2.6 - Exploring Data: Linear Models and Scatter Plots

This section ties in heavily with the notes for a statistics class. In particular, look at the Introduction to Statistics and Lists on the TI82 and Scatter Plots and Regression Lines on the TI82. With that said, I will try to convey most of that information here, also.

I have included notes for the TI82, TI83, and TI85 calculators. I don't normally do the TI85, but the statistics mode on it is fairly difficult to grasp. I do not have notes on how to use any of the other calculators, like the Casio or Sharp. If you would rather see all of the notes for your calculator in one location, then see the lecture notes for either the TI82 / TI83 or the TI85. Everything here is included there, so you only need to print one or the other.

TI83 Users Only

There is a correlation coefficient which is mentioned in the book. By default, the TI83 does not give this to you. You can enable it (you only need to do this once, and then it's done forever more until you lose power or reset your calculator) by going [Catalog] (2nd zero). Then, scroll down to DiagnosticOn (hit D [the calculator is already in alpha mode, so just press the inverse key] to get close quickly) and press enter twice until the calculator says Done.

Correlation and Regression

The Linear Correlation Coefficient (r) is a measure of the strength and direction of a relationship between two variables. If the y gets larger when the x gets larger, the coefficient is positive and if the y gets smaller when the x gets larger, the coefficient is negative. If there is no linear relationship between the two variables, then the coefficient is zero. If all the data exactly lies on a line, then it is called perfect correlation and the value will either be 1 or -1. The closer the value is to 1 or -1, the closer the points are to the line and the stronger the linear relationship. The same concept applies for other types of regression (the TI82, TI83, and TI85 will do linear, logarithmic, exponential, power, quadratic, cubic, and quartic regression).

The Regression Equation or Regression Line is the equation of the line which best fits the data.

One of the uses of regression is to see if there is a correlation between the two variables. This is indicated by the value of the correlation coefficient r. Another use of regression is to predict values.

Clearing Existing Data in the Calculator

TI82 / TI83

I'm going to assume that you're putting the x-coordinates into List 1 and the y-coordinates into List 2. Make appropriate modifications to the instructions if you use different lists.

  1. Hit the [Stat] key.
  2. Choose the Edit option.
  3. Arrow all the way to the top (above the line) of the list that you're going to use. Usually this will be L1.
  4. Press the [Clear] key
  5. Press [Enter]
  6. Arrow to the next list to clear out and repeat steps 3-5.

TI85

I'm going to assume that you're putting the x-coordinates into xStat and the y-coordinates into yStat. Make appropriate modifications to the instructions if you use different lists.

  1. Hit the [Stat] key.
  2. Choose Edit [F2]
  3. [Enter] twice to accept the lists xStat and yStat (you can change names if you want to).
  4. Choose CLRxy [F5]

Entering Data into the Calculator

TI82 / TI83

With the TI82 or TI83, you enter all the x-coordinates first and all the y-coordinates second. If you have just cleared out your data, skip steps 1 and 2 and begin on step 3.

  1. Hit the [Stat] key
  2. Choose the Edit option
  3. Arrow to the first element in the list to be used to hold the x-coordinates. This is the first element below the line.
  4. Type in the first x-value and hit [Enter].
  5. Repeat for each additional x-value.
  6. Arrow to the list to be used for the y-coordinates. Repeat steps 4 and 5 for the y-coordinates.
  7. Choose [Quit] (2nd Mode)

TI85

With the TI85, you enter pairs of x-coordinates and y-coordinates. If you have just cleared out your data, skip steps 1 through 3 and begin on step 4.

  1. Hit the [Stat] key.
  2. Choose Edit [F2]
  3. [Enter] twice to accept the lists xStat and yStat (you can change names if you want to).
  4. Enter the x-coordinate of the first point and hit [Enter]
  5. Enter the y-coordinate of the second point and hit [Enter]
  6. Repeat steps 4 and 5 for each additional point
  7. [Exit] twice to get back to the main screen.

Scatter Plots

TI82 / TI83

  1. Turn off all regular y= plots by going into [y=] and clearing them out (or move to the equal sign and hit enter).
  2. Hit [StatPlot] (2nd y=).
  3. Choose Plot1. You can use any of the three, but for simplicity sake, I'm going to use plot 1.
  4. Turn it ON.
  5. Choose the Scatter Plot (first plot type - with a bunch of dots)
  6. For the Xlist, choose List 1 (L1). On the TI82, just press enter. On the TI83, you will need to actually put in List 1 by pressing [2nd] and [1].
  7. For the Ylist, choose List 2 (L2).
  8. The Mark can be anything. I recommend the little square as it shows up better than the cross or dot.
  9. Hit [Zoom]
  10. Choose the Statistics option. Option 9 (just hitting 9 is quicker than scrolling down).

TI85

On the TI85, you must set up the viewing window before you view the scatter plot. Also, the ZoomFit option does not work to automatically set the window for the data. However, the TI85 does provide an easier way to graph the regression equation

  1. Turn off all regular y= plots by going [Graph], y(x) [F1] and either clearing or deleting the functions. You may also choose to leave the graphs but turn them all off by using [More] and All- [F2]
  2. Set your viewing window by choosing Range [F2]
  3. Set the values so that all the data values are displayed.
  4. Hit [Stat]
  5. Choose Draw [F3]
  6. Choose Scatter Plot [F2]

It will be difficult to see the dots on the scatter plot because they are just individual pixels, not boxes or crosses like with the TI82 or TI83.

Least Squares Regression Line

The least squares regression model is the line which minimizes the variation. Ok, if you really want to know, take a statistics class. In general terms it is the line that best fits the data. That is why it is sometimes called the best fit line.

TI82 / TI83

If you have a TI83, make sure you enable Diagnostics as explained at the top of this document. You need to only do that once.

The TI82 and TI83 return the regression equation y=ax+b. If you want the regression equation y=a+bx, then you should choose LinReg(a+bx) (option 9 or 8) instead.

  1. Hit the [Stat] key
  2. Arrow right to the Calc screen
  3. Choose the "LinReg (ax+b)" option. It's #5 on the TI82 and #4 on the TI83 .
  4. Tell the calculator where you put the data. Enter L1,L2 so that the screen says "LinReg(ax+b) L1,L2". To enter L1, hit [2nd] [1]. Be sure you get the comma between L1 and L2. The comma is right above the 7.
  5. Press [Enter] to execute the command.
  6. The calculator will return values for a, b, and r (and r^2 on the TI83). Write out the equation of the regression line as y = ax + b, but actually write in the values for a and b.

TI85

Warning! The TI85 returns the regression equation y=a+bx. The book and the other TI calculators will be using y=ax+b instead. Be sure to adjust appropriately.

If you have just plotted your scatter plot, then skip to step 2.

  1. Hit [Stat]
  2. Choose Calc [F1]
  3. Hit [Enter] twice to accept the lists containing your data
  4. Choose Linear Regression [F2]
  5. The calculator will return the y-intercept (a), slope (b), and the correlation coefficient r (corr). The TI85 also returns the number of values (n).

Drawing the Regression Line on the Scatter Plot

TI82 / TI83

You must compute the regression equation before doing this.

  1. Hit [y=]
  2. You could re-enter the equation that you wrote down, earlier, but it is a pain to do, and you're likely to lose some accuracy when you do that. While still waiting to enter the equation to plot, hit the [Vars] key.
  3. Choose Statistics (5)
  4. Arrow right to EQ
  5. Choose RegEQ. This is option 7 for the TI82 and option 1 for the TI83
  6. Hit [Graph]

TI85

You must compute the regression equation before doing this. If you just computed the regression equation, skip to step 2

  1. Hit [Stat]
  2. Choose Draw [F3]
  3. Choose Draw Regression Equation (DDREG) [F4]

Predicting Values

One of the uses of the regression equation is to predict values.

TI82 / TI83

You must have computed the regression equation and put it into y1 to use this.

  1. Hit [TblSet] (2nd Window)
  2. The easiest way is to set TblMin to be the value of x you wish to predict y for.
  3. Set delta-Tbl to be 1
  4. Hit [Table] (2nd Graph)
  5. You can scroll through and see other values if you wish.

If you wish to find the value of x that gives a specific y-value, then you need to solve the regression equation for x and then substitute the known value of y.

TI85

You must have computed the regression equation, but you do not have to graph it to use this. If you have just completed the Linear Regression, then skip to step 2.

  1. Hit [Stat]
  2. Choose Forecast (FCST) [F4]
  3. Enter the value for x you wish to use and press [Enter].
  4. With the cursor sitting on the y= line, press Solve [F5]

The TI85 can be used to find the values of x that give a particular value of y. To do that, enter the value for y, position the cursor on the x= line and hit solve.

When you're all done

Be sure to turn off the statistical plots and clear out the regression equation when you're done

TI82 / TI83

  1. Hit [StatPlot] (2nd y=)
  2. Choose PlotsOff (4)
  3. Press [Enter]

TI85

You don't have to do anything special. The statistical plots and y= plots are kept separate from each other. If you want to, however, you may clear your drawing

  1. Hit [Graph], [More]
  2. Choose Draw [F2]
  3. Hit [More] ClrDrw [F5]

Alternatively, and probably quicker ...

  1. Hit [Stat]
  2. Choose Draw [F3]
  3. Choose ClrDrw [F5]