Appendix A - 1
Graphing and Analyzing Linear Data[1]
A graphing calculator can be used both to make a scatter plot of data and to fit an equation to the data. The simplest type of relationship between variables that can be observed is linear. In a linear relationship, the prediction equation which relates one variable to the other is the equation of a line. Once the data have been entered into the data lists in the statistics environment of the calculator, the calculator can be used to determine the equation which best matches the data. This equation is called a regression equation.
The following data which show the mass and weight of an object can be used to find the equation of the best fit line.
Mass (kg) / 0.100 / 0.200 / 0.300 / 0.400 / 0.500 / 0.600Weight (N) / 1.00 / 2.00 / 3.00 / 3.90 / 5.00 / 5.90
©Modeling Workshop Project 20021Appendix A v2.0
First, the data must be stored in the calculator. This is accomplished in the STATistics area of both the TI-82 and 83 calculators. The screens which are shown are from the TI-82. TI-83 users will notice some screen differences, but the instructions are the same unless otherwise noted.
Press the [STAT] key. Since EDIT is already selected, press [ENTER], which takes you to the calculator’s data lists.
If there are data in the lists labeled L1 and L2, then clear these lists. To clear L1, move the cursor to the topmost cell, which contains the label L1.
Press [CLEAR] and then [ENTER]. The data stored in the list will be erased. Repeat this process for L2.
Enter the independent variable data into the L1 list, pressing [ENTER] after each value. Enter the dependent variable data into the L2 list.
Once data have been entered into the lists, you must tell the calculator what lists to plot. To do this, you need to enter the area where the statistics plots are defined. To select STAT PLOT, press [2nd] [Y=]. A screen similar to the one at the right is displayed.
In order to select Plot1 for setup, press [ENTER]. Adjust the settings to match those shown at the right. You have now instructed the calculator to make a scatter plot using the values stored in L1 as the x- values and those stored in L2 as the y-values. [TI-83 Users: The equivalent 83 screen has more choices for the plot type and also provides a shortcut to screens on which additional plots can be defined. The Xlist and Ylist are defined by moving the cursor to the appropriate field and then using the [2nd] key in conjunction with one of the number keys, 1 - 6, to select a list from L1 to L6. The screen should be adjusted to match the TI-83 screen shown above.
The calculator can be instructed to set the graphing WINDOW so that all the data are displayed. Press [ZOOM] and then move the cursor down to 9:ZoomStat.
Press [ENTER]. The data will be displayed as shown here. If extraneous lines/curves appear on your screen, you need to clear out previously defined functions. To enter the function definition environment, press [Y=] and use [CLEAR] to remove the expressions.
Frequently, ZoomStat produces a viewing window which does not include the origin. To view the entire first quadrant, press [WINDOW] and change the Xmin and Ymin values to 0. The Xscl and Yscl values control the spacing between the hashmarks on the axes and should be set to about one-tenth the Xmax and Ymax values.
Pressing [GRAPH] yields the graph shown at the right. The data appear to follow a fairly linear distribution.
The calculator can compute the slope and the y-intercept of the best-fit line. Press [STAT]. This time, move the cursor to the CALC field which displays the various calculation options.
Choose (by moving the cursor to 5:LinReg(ax+b) and pressing [ENTER] or by pressing 5) 5:LinReg(ax+b) which tells the calculator to compute the equation of the best fit straight line, using a statistical technique called linear regression. On the TI-83, this option is number 4. Note: the expression, LinReg(ax+b) will be pasted to the home screen. The calculator is waiting to be told which lists contain the x (independent) and the y (dependent) variables.
Recall that the independent variable data are stored in L1 and thedependent variable data are stored in L2. Press [2nd] [1] to choose L1 followed by [,] then [2nd] [2] to choose L2.
(TI-83 users can paste the regression equation into the Y1= field at the same time by pressing [,] [VARS] and then using the right arrow to highlight Y-VARS to drop the y-variables menu. Press [ENTER] [ENTER] the expression Y1 is copied to the screen. Skip the next 3 paragraphs.)
Press [ENTER] and, the equation will be displayed, along with the correlation coefficient which indicates how well the equation fits the data. The closer |r| is to 1, the better the equation fits the data. TI-83 users will not see the correlation coefficient unless the Catalog has been used to select DiagnosticOn. Press [2nd][0] to select Catalog. Then press
[x-1] to select D and scroll down to DiagnosticOn. Press [ENTER] to select that setting, and the correlation will automatically be displayed when the regression line is computed.
To visually inspect the fit of the line, the regression equation can be plotted, superimposed on the scatter plot. This takes several key strokes but is well worth the effort. Press [Y=] to enter the function window. You want to define the Y1 function to be the regression equation. Move the cursor to the Y1= field.
Next, press [VARS] (to select the variables menu) and select 5:Statistics.... Use the right arrow to move the cursor to EQ and then select RegEQ, which is option 7 on the TI-82 and option 1on the TI-83. The regression equation is then pasted to the Y1= field.
Press [GRAPH] to see both the data and the line.
Returning to the home screen by pressing [2nd] [MODE], you can note the slope (a) and the y-intercept (b) of the line.
In this case, the slope of the graph is about 9.8 N/kg, and the y-intercept is about 0.027 N. What is the physical meaning of the slope and the y-intercept?
APPENDIX A-2
Using the Graphing Calculator to Linearize Data
Often the data shows variables which are related to one another in a nonlinear way. That is, a graph of one variable against the other does not produce a straight line. The graphing calculator can be utilized to determine the relationship between the variables. Consider the following data set.
Mass(kg) / 0.50 / 1.00 / 1.50 / 2.00 / 2.50 / 3.00Distance(cm) / 99.7 / 54.8 / 36.5 / 27.8 / 22.6 / 18.8
Press the [STAT] key and press [ENTER] in order to access the data lists, as described on the first page of this appendix. Enter the mass data into the L1 list and the distance data into the L2 list.
Press [2nd] [Y=] and then press [ENTER] to select Plot1. Define Plot1 as shown at the right. Recall that the TI-83 screen is somewhat different, but that the settings should be the same.
Press [ZOOM] and select 9:ZoomStat to view a graph of the data in a window chosen by the calculator to display all the data.
Since the y-variable decreases as the x-variable increases, an inverse proportion may fit the data. The calculator can be used to plot a graph of y versus 1/x by storing the 1/x values in another list.
1.To enter the list editing environment, press [STAT] and press [ENTER]. Move the cursor to the L3 column label and press [ENTER]. A flashing cursor will be seen at the bottom of the screen.
2.To instruct the calculator to store the reciprocal x values in the L3 column, press [2nd] [1] to tell the calculator to use values from the L1 list and then [x-1] to instruct the calculator to store the reciprocals of the L1 values in the L3 list.
3.Pressing [ENTER] will cause the reciprocals of the L1 values to be stored and displayed in the L3 list.
4.The calculator must now be told to plot a graph of the L2 values (distance) versus the L3 values (1/mass). Press [2nd] [Y=] to view the STAT PLOTS screen. Plot1 must be redefined. Press [ENTER] in order to edit Plot1. Adjust the settings to agree with those shown at the right. (Recall that the TI-83 plot setup screen is somewhat different from the TI-82 screen. Refer to page 2 of this appendix.)
5.Press [ZOOM] and select option 9:ZoomStat in order to adjust the WINDOW settings to fit the data.
Since this relationship appears to be linear, we can compute the equation of the best fit line (regression line) and superimpose this line on the data plot using essentially the same steps as those used in analyzing the data on pages 1 - 3 of this appendix. The key difference is that the horizontal axis data are now located in L3 instead of L1.
The calculator can compute the slope and the y-intercept of the best-fit line. Press [STAT]. This time, move the cursor to the CALC field which displays the various calculation options.
Choose 5:LinReg(ax+b) (or 4:LinREg(ax+b) on the TI-83) which tells the calculator to compute the equation of the best fit straight line, using a statistical technique called linear regression. Note: the expression, LinReg(ax+b) will be pasted to the home screen. The calculator is waiting to be told which lists contain the quantities to be plotted on the x and y axes. Recall that the reciprocal mass values are stored in L3 and thedistance data are stored in L2. Press [2nd] [3] to choose L3 followed by [,] then [2nd] [2] to choose L2.
Press [ENTER] to view the regression equation.
Recall that ZoomStat frequently produces a view that does not include the origin. Press [WINDOW] and reset Xmin and Ymin to 0. You may also need to adjust the Xscl and Yscl values to about one-tenth of the maximum x and y values.
Press [GRAPH] to view the graph in the modified window.
To visually inspect the fit of the line, the regression equation can be plotted, superimposed on the scatter plot. Press [Y=] to enter the function window. You want to define the Y1 function to be the regression equation. Move the cursor to the Y1= field. Next, press [VARS] (to select the variables menu) and select 5:Statistics.... Move the cursor to EQ and then select RegEQ, which is option 7 on the 82 and option 1on the 83. The regression equation is then pasted to the Y1= field.
Press graph to see both the data and the line. The line appears to be a good match to the data. We conclude that the equation which relates the distance to the mass is given by:
The y-intercept is less than 5% of the maximum y-value so we can conclude that the y-intercept is essentially 0. Therefore the relationship can be written as
Linearizing Data -- Another Example
The technique for linearizing data described on pages 4 - 5 of this appendix can be used whenever the data must be manipulated in order to produce a straight line plot. As an example, consider the following data set.
Stretch Distance (cm) / 2.0 / 4.0 / 6.0 / 8.0 / 10.0 / 12.0 / 14.0 / 16.0Distance up Ramp (cm) / 4 / 8 / 17.5 / 30 / 48.5 / 69 / 93 / 117
Enter the data into the data lists and display a graph of the data following the instructions on page 5. Note that the y-variable increases more rapidly than the x-variable in this data set. To linearize this data, the x-values must be squared.
Return to the [STAT] EDIT screen and store x2 values in L3 using the strategy described on page 5 of this appendix. Go to the STAT PLOTS definition screen by pressing [2nd][Y=] . Define Plot1 to be a scatter plot of L2 versus L3. Then, use [ZOOM]9:ZoomStat to plot a graph of the manipulated data. The graph WINDOW can be fine-tuned by pressing [WINDOW] and setting Xmin and Xmax to 0 and setting Xscl ~ Xmax/10 and Yscl ~ Ymax/10. Press [GRAPH] to view the graph.
Since the graph appears to be linear, the equation of the best fit line should be computed using the LinReg(ax+b) option from the [STAT] CALC menu. (For detailed instructions refer to page 6 of this appendix.) Remember to tell the calculator that L3 and L2 contain the x and y variables, respectively.
To visually inspect the fit of the line to the data, the regression equation can be pasted into the Y1= field of the [Y=] screen. Press [Y=] and clear out the contents of Y1 by moving the cursor to the Y1= field and pressing[CLEAR]. To paste the equation into the space, requires the following sequence: press [VARS] (to select the variables menu) and select 5:Statistics.... Move the cursor to EQ and then select RegEQ, which is option 7 on the 82 and option 1on the 83. The regression equation is then pasted to the Y1= field. Press [GRAPH] to view the line superimposed on the plotted points. The line appears to be a good fit to the data. We can write the equation as:
Since the y-intercept (1.61) is small (<5%) compared to the maximum y-value (117) , the equation can be written as
Appendix A-3
Using the Get Program (TI-82) orGetCalc( (TI-83) to Share DataTI-82
Data can be transferred to another calculator without leaving the data collection program and without using the LINK environment. Assume that data have been collected and are residing in student A’s calculator and that the students want to save this data before performing another trial. The following program, GET, should stored in student B’s calculator.
GET
ClrHome
ClrList LÏ,LÌ
0ÍLÏ(1)
Disp "LINK CALCULATORS"
Disp "AND HIT [ENTER]."
Pause
Lbl æ
ClrHome
Disp "GETTING LISTS..."
Get(LÏ)
Get(LÌ)
Get(LÓ)
Get(LÔ)
Get(L�)
Get(LÒ)
If dim LÏ=1 and LÏ(1)=0
Then
ClrHome
Disp "*TRANSMIT ERROR*"
Disp ""
Disp "CHECK LINK CORD"
Disp "CONNECTION AND"
Disp "HIT [ENTER] TO"
Disp "TRY AGAIN."
Pause
Goto æ
End
ClrHome
PlotsOff
FnOff
Plot1(xyLine,LÏ,LÌ,Å)
ZoomStat
Stop
Connect calculator A to calculator B with the black link cable. Execute the GET program on calculator B by pressing [PRGM],scrolling down to find GET, and pressing [ENTER]. Calculator B will then get the data in the lists of calculator A and will display a graph of the data in lists 1 and 2. Calculator B can now be used to store the data as a program for later analysis, as described in the next section.
TI-83
The TI-83 has a GetCalc( function which can be used to “grab” data from another TI-83 calculator one list at a time. Connect the two calculators (both must be TI-83’s) together with the black link cable.The GetCalc( command is accessed from the Catalog environment by pressing [2nd][0] and then [tan] to tell the calculator to scroll to the G section of the catalog. Use the down arrow to move the cursor to GetCalc( and press [ENTER]. The expression, GetCalc( will be pasted to the home screen. Tell the calculator what list should be transferred by pressing [2nd] followed by the appropriate number key and close the parentheses. Press [ENTER] and the list will be transferred. Repeat the steps for each list to be transferred.
Appendix A-4
Storing Data Lists in Programs
Both the TI-82 and the TI-83 have six different lists for storing data. Sometimes it is necessary to clear data from a list yet maintain the data in the calculator’s memory. The programming area of the TI-82/83 provides a nice vehicle for saving the data for later recall into the lists.
Enter the following data from page 5 of this appendix into lists L1 and L2.
Mass(kg) / 0.50 / 1.00 / 1.50 / 2.00 / 2.50 / 3.00Distance(cm) / 99.7 / 54.8 / 36.5 / 27.8 / 22.6 / 18.8
Press [PRGM] and move the cursor to the NEW field. Press [ENTER] to Create New program. You will be asked to type the name of the program to be created. The input mode is set automatically to ALPHA-LOCK. Type in a name (8 characters or less) for your program (e.g. DATA) and press [ENTER].
You are now in the program editing environment and can write your data storage program in a few easy steps.
1. To retrieve the contents of L1, you need to recall (RCL) L1 by pressing [2nd] [STO][2nd] [1] which should produce the screen at the right.
2.Press [ENTER] to paste the contents of L1 into the program.
3.Tell the calculator to store those numbers in L1 by pressing [STO][2nd] [1]. Then, press [ENTER].
4.Repeat steps 1 - 3 for L2, being careful to replace L1 with L2. Your program should appear as shown at the right.
QUIT the program editor by pressing [2nd] [MODE]. Your program has been automatically stored in the calculator’s memory, as you composed it.
©Modeling Workshop Project 20021Unit I Teacher Notes v2.0
[1] The information in this set of appendices was written 6 years ago. By and large, the procedures for data manipulation and graphing are still correct, but the information about data transfer via GraphLink needs to be updated.