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

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 = 0.997. But then, when you plot the residuals, you get this. Uh-oh, now what do we do???

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…

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…

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.

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…

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

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

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