11.4 Regression on the calculator

Regression on the calculator

What Kepler did is an example of“regression”: finding an equation that models a particular set of data.

Kepler became famous because regression is hard. Who would have thought to look for $T^2=k{a}^3$ ? Especially when you consider all the other equations that would still have a“this goes up, that goes up”relationship, such as $T=2^a+7$ , or $T=\frac{1}{10-a}$ , or maybe $T=3\text{log}5a$ ?

Fortunately, we have a tool that Kepler did not have: the modern computer. Mathematical programs and graphing calculators can take a set of points, and find the line or curve of“best fit”to model the data.

As an example of this process, suppose that you have run an experiment and generated three data points: (2,1), (4,3), and (5,8). What function might model those data?

Entering the data

• Hit STAT to go into the Statistics menu.
• Choose Edit... This brings you to a screen where you enter a bunch of L1 and L2 values.

• Hit 2nd QUIT return to the main screen.

Viewing the data, and guessing at the shape

Once you have entered your points into the L1 and L2 lists, your calculator can show you a“scatterplot”—which is a pointlessly fancy word for“a graph of a bunch of points,”like you used to make when you were first learning what graphing was.

• Hit WINDOW (near the upper-left-hand corner of the calculator).
• Set the variables Xmin, Xmax, Ymin, and Ymax appropriately. For instance, for the three points shown above, the x-values are 2, 4, and 5, and the y-values are 1, 3, and 8. So it might make sense to set Xmin=0, XMax=10, Ymin=0, and Ymax=10—and that is how I did it in the drawing below. Of course, there are many other settings you could use. But if you go through the whole process and don’t see any points, it’s a reasonable guess that your window is not set properly.
• Hit Y= (upper-left-hand corner of the calculator).
• Then, hit the up-arrow key, so the focus moves to Plot1 (which will start blinking).
• Hit ENTER . It’s actually impossible, at this point, to see that anything has happened. But if you down-arrow away from Plot1 , you should see that it remains darkened (white letters on a black background, instead of the other way around). This indicates that it has been selected.

At this point, looking at the data, it is often useful to categorize it in two ways.

First: is it increasing or decreasing? In our example, of course, the points are increasing. (Some data, of course, may be doing both at different times: consider, for instance, a parabola.)

Now, in the case of an increasing function, you can categorize it as one of the following.

Decreasing functions can be categorized similarly, of course. If a function decreases and then increases, a parabola is probably the best fit. Functions that go up, then down, then up again, are most likely to be higher-order polynomials.

Finding the formula

Once you have decided on the right shape, the hard work is done: the calculator takes care of the rest.

• Hit STAT to return again to the Statistics menu.
• Hit the right-arrow key to go to CALC .
• At this point, you have several choices. LinReg will give you a line that best fits your points. QuadReg will give you a quadratic function, aka a second-order polynomial. There are also options for "cubic" (third-order polynomial), "quartic" (fourth-order polynomial), logarithmic, or exponential curves. Choose the one you want and hit ENTER .

The calculator does not graph your curve for you, but it does tell you what the curve is. For instance, if I run a QuadReg on the data above, the calculator gives me:

This tells me that the best quadratic fit for my data is the curve $y=1.33x^2-7x+9.67$ . One way to double-check this, of course, is to enter $Y1=1.33x^2-7x+9.67$ and then graph it, and see how closely it approximates the points!

Starting over

There's just one more thing you have to know: once you've done this once, how do you clear out the lists to enter new ones? Here is one way to do it.

• Hit MEM (you do this by hitting the yellow 2nd key, and then hitting + ).
• This brings up a menu. Choose ClrAllLists .
• Then—after you return to the main screen—hit ENTER and the lists are emptied out.