
data)
Report data in original format so it makes sense to the everyday person
For Example: If the response variable is “weight”, then the predictions (values) provided
must represent “weight”, not “log(weight)”.
To get back to the correct equation, apply inverse operations of logarithms to the
regression line.