1. Straightening a curved scatterplot
Re-expressing Data
Straightening a curved scatterplot

2. Suppose this happens to you…
You make a scatter plot on your calculator and plot the LSRL. It looks like a pretty good fit, and r =
But then, when you plot the residuals, you get this.
Uh-oh, now what do we do???

3. What do you do when a linear model is NOT appropriate?
We transform our data values using mathematical functions to see if they fit better than a straight line.
You can take the logarithm of the x values, take the logarithm of the y values, take reciprocals, take square roots, etc.
To keep you organized, your textbook gives some guidelines…

4. Using logarithms to straighten out a scatter plot
If the data looks exponential… X
log(Y)
If the data looks logarithmic…
log(X) Y
If the data doesn’t straighten out with either of these…

5. The Ladder of Powers Other functions might work too…2
Square the data values, or y2 1
The original data values
1/2
The square root of the data values
Not the zero power, but the logarithm of either x or y or both
-1/2
The reciprocal square root of y -1
The reciprocal of y.

6. Remember…
Finding a good functional model is a guess-and-check process.
There is no such thing as a perfect model.
You are looking for one that is simply good enough…
Let’s try an example…

7. Linearize the data and make a model for world population…
Enter this data in L1 and L2 and plot the points. What do you notice?
Now try making L3 by using log(L2).
Plot L1 and L3 and now make the LSRL.
What is the predicted population for 2010?
Year
Pop. (millions)
1950
2519
1955
2755
1960
3020
1965
3334
1970
3691
1975
4066
1980
4430
1985
4825
1990
5255
1995
5662
2000
6057

8. Linearize the data and make a model for light intensity…
Enter this data in L1 and L2 and plot the points. What do you notice?
Now try making L3 by using a function of L2. What should we use?
Plot L1 and L3 and now make the LSRL.
What is the predicted intensity for 12 feet?
Distance (ft)
Candlepower 2
531.2 5
84.3 8
33.6 10
21.1 15
9.5 20
5.3 25
3.4

9. Linearize the data and make a model for mortgage amounts at Republic National Bank…
Enter this data in L1 and L2 and plot the points. What do you notice?
Now try making L3 by using a function of L2. What should we use?
Plot L1 and L3 and now make the LSRL.
What is the predicted mortgage level for 1990?
Year
$ (millions)
1970
1.2
1972
2.5
1974
2.9
1976
3.1
1978
5.8
1980
8.3
1982
10.8
1984
14.7
1986
21.8
1988
29.7