Fitting Formulas to Data: TI83

The TI83 offers the following options for curve fitting. These can all be found by pressing [STAT], and then by moving the cursor across the top of the screen to option CALC.

Description	Command	Formula
Median-median fit	`Med-Med`	y = a x + b
Linear regression	`LinReg(ax+b)`	y = a x + b
Quadratic regression	`QuadReg`	y = a x² + b x + c
Cubic regression	`CubicReg`	y = a x³ + b x² + c x + d
Quartic regression	`QuartReg`	y = a x⁴ + b x³ + c x² + d x + e
Linear regression	`LinReg(a+bx)`	y = a + b x
Logarithmic regression	`LnReg`	y = a + b ln(x)
Exponential regression	`ExpReg`	y = a b^x
Power regression	`PwrReg`	y = a x^b
Logistic regression	`Logistic`	y = c (1 + a e^{-b x})^-1
Sinusoidal regression	`SinReg`	y = a sin(b x + c) + d

The steps for fitting a regression curve to a collection of data points are as follows:
1. Enter the data
2. View a scatter plot
3. Obtain an appropriate regression equation
4. Plot the regression equation along with the scatter plot
Example 1: Our first example will be represented by the following table:

a 3 4 5 6

S 105 117 141 152

We wish to find a curve that best fits this set of points.
1. Enter the data
  
  The calculator stores data in lists. We will use the lists specified L₁ and L₂. The steps are:
  1. Clear the lists:
    
    We have two methods of clearing the lists.
    1. Press [STAT]. With the cursor on option 1:EDIT, under EDIT, press [ENTER]. The lists will then appear on the screen. If L₁ has data already, move the cursor up to the symbol L₁ at the top of the list. Then press [CLEAR] followed by [ENTER]. Do the same for L₂.
    2. Press [STAT], move the cursor down the list to 4:ClrList, and press [ENTER]. On the screen, ClrList will appear. Add to this line a list of the list names that you wish to clear, such as the lists L₁ and L₂ (which would be entered as [2nd][1][,][2nd][2]). Then press [ENTER].
  2. Now enter the list screen by pressing [STAT] and selecting option 1:EDIT, under EDIT. Move the cursor into the L₁ column. From the bottom of the screen, below the lists, enter each data item from the line of the table that corresponds to the values of a, followed by [ENTER].
    
    Next move the cursor into the L₂ column and repeat for each element in the line of the table that corresponds to the values of S.
    
    Exit the lists by pressing [QUIT] (which is [2nd][MODE]).
2. View a scatter plot:
  1. Any undesired functions entered on the \Y₁=, \Y₂=, ... lines of the [Y=] window will have to be cleared out. To do this, press [Y=], move the cursor to each appropriate line and press [CLEAR]. Exit by pressing [QUIT].
  2. Plot1 must be turned On. There are two methods of doing this:
    1. Press [Y=]. If Plot1 is encased in a dark rectangle, then it is On. It is Off otherwise. To change the On/Off status of Plot1, move the cursor to the top of the screen, to Plot1, and toggle it by pressing [ENTER]. Once the proper status is designated, press [QUIT].
    2. Go to the [STAT PLOT] menu (press [2nd][Y=]). There you can see whether or not Plot1 is turned On. To turn a given Plot On, move the cursor down the list to the designated Plot and press [ENTER]. In this new screen, move the cursor to On and press [ENTER]. To exit, press [QUIT]
  3. To change the type of points that Plot1 is designated to produce, go to the [STAT PLOT] menu, and with the cursor on option 1:Plot1, press [ENTER]. From this screen, move the cursor down the list to the option Mark:, and then move it over to the point option desired--either a small square, a plus sign, or a dot. Once the cursor is in position, press [ENTER]. Press [QUIT] to exit.
  4. There are a number of methods of generating a scatter plot. We shall present three methods, each requiring that Plot1 is On. Of the three, we recommend the first.
    1. Press [ZOOM]. Move the cursor down the list to option 9:ZoomStat, and press [ENTER]. The scatter plot will then appear.
      
      The x- and y-dimensions of the sceen window will be set automatically. Look at the [WINDOW] settings to see which parameters were chosen.
    2. Press [ZOOM]. With the the cursor on option 1:ZBox, press [ENTER]. A scatter plot of the data will then appear.
      
      The x- and y-dimensions of the sceen window will not be set automatically, and may need to be altered manually to get a good picture of the data (press [WINDOW]).
    3. A scatter plot of the data will also appear immediately after pressing [GRAPH].
      
      The x- and y-dimensions of the sceen window may need to be altered manually (press [WINDOW]).
3. Obtain an appropriate regression equation:
  1. The scatter plot suggests that the relationship between measurements a and S is approximately linear, so we will choose a linear regression to obtain a line to best represent the points. Note that the median-median fit will also yield a line, but this means of curve fitting is not commonly used in this type of situation.
  2. Press [STAT]. Move the cursor across the top of the screen to option CALC. Then move the cursor down the list to option 4:LinReg(ax+b), and press [ENTER]. On the screen, LinReg(ax+b) will appear. At this point, press [ENTER] once again.
    
    This yields the values a=16.5 and b=54.5 for the equation y=ax+b, so that our regression equation is
    
    y = 16.5 x + 54.5.
    The correlation coefficient, r=.9886267625, indicates how well the regression curve fits the data. A correlation that is close to 1 is good.
  3. We can store the regression equation as Y₁ on the [Y=] screen. There are a number of ways to do this.
    1. On your screen, next to LinReg(ax+b), before evaluating this command, type: [L₁][,][L₂][,][Y₁]. This will store the regression equation automatically.
      
      The symbols for L₁ and L₂ are on the keyboard as [2nd] [1] and [2nd] [2], respectively. To obtain the symbol for Y₁, press [VARS], move the cursor across the top of the screen to option Y-VARS, then down the list to option 1:Function, and press [ENTER]. This gives a list of symbols, from which you must choose option 1:Y₁ (press [ENTER]).
    2. Press [Y=] and move the cursor to a line that has already been cleared. From this screen, press [VARS], move the cursor down the list to the 5:Statistics... option, and press [ENTER]. Move the cursor across the top of the screen to option EQ. From here move the cursor down the list to the option 1:RegEQ, and press [ENTER]. The equation will then appear in the [Y=] screen.
    3. The resulting regression equation can also be typed in manually.
4. Plot the regression equation along with the scatter plot:
  
  Plot1 has already been turned on, and the regression equation has been input into the [Y=] window, so all we have to do is press [GRAPH], and the graph appears on the screen, along with the scatter plot.
Example 2: Here we use the data from Problem 4 on page 5, which gives the profit (in millions of dollars) for The Gap, Inc. for the years from 1990 to 1997.

Year (since 1990) 0 1 2 3 4 5 6

Sales ($ million) 144.5 229.9 210.7 258.4 350.2 354.0 452.9

Once again, we wish to find a curve that best represents the relationship between the two variables.
1. Enter the data as we did earlier.
2. View the corresponding scatter plot.
  
  The scatter plot looks fairly linear, but, since this is financial data, we want a growth rate, so we will look for an exponential model.
3. Follow the same steps to obtain and store the regression equation, but this time choose 0:ExpReg, in the [STAT] CALC menu.
  
  This yields the values a=160.7838984 and b=1.186937869 for the equation y=a*b^x, so that our regression equation is approximately
  
  y = 160.8 (1.187)^x.
  Recall that our model is
  
  P = P₀ (1 + r)^t.
  We have found the growth factor 1 + r to be approximately 1.187. This means that the growth rate r = 0.187. We can conclude that the profits for The Gap, Inc. were growing at 18.7% for this time period.
  
  To obtain the r= and r²= output values in the regression equation screen, go to [CATALOG] (which is [2nd] [0]), scroll down to DiagnosticOn, and press [ENTER] twice. Then perform the regression. In this case, r=.9620515412.
4. Finally, press [GRAPH] to view the exponential graph with the scatter plot.

Last modified: Wed Mar 27 2002

Year (since 1990)	0	1	2	3	4	5	6
Sales ($ million)	144.5	229.9	210.7	258.4	350.2	354.0	452.9

a	3	4	5	6
S	105	117	141	152