Regression (Curve Fitting)
with the TI-83/84 Calculator

Have Your Calculator Ready and Follow Along!

The ordered pairs (t, N) of data values are obtained from some measuring process. Using these data, lets find a formula between t and N by a linear regression.

Regression is a statistical method that is used to do just that. The purpose of regression is to construct a rule or formula (function) that "best" explains the observed data. It is always possible to reproduce the data exactly by choosing a sufficiently complicated formula, e.g., an (n-1)^th degree polynomial. In practice, we try to find a simpler relationship that "fits" most of the data.

The TI-83/84 can be used to fit various situations: linear (using least squares or median-median regression), polynomial (quadratic, cubic, and quartic), exponential, logarithmic, power, logistic, and sinusoidal. The examples below show how we fit linear and polynomial models to data and plot the results.

Entering the Data

1. Press the key and choose [EDIT] then [1:Edit] … (Fig. 1).
   • This brings up the edit window (Fig. 2).
   • On the TI-83, data are entered into "list" variables.
   • There are six list variables available, called L1, L2, L3, L4, L5, and L6.
     • Their names appear as the 2^nd functions of the numeric keys 1, 2, 3, 4, 5, and 6.
     • To use these list variables, you must choose them from the keyboard; they cannot be typed using ordinary alphanumeric characters.
     • You may also choose your own names for the lists by inserting a column and then typing the name at the Name= prompt. 
     • Be careful with lists. Do not accidentally delete a list column! If you do, you can go to the MEM reset option and reset the calculator to get all of the columns back, but the data is lost!
2. Enter the data values one column at a time pressing after each data value.
   • Place the t-values in L1 and N-values in L2.
   • Fig. 3 shows the first seven observations.
   • Make sure you have as many t's as N's. If you don't you will get a "Dimension Error."

Plotting the Data (Scatter Plots)

To decide with which model to fit, create a scatter plot of the data.

1. Begin by turning Stat Plots on. Press using to bring up the display shown in Fig. 4.
   • With the first plot selected, press to bring up the settings for this plot (Fig. 5).
   • Change the first option to On.
   • The default settings for the remaining options will produce a scatter plot using L1 as the x-variable and L2 as the y-variable. Mark defines the symbol type used in the plot, a box, or plus or dot are the options. Typically we just leave it as the box since it is easy to see.     
2. Define the data range for the plot.
• Press and enter the settings shown  in Fig. 6.
• The nice thing about working with stats, you know the domain and range that best fits the data by looking for the extremes in input and output in the data set.
3. Next press . If there are any functions displayed already, be sure they are deselected. They do not need to be deleted!
   1. A function is selected if the "=" sign is displayed in reverse video . (In Fig. 7, Y1 is selected while Y2 is not.)
   2. To deselect a function, use the arrow keys to move the blinking cursor directly on top of the = sign of the selected function. Press to deselect the function. (Pressing again will reselect it.)
4. Finally, press to produce the scatter plot (Fig. 8).

The scatter plot in Fig. 8 suggests that a straight line relationship is not too unreasonable. Our next step is to get the formula that would help us in approximating values of N given t. The process is tedious, but beats hand plotting and calculation!

1. Press the key and choose [CALC] then [4:LinReg(ax+b)] as shown in Fig. 9.
   • (Note: choosing option [8: LinReg(a+bx)] will produce the same results as choosing option 4 except that the labels for the intercept and slope will be reversed. It doesn’t matter which of these two options you choose.)
   • This puts LinReg(ax+b) on the home screen.     
2. By default it is assumed that the x-variable is in L1 and the y-variable is in L2, so pressing at this point will produce the correct results. A better choice is to list L1 and L2 as a part of the command.
   • The order of arguments is x-variable, y-variable, storage location (Fig. 10).
   • So the command LinReg(ax+b) L1,L2,Y1 will use L1 as the x-list, L2 as the y-list, and overwrite the contents of Y1 with the regression formula and automatically selects Y1 for plotting! Wow!
   • Notice that the arguments are separated by commas.
3. Also remember, L1, L2, and Y1 are special characters and cannot be entered by typing ordinary letters and numbers.
   • L1 is the 2^nd function of numeric key ,
   • L2 is the 2^nd function of numeric key
   • Y1 is obtained by pressing , choosing [Y-VARS]  then [1:Function], and then [FUNCTION] then [1:Y1]. (See Fig. 11 and 12.)
4. Once the command and its arguments are pasted to the screen, press to produce the regression results shown in Fig. 13.
   • The values of a and b are displayed on the screen along with the from of the model that was chosen.
   • Based on the output the fitted model is N(t) = -130.17 + 23.374t.
   • Since the regression function is now stored in Y1 and is selected, pressing will produce a scatter plot with the regression line superimposed (Fig. 14).

     

Interpreting the Results -  How Good is the "Fit?"

Qualitatively it would appear from the graph in Fig. 14 that a linear function is a "reasonable" model. The standard quantitative measure of the usefulness of the regression model is , the coefficient of determination (also, correlation coefficient) .   can take on values between 0 and 1. The higher the value the better the fit. So a 1 would be perfect. The TI-83 calculates this quantity automatically.

1. Confusingly, for simple straight line models such as this one, the TI-83 stores the coefficient of determination in a variable it calls r ². For more complicated models, it stores the coefficient of determination in the variable R².
2. To access either one press and then select from the [VARS] submenu option [5:Statistics] (Fig. 15).
3. In the Statistics window that appears move the cursor to the third column to display the [EQ] menu. The eighth option, [8:r ²], is where the coefficient of determination is stored for this model (Fig. 16).
4. Press to paste the value to the screen and then again to see its contents (Fig. 17).

We can conclude that the fit from this linear regression has an r ² value of about 92%. Without reference to a different model, we really don't know if this is the best fit we can achieve short of a a 9^th degree polynomial. As you will see a little later, it isn't all that great!

Fitting a Quadratic Function

The scatter plot in Fig. 8 or 14 reveals a slight curve in the data trend. A higher degree polynomial model might be appropriate. The nice thing about calculators, they allow exploration without doing a lot of extra work.

1. Press the key and choose CALC 5:QuadReg (Fig. 18). This puts QuadReg on the home screen. (QuadReg fits a second degree polynomial. Third and fourth degree polynomials can be fit by choosing CubicReg and QuartReg respectively.)
2. Let's modify the command in the same way as before to create the command QuadReg L1,L2,Y2 will use L1 as the x-list, L2 as the y-list, and overwrite the contents of Y2 with the regression function and automatically select Y2 for plotting.
3. As was described for the linear model, L1, L2, and Y2 must be pasted in by making the appropriate keyboard and menu choices. Press to produce the regression results shown in Fig. 20.
4. The values of a, b, and c are displayed on the screen along with model that was fit. Based on the output, the fitted model is using some rounding.
5. Since the quadratic regression function is now stored in Y2 and is selected, and the straight-line regression function is still stored in Y1, pressing will produce a scatter plot with the quadratic regression function and linear regression function superimposed on a scatter plot of the data (Fig. 21).
6. Now go to the [VARS]  menu again and find the values for R². The coefficient of determination has jumped to about 99%! Also, the graph of the Y2 formula is significantly better is a fit to the data.

Try other choices in the Regression menu. While the quadratic is a pretty good, one of the other choices may do even better. The data has the appearance of an exponential and some characteristics of a logistic model.