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# Transformations to Achieve Linearity for AP Statistics

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By — McGraw-Hill Professional
Updated on Feb 5, 2011

Practice problems for these concepts can be found at:

Until now, we have been concerned with data that can be modeled with a line. Of course, there are many two-variable relationships that are nonlinear. The path of an object thrown in the air is parabolic (quadratic). Population tends to grow exponentially, at least for a while. Even though you could find a LSRL for nonlinear data, it makes no sense to do so. The AP Statistics course deals only with two-variable data that can be modeled by a line OR non-linear two-variable data that can be transformed in such a way that the transformed data can be modeled by a line.

example: Let g(x) = 2x, which is exponential and clearly nonlinear. Let f(x) = ln(x). Then, f[g(x)] = ln(2x) = xln(2), which is linear. That is, we can transform an exponential function such as g(x) into a linear function by taking the log of each value of g(x).

example: Let g(x) = 4x2, which is quadratic. Let f(x) = . Then f[g(x)] = =2x, which is linear.

example: The number of a certain type of bacteria present (in thousands) after a certain number of hours is given in the following chart:

What would be the predicted quantity of bacteria after 3.75 hours?

solution: A scatterplot of the data and a residual plot [for Number = a + b(Hour)] shows that a line is not a good model for this data:

Now, take ln(Number) to produce the following data:

The scatterplot of Year versus ln(Population) and the residual plot for ln(Number) = –0.0047 + 0.586(Hours) are as follows:

The scatterplot looks much more linear and the residual plot no longer has the distinctive pattern of the raw data. We have transformed the original data in such a way that the transformed data is well modeled by a line. The regression equation for the transformed data is: ln(Number) = –0.047 + 0.586 (Hours).

The question asked for how many bacteria are predicted to be present after 3.75 hours. Plugging 3.75 into the regression equation, we have ln(Number) = –0.0048 + 0.586(3.75) = 2.19. But that is ln(Number), not Number. We must back-transform this answer to the original units. Doing so, we have Number = e2.19 = 8.94 thousand bacteria.

It may be worth your while to try several different transformations to see if you can achieve linearity. Some possible transformations are: take the log of both variables, raise one or both variables to a power, take the square root of one of the variables, take the reciprocal of one or both variables, etc.

Practice problems for these concepts can be found at:

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