Histogram -- Frequency Polygon -- Ogive

This page is devoted to presenting, in a step by step fashion, the keystrokes and the screen images for doing a generating a Histogram -- Frequency Polygon -- Ogive on a TI-83 (TI-83 Plus, or TI-84 Plus) calculator. We will use the histogram feature on the calculator first, but this does not give us as much as we need. Then we will use the COLLATE3 program to help generate some required values. Then we can return to using the calculator to produce some of the required images.

We need to start with some data. We will generate a list of data on the calculator using GNRND4 with Key 1=1106176804 and Key 2=38403258. That list will be the same numbers that appear in the following table: Thus, our problem will be to generate a histogram for the data in the list above.

Note that the earlier page related to frequency tables for continuous data went through the steps that you need for creating a histogram. We will start by repeating those steps.

Figure 1 We begin by running the GNRND4 program with the given key values.

Figure 2 This produces the list of values that we want. We recall that those values are stored in L₁.

Figure 3 Use to open the STAT PLOTS menu. Here we see that Plot1 is set to display a histogram using the data in L₁. This is just what we want.
Press to move to the ZOOM menu.

Figure 4 Now we move down the menu to find the desired ZoomStat option. Once it is highlighted, press to perform that option.

Figure 5 The calculator examines the data in L₁ and then it creates a histogram based on those values. We can inspect the valeus that the calculator used by moving to the WINDOW screen (press ).

Figure 6 Here ae the values being used. In particular, the lower limit of the first class of values is 232.1 (the leading 2 is covered by the blinking cursor). the width of each class is 24.385714... and, if there were an additional class beyond the 8 needed to cover the values in L₁ then that additional class would start at 427.18571... That value is the effective upper limit of the last class in our histogram.
We can decide to change these values if we want a slightly different structure for our histogram.

Figure 7 We have set these values so that the start of the historgram is at 230, the width of each class is 20, and the effective upper limit of the final class is 410.

Figure 8 Having set new WINDOW values in Figure 7, we press to get the new histogram shown in Figure 8. Both the histogram of Figure 5 and the histogram of Figure 8 are correct. The difference is the choice of lower limits and the class width. The only advantage related to the histogram of Figure 8 is that the lower limit and width are nice, even values. As demonstrated in earlier pages, we could actually shift into "trace" mode and find the number of items in each of the classes (columns) shown in Figure 8. However, we will take the slightly easier approach of using the COLLATE3 program to do this for us.
We have conveniently skipped over the real problem here. We have had the calcualtor produce a histogram but we have not talked about our producing one. We will return to this issue in Figure 33.

Figure 9 We exit the graph mode and select the COLLATE3 program from the list of programs.

Figure 10 When we run the program it asks us for the location of the data. We give it L₁.

Figure 11 The program gives us the low and high values in our data and asks for the starting point of the histogram. As we did with the earlier histogram we will set that starting point to be 230.

Figure 12 The progream gives a suggested width, but we had already made up our mind to use the value 20. We respond with that value.

Figure 13 Figure 13 shows some of the display output from the program.

Figure 14 Figure 14 shows the rest of the display output from the program.

Figure 15 We have seen the COLLATE3 program before. We know that it creates other lists while it does its work. We move to the STAT menu and then open the Stat Editor to see those values. This gives, for each class, the LOW values, the count of values in the class (i.e., the frequency), and the relative frequency of those counts.

Figure 16 We can run down the list to see the remaining class values. Note that the LOW list has an extra element holding the lower limit of the next class if it were there.

Figure 17 Moving the cursor to the right allows us to see two of the remaining lists in the display. CMCNT holds the cumulative count of values, while CRFRQ holds the cumulative relative frequencies.

Figure 18 As you may have noted in our earlier work, the histogrm is not the only style of graph that the calculator can do. We will create a relative frequency graph from the output lists of COLLATE3. Since those lists are already in the calculator, we return to the STATS PLOT menu via . We want to change Plot1 and that the choice that is highlighted so we just press .

Figure 19 Figure 19 shows the Plot1 screen after we changed it to select the line chart (second chart option on the top line of charts) and we have moved the cursor to follow the Xlist: field. The current value of that is L₁. We want to use the LOW values as the Xlist:. To do this we can either type the characters of the name or move to the LIST menu and select the LOW entry there. This latter option is shown in Figure 19a.

Figure 19a Selecting LOW from the list of lists.

Figure 20 We want the Ylist: to be RFREQ so that we have a graph that shows the relative frequency. Again, we can type the name, note that the calculator os already in "alpha" mode, or we can return to the LIST menu and select the desired name.

Figure 21 Figure 21 shows the screen with all the changes made. Press to move to the ZOOM menu.

Figure 22 On the ZOOM menu, move down to highlight the ZoomStat option. Press to perform that option.

Figure 23 The result is an error! We read the error message, ERR:DIM MISMATCH and press to choose the Quit option, there being no other choice.
So what went wrong? We have actually seen the cause of the error, and even commented on it, although we did not recognize it as a problem at the time. We have asked the calculator to create a graph based upon pairs of values in two lists, _LLOW and _LRFREQ. The problem is that the program COLLATE3 put an extra value into _LLOW. We noted this in Figure 16. When I wrote COLLATE3 I thought that it would be helpful to add that extra value so that it would be easy to find the upper limit of the last class. However, that means that the two lists, _LLOW and _LRFREQ have different numbers of elements. This is the DIM MISMATCH of the error message. We no longer need the extra value in LOW. Therefore, the easiest fix to this problem is to remove that value.

Figure 24 We return tot he Stat Editor, move down to the extra value in LOW, and then press to move to Figure 25.

Figure 25 Having fixed the problem, we return to the ZoomStat option, shown in Figure 26.

Figure 26 Press to have the calculator perform the action. The result is a relative frequency chart as chown in Figure 27.

Figure 27 This is the relative frequency chart that we wanted.
If, on the other hand, we want a chart of the cumulative relative frequency, an Ogive chart, then all we need to do is set Plot1 to use the list CRFRQ.

Figure 28 To do this we return to the STAT PLOTS menu and select Plot1.

Figure 29 On the Plot1 screen we change the Ylist: value to be CRFRQ.

Figure 30 We will need to have the calcualtor redo its analysis and reset the appropriate WINDOW values. To accomplish this we return to the ZOOM menu and again select and perform ZoomStat.

Figure 31 Here we have a cumulative frequency chart for our data, remembering that we really have the cumulative relative frequency of the classes into which we divided all of the original data.

Figure 32 If we press to enter trace mode, and then move to the right three times, we have highlighted a point on the chart. That point corresponds to the value 290 with a cumulative relative frequency of 0.4057971. We should recall that we chose the Xlist: to be the values in LOW and that is what the calculator is reporting. However, we are really looking at 0.4057971 as the relative frequency of original data points through the fourth class, the class that runs from 290 to just before 310.

Figure 33 As indicated above in the discussion tied to Figure 8, although we have produced a histogram on the calculator, we do not know, at this point, how high to make each bar if we are actually drawing a histogram. We did look at this problem in a much earlier page dealing with bar charts. The same logio applies here. If we pick any particular column, say the column representing the class from 310 to just before 330, and we decide to have that column in the histogram be 7 centimeters tall, then all other columns must be in proper proportion given their respective frequencies. The class from 310 to almost 330 has 16 values in it. Therefore, for any other class we can find its required height by taking its frequency times 7 and dividing by 16. We can do this all at once with the command _LICNT*7/16 We have constructed such a command at the bottom of Figure 33 and then assigned the result to be stored in L₂. (We get the name of the list from the LIST menu and then complete the command with the appropriate keys.)

Figure 34 After performing the command that we created in Figure 33 we return to the STAT menu to get to the SetUpEditor command, which we then select.

Figure 35 In this case we want to create the command SetUpEditor _LLOW,L₂ We return to the LIST menu to get the _LLOW name.

Figure 36 We find the LOW name and press .

Figure 37 Then we complete the command via . Finally, press to perform the command.
Then when we re-open the Stat Editor it will show those two lists.

Figure 38 Here we see the start of the two lists. From this we can tell that the height of the claas from 230 to just before 250 needs to be 1.3125 centimeters. The height of the claas from 250 to just before 2570 needs to be 1.75 centimeters. All of this based upon our first establishing that we wanted the height of the class from 310 to just before 330 to be 7 centimeters, and that is the value shown here.

Figure 39 Finally, we move down to see the required height of each of the remaining classes.

Figure 1	We begin by running the GNRND4 program with the given key values.
Figure 2	This produces the list of values that we want. We recall that those values are stored in L₁.
Figure 3	Use to open the STAT PLOTS menu. Here we see that Plot1 is set to display a histogram using the data in L₁. This is just what we want. Press to move to the ZOOM menu.
Figure 4	Now we move down the menu to find the desired ZoomStat option. Once it is highlighted, press to perform that option.
Figure 5	The calculator examines the data in L₁ and then it creates a histogram based on those values. We can inspect the valeus that the calculator used by moving to the WINDOW screen (press ).
Figure 6	Here ae the values being used. In particular, the lower limit of the first class of values is 232.1 (the leading 2 is covered by the blinking cursor). the width of each class is 24.385714... and, if there were an additional class beyond the 8 needed to cover the values in L₁ then that additional class would start at 427.18571... That value is the effective upper limit of the last class in our histogram. We can decide to change these values if we want a slightly different structure for our histogram.
Figure 7	We have set these values so that the start of the historgram is at 230, the width of each class is 20, and the effective upper limit of the final class is 410.
Figure 8	Having set new WINDOW values in Figure 7, we press to get the new histogram shown in Figure 8. Both the histogram of Figure 5 and the histogram of Figure 8 are correct. The difference is the choice of lower limits and the class width. The only advantage related to the histogram of Figure 8 is that the lower limit and width are nice, even values. As demonstrated in earlier pages, we could actually shift into "trace" mode and find the number of items in each of the classes (columns) shown in Figure 8. However, we will take the slightly easier approach of using the COLLATE3 program to do this for us. We have conveniently skipped over the real problem here. We have had the calcualtor produce a histogram but we have not talked about our producing one. We will return to this issue in Figure 33.
Figure 9	We exit the graph mode and select the COLLATE3 program from the list of programs.
Figure 10	When we run the program it asks us for the location of the data. We give it L₁.
Figure 11	The program gives us the low and high values in our data and asks for the starting point of the histogram. As we did with the earlier histogram we will set that starting point to be 230.
Figure 12	The progream gives a suggested width, but we had already made up our mind to use the value 20. We respond with that value.
Figure 13	Figure 13 shows some of the display output from the program.
Figure 14	Figure 14 shows the rest of the display output from the program.
Figure 15	We have seen the COLLATE3 program before. We know that it creates other lists while it does its work. We move to the STAT menu and then open the Stat Editor to see those values. This gives, for each class, the LOW values, the count of values in the class (i.e., the frequency), and the relative frequency of those counts.
Figure 16	We can run down the list to see the remaining class values. Note that the LOW list has an extra element holding the lower limit of the next class if it were there.
Figure 17	Moving the cursor to the right allows us to see two of the remaining lists in the display. CMCNT holds the cumulative count of values, while CRFRQ holds the cumulative relative frequencies.
Figure 18	As you may have noted in our earlier work, the histogrm is not the only style of graph that the calculator can do. We will create a relative frequency graph from the output lists of COLLATE3. Since those lists are already in the calculator, we return to the STATS PLOT menu via . We want to change Plot1 and that the choice that is highlighted so we just press .
Figure 19	Figure 19 shows the Plot1 screen after we changed it to select the line chart (second chart option on the top line of charts) and we have moved the cursor to follow the Xlist: field. The current value of that is L₁. We want to use the LOW values as the Xlist:. To do this we can either type the characters of the name or move to the LIST menu and select the LOW entry there. This latter option is shown in Figure 19a.
Figure 19a	Selecting LOW from the list of lists.
Figure 20	We want the Ylist: to be RFREQ so that we have a graph that shows the relative frequency. Again, we can type the name, note that the calculator os already in "alpha" mode, or we can return to the LIST menu and select the desired name.
Figure 21	Figure 21 shows the screen with all the changes made. Press to move to the ZOOM menu.
Figure 22	On the ZOOM menu, move down to highlight the ZoomStat option. Press to perform that option.
Figure 23	The result is an error! We read the error message, ERR:DIM MISMATCH and press to choose the Quit option, there being no other choice. So what went wrong? We have actually seen the cause of the error, and even commented on it, although we did not recognize it as a problem at the time. We have asked the calculator to create a graph based upon pairs of values in two lists, _LLOW and _LRFREQ. The problem is that the program COLLATE3 put an extra value into _LLOW. We noted this in Figure 16. When I wrote COLLATE3 I thought that it would be helpful to add that extra value so that it would be easy to find the upper limit of the last class. However, that means that the two lists, _LLOW and _LRFREQ have different numbers of elements. This is the DIM MISMATCH of the error message. We no longer need the extra value in LOW. Therefore, the easiest fix to this problem is to remove that value.
Figure 24	We return tot he Stat Editor, move down to the extra value in LOW, and then press to move to Figure 25.
Figure 25	Having fixed the problem, we return to the ZoomStat option, shown in Figure 26.
Figure 26	Press to have the calculator perform the action. The result is a relative frequency chart as chown in Figure 27.
Figure 27	This is the relative frequency chart that we wanted. If, on the other hand, we want a chart of the cumulative relative frequency, an Ogive chart, then all we need to do is set Plot1 to use the list CRFRQ.
Figure 28	To do this we return to the STAT PLOTS menu and select Plot1.
Figure 29	On the Plot1 screen we change the Ylist: value to be CRFRQ.
Figure 30	We will need to have the calcualtor redo its analysis and reset the appropriate WINDOW values. To accomplish this we return to the ZOOM menu and again select and perform ZoomStat.
Figure 31	Here we have a cumulative frequency chart for our data, remembering that we really have the cumulative relative frequency of the classes into which we divided all of the original data.
Figure 32	If we press to enter trace mode, and then move to the right three times, we have highlighted a point on the chart. That point corresponds to the value 290 with a cumulative relative frequency of 0.4057971. We should recall that we chose the Xlist: to be the values in LOW and that is what the calculator is reporting. However, we are really looking at 0.4057971 as the relative frequency of original data points through the fourth class, the class that runs from 290 to just before 310.
Figure 33	As indicated above in the discussion tied to Figure 8, although we have produced a histogram on the calculator, we do not know, at this point, how high to make each bar if we are actually drawing a histogram. We did look at this problem in a much earlier page dealing with bar charts. The same logio applies here. If we pick any particular column, say the column representing the class from 310 to just before 330, and we decide to have that column in the histogram be 7 centimeters tall, then all other columns must be in proper proportion given their respective frequencies. The class from 310 to almost 330 has 16 values in it. Therefore, for any other class we can find its required height by taking its frequency times 7 and dividing by 16. We can do this all at once with the command _LICNT7/16 We have constructed such a command at the bottom of Figure 33 and then assigned the result to be stored in L₂. (We get the name of the list from the LIST* menu and then complete the command with the appropriate keys.)
Figure 34	After performing the command that we created in Figure 33 we return to the STAT menu to get to the SetUpEditor command, which we then select.
Figure 35	In this case we want to create the command SetUpEditor _LLOW,L₂ We return to the LIST menu to get the _LLOW name.
Figure 36	We find the LOW name and press .
Figure 37	Then we complete the command via . Finally, press to perform the command. Then when we re-open the Stat Editor it will show those two lists.
Figure 38	Here we see the start of the two lists. From this we can tell that the height of the claas from 230 to just before 250 needs to be 1.3125 centimeters. The height of the claas from 250 to just before 2570 needs to be 1.75 centimeters. All of this based upon our first establishing that we wanted the height of the class from 310 to just before 330 to be 7 centimeters, and that is the value shown here.
Figure 39	Finally, we move down to see the required height of each of the remaining classes.