Mean, Median, and Mode on the TI-89

This page will use the TI-89 to find the mean, median, and mode of two sets of data. As a side benefit, we will also find the range, the quartile points, and the standard deviation of the data. The first set of data is 1, 2, 3, 5, 1, 5, 1, 4, 3, 6, 1, 3, 4, 5, 6, 4, 1, 5, 2, 6, 2 We can use the built-in statistical capabilities of the TI-89 to do some of the work. We will want the TI-89 to perform what it calls a One Variable analysis of the data. There are a number of different forms that we can use to represent the data for the One Variable analysis on the TI-89. However, for now, we will preform that analysis on the data as it has been given above. To do this we will create a list that will hold the numeric values.

Figure 1 Figure 1 represents a "clean" TI-89 home screen. That screen has a command entry line, almost at the bottom, and a history area for most of the screen (between the menu items at the top and the command entry line at the bottom). Pressing the key will clear the command line. It is not necessary to clear that line, but doing so does remove some of the clutter.
In the same way, we can clear the history area by using the two key sequence . This will open a small window and select the Clear Home option. Again, this is not necessary.

Figure 2 We will start the problem by creating a list of values. We start a list with the { character. We press to generate that curly brace in the data line. Then we enter each of the numbers in the list, separating numbers with a comma. The keys to complete Figure 2 are and . This takes us to the right edge of the screen. We are not done entering the list, however.

Figure 3 In Figure 3 we continue entering the numbers in our list. Note that the command area display scrolls to the right so that we can see the new numbers and the commas. The remaining numbers are generated via and . We complete the list by entering the right curly brace, }, via the sequence . Finally, we want to store this list under the name l1 (note that this is a lowercase letter "l" and the digit "1"). The keys complete the input line from Figure 3.

Figure 4 We left Figure 3 with the complete command formulated in the command entry line. However, we still have to tell the TI-89 to perform the command. We do this by pressing the key. The TI-89 responds by first echoing the command in the history area and then displaying the result of the command in that same history area. We can see this in Figure 4 where we have two new lines in the history area. Note that the result of our creation of a list called l1 is an echo of that list. At the same time, the entire command input line has been highlighted.

Figure 5 We can move from the input line into the history area to inspect the information that is displayed there. To do this we press the key. Note in Figure 5 that the input area is no linger highlighted, and that the bottom line in the history area is now highlighted. The right-pointing arrow in that line indicates that we can shift the display to the right to see more of that information.

Figure 6 Figure 6 is the result of pressing the key 6 times, shifting the display for the highlighted line. First, note that the other lines are not affected. Second, we observe that the highlighted line now has both a left-pointed and right-pointed arrow at the respective edges of the list. These indicate that there are more values to be viewed at either end of the list.

Figure 7 We can move back to the input line area by pressing the key. Note, in Figure 7, that we have moved back to the input line, but that the shift in the bottom line of the history has not been changed.
Now that the desired list has been created in the calculator, we can proceed to doing the statistical computations. As noted above, we want to use the OneVar command. We could type this command. However, that would require a number of keystrokes and it would require that we completely remember the command that we want to use.

Figure 8 A method for manufacturing the OneVar command is to find that command in the catalog, a list of commands on the calculator. We open that catalog by pressing the key. Figure 8 shows the catalog window. The contents of the catalog are arranged alphabetically. The pointer is at the first entry, abs(. We need to move that pointer down the list to find our desired command.

Figure 9 One way to jump down, or up, the list is to press the key associated with the first character of the command we want. In our case, we want the OneVar command. Therefore, we press the key that is associated with the letter "o", namely the key. Pressing that one key causes the calculator to jump down the catalog names to the o's. Fortunately for us, the first "o" command is the one we want, OneVar. Figure 9 shows that command at the top of the list with the selector arrow pointing to it.
A careful examination of the screen in Figure 9 reveals that the calculator gives us some helpful hints about the appropriate use of the OneVar command. At the very bottom left of the screen we find the new message . This is information about the required and the optional parameters that are to follow the OneVar command. In this case we note that the command is to be followed by at least one list, and possibly by as many as four lists.

Figure 10 We press the key to actually select the OneVar command from Figure 9 and to paste it into the command line, replacing our entire former command, as is shown in Figure 10.
As noted in the discussion following Figure 9, the OneVar command needs to be followed by the name of a list. Quite by accident, our list is named "l1" much like the "l1" suggested by the calculator in Figure 9. We will enter our name for our list in Figure 11.

Figure 11 We enter our list name by pressing the keys. Figure 11 shows our complete command.

Figure 12 Now we press to perform the OneVar command. Figure 12 illustrates the results of that command. It almost appears as if the command did nothing. The only thing that has changed is that we have the command added to the history area, at the bottom left, and the word "Done" has been added to the lower right of the history area. Unfortunately, the OneVar command instructs the calculator to do certain computations, but it does not cause the TI-89 to display those results. The "Done" is our only indicator that the command was successfully completed.

Figure 13 We still need to find some way to see the results of the OneVar command. The command to show those results is the ShowStat command. Again, we can find it in the catalog, which we re-open via the key. Note that the catlog appears just as we left it in Figure 9.

Figure 14 We are looking for the ShowStat command. To move down the command list to the "S's" we press the key. The calculator display shifts to that shown in Figure 14. Unfortunately, there are many commands that start with the letter S. Our desired command is not even on the screen yet.

Figure 15 We can press the key 11 times to move down to the ShowStat command, as has been done in Figure 15. Note that we merely want the selector arrow to be pointing to the ShowStat command.

Figure 16 Having selected the ShowStat command in Figure 15, we press to place that command into the command line in Figure 16.

Figure 17 Once the command is in place in Figure 16, we press again to perform the command. Figure 17 displays the window that the ShowStat command opens. That window indicates that the mean of the data is 3.333333, that the sum of the values in our "l1" list is 70, and that the sum of the squares of the values in the list is 300. The fourth value, gives an estimate of a standard deviation based on a sample. This statistic is beyond the scope of the material presented here. The next three lines in the window indicate that the number of items in the list is 21, that the lowest value in the list is 1, and that the first quartile point is at 1.5. The final line in the window, , gives us the median value of the list as being 3. Although, at first, there is no special indication of the fact, we might suppose that there is even more to see in this window. After all, we have seen the first quartile, but not the third, and we have seen the minimum value, but not the maximum value in the list. A closer examination shows that the equal sign in the line is really a tiny arrow pointing down. This seems to be meant as our only on-screen indication that more information is available.

Figure 18 We press the key 2 times to shift the display to that shown in Figure 18. Here we see the final two values: the third quartile is at 5 and the maximum value is 6. At the same time, note that the equal sign for the median, the medStat line, is now correctly formed, whereas, the equal sign in the top output line of the window now has its equal sign contorted to be an up-arrow.

Figure 19 We have seen all that the ShowStat command has to offer. We press the to close that window.
We do need to recognize that ShowStat displays the results computed by the OneVar command issued earlier. The two commands need to be given to compute the values and then to display those values.
We might have noticed that the ShowStat command does not display the population standard deviation for our values. This is a value that is computed by the OneVar command. It is just not displayed by ShowStat on the TI-89.

Figure 20 We would like to see the value of the population standard deviation. It is stored in a system variable called . We will need to generate that name. We can use the TI-89 menu system to do this. We press to open the CHAR window, shown in Figure 20.

Figure 21 In Figure 20, the Greek option was highlighted, and that is the option we want. We open the Greek sub-window by pressing the key. Figure 21 shows that window. Our desired Greek character, , is not in that window. However the down-arrow on the sixth line suggests that more characters follow.

Figure 22 We move down the sub-window by pressing the key until we find the desired character, as shown in Figure 22.

Figure 23 Now we paste that character into the command line by pressing the key.
(Note: we have used the TI-89 menus to generate the Greek letter sigma, . An alternative method to generate the sigma character is to remember the sequence .)

Figure 24 Of course, we wanted not just , so we need to complete our command by pressing . This produces the command line shown in Figure 24.

Figure 25 Press to perform that command. The result is shown in Figure 25 where the history area holds the command and the associated value, 1.78174161275. This is the population standard deviation of the numbers in our original list.

Now we will turn our attention to finding the mode of the data. Unlike the finding the mean, the TI-89 does not have a built-in process for finding the mode. We have a program written for the TI-89 that will help in finding this value. That program is called COLLATE3. You can get a copy of the program from another TI-89 that has it, or you can use the TI-Graph Link program to transfer COLLATE3 from a PC that is storing it. The page collate3.htm holds a listing of the program (in case you want to type it into your calculator) and it has a link that will allow you to download the program to a PC (for subsequent transfer via TI-Graph Link).

Figure 26 In order to run the program COLLATE3 we need to manufacture that name in the command line. We could do this by typing the characters, or we could find the program in the variable listing on the TI-89. We will choose this second approach. We open the VAR-LINK window by pressing the keys. Note that the calculator used here has many programs and variables defined on it. If you are following along on a different calculator, then your display will be different, reflecting your different list of available programs and variables.

Figure 27 As before, we can move to the names that start with the letter "C" by pressing the key. On this calculator that action took our selection highlight directly to the collate3 program item.

Figure 28 Pressing copies collate3 to the command line, along with a left parenthesis. Programs on the TI-89 are followed by a possible list of parameters enclosed in parentheses. COLLATE3 does not have any parameters. nonetheless, the TI-89 will insist that the name be followed by a pair of parentheses enclosing whatever parameters there may be.

Figure 29 We complete our command by adding the required closing parenthesis by pressing the key.

Figure 30 In Figure 29 we completed forming the desired command. We press the key to start the program. Figure 30 shows us the input/output screen for the running program. Note how the TI-89 hazes out the menu area. Those menu items are no longer available.
The program has prompted us for the name of the list of values that we wish it to process.

Figure 31 We respond to the prompt of Figure 30 by pressing the keys to generate our list name, l1.

Figure 32 Having typed the list name in Figure 31, we use to submit that name to the program. The collate3 program will process that list of values and it will produce many lines of output that just flow across the input-output screen. Finally, when the entire l1 list has been processed, the collate3 program pauses as shown in Figure 32. Note the indicator in the lower right corner of the screen.

Figure 33 We press to continue the program. The output gives us the details on the frequency distribution of the values in the original list. From Figure 33 we can see that the number 1 appears 5 times in l1. The numbers 2, 3, 4, and 6 each appear 3 times, while the value 5 appears 4 times. Therefore, the mode value is 1.
Again, the calculator is paused, waiting for our command to go on to its next step.

Figure 34 Pressing the key moves us from Figure 33 to Figure 34. The collate3 program concludes by producing a histogram of the input data values.
In a subtle change, the calculator has also moved from the input-output screen of Figure 33 to the graphics screen of Figure 34. Note that an entirely new set of menu options has appeared for Figure 34.

Figure 35 So far we have seen a three different kinds of screens. We started on the HOME screen. The programs accept input and produce output on their own screen. And, there is a graphics screen. To continue our work we will want to return to the HOME screen. We do this by pressing the key. For Figure 35 we have returned to the HOME screen.

The first 35 Figures demonstrate statistical processing for a 21-element data set. It is nice to see the TI-89 do all of the computations, but the process seems to take many steps just to process those 21 values. The real power of the TI-89 can be seen if we look at processing a much larger set of data. For example, consider the following table of numbers taken from a sample test on this material.

-113 -133 -132 -91 -123 -121 -93 -103 -104 -102 -106 -126 -136 -90 -120

-105 -140 -125 -127 -110 -109 -109 -128 -88 -114 -133 -143 -120 -97 -108

-102 -107 -96 -108 -91 -115 -122 -122 -82 -111 -130 -116 -97 -122 -107

-85 -135 -116 -116 -94 -91 -142 -119 -119 -121 -115 -117 -120 -136

We could process this data using exactly the same steps that we used before. The first step will be to get this list into the calculator. Even that seems to be a formidable task. However, in this case, you can generate this same list on your calculator as as l1 via the gnrnd9 program on the TI-89 with
SEED 1= 54365448139 and SEED 2= 5391885598 and CHECKSUM=232. We will demonstrate using the program to do this.

Figure 36 We open the VAR-LINK window via th keys.

Figure 37 We want to find the gnrnd9 program in the list. We press to jump to the start of the "G" entries in the window. On the calculator used to make these images, the first such "G" entry is "game". This is the situation shown in Figure 37.

Figure 38 We move from Figure 37 to Figure 38 by pressing the key 3 times to move the highlight to the gnrnd9 program entry.

Figure 39 Presing the key will paste gnrnd9 and the left parenthesis onto the command line. Then we press to enter the closing parenthesis. Again, the gnrnd9 program does not use parameters.

Figure 40 Pressing the key again starts the program. The TI-89 shifts back to the input-output screen. Here, in Figure 40, we see the remnants of the earlier program, and the new output from our program. The GNRND5 is text produced by the program (it should be changed to GNRND9, but the original version of the program was written on a TI-85 and the text output was not changed in the conversion to TI-89 code). The program has prompted us for the first seed value.

Figure 41 In Figure 41 we have pressed the keys needed to generate that seed value, as given above, 54365448139.

Figure 42 Press to accept that value. The program then asks for the second seed value. Enter the number 5391885598 and press . The program asks for the checksum value. Enter the value 232 as given in the problem above.

Figure 43 Press to accept the checksum. Assuming that we entered the correct values, the program works on the task for a while. When it has completed the task of generating all of the numbers that were in the problem table, the program displays those values, and pauses while we inspect the values. Note the PAUSE.

Figure 44 We can use the key to shift the display so that we can see more of the generated list. For Figure 44 we have pressed that key 6 times.

Figure 45 Remembering that the GNRND9 program is still running, we press to allow the program to run to completion. The only noticeable change in Figure 45 is that the indicator has disappeared. Figure 45 continues to reflect the input-output program screen. If we are to do more work we will have to return to the main HOME screen.

Figure 46 We press to return to the HOME screen. It appears just as we left it, expect that we can see that gnrnd9 has run to completion.
GNRND9 produced our desired list of values in a list called l1. Our next step is to do the OneVar analysis on the values stored in l1. We could go back to the catalog to find the OneVar command, and then we could append the name of our list. However, this is exactly the command that we gave earlier. In fact, we can see that earlier command at the top of our history area. Our approach will be to retrieve that command and to re-issue it to do a new analysis on the new data stored in l1.

Figure 47 We can move the highlight up to the OneVar command by using the key 10 times. Figure 47 shows the result of doing this.

Figure 48 We press to retrieve that command and to paste it into the command line, as shown in Figure 48.

Figure 49 We press again to perform the action in the command line. The result is the simple acknowledgement that the task was DONE. Recall that we need to issue the ShowStat command to see the results. That command is in the history area. Let us return there to retrieve it.

Figure 50 Press 10 times to move the highlight to the desired command..

Figure 51 Press to paste that command into the command line, as in Figure 51.

Figure 52 Then we press to perform the command. This opens the new window, seen in Figure 52. We can read the most of our desired values from this table.

Figure 53 To see the remaining values, the 3rd quartile point and the maximum value, we press the key 2 times. Figure 53 shows the change in the display window.

Figure 54 We press the key to leave the statistics display window and return to the HOME screen. We still want to find the population standard deviation. Again, the desired command, , is in the history area.

Figure 55 Press the key 10 times to move the highlight to the command in the history area.

Figure 56 Press to retrieve the command and to perform it. The result is shown in Figure 56.
Continuing to repeat our earlier actions, we want to run the COLALTE3 program so that we can examine the frequency distribution of the values in the original list. That way we can identify the mode value in the list.

Figure 57 We return to the history area and select the old command by pressing 10 times.

Figure 58 Paste that command into the command line by pressing the key.

Figure 59 We start the program by pressing . The program asks for the name of our original list. We respond by pressing to indicate that the name of the list is l1.

Figure 60 We press to accept the name l1 from Figure 59. The program processes the values in that list, producing lines of output as it goes along. The program pauses, as shown in Figure 60, when it is ready to give us the results.

Figure 61 Pressing has the calculator display the first six values, and their associated frequencies.

Figure 62 Pressing causes the next group of values to be shown.

Figure 63 Another brings up another set of values and their associated frequencies. Note that the output on this screen indicates both ^–122 and ^–120 appear 3 times in the original list.

Figure 64 Yet another causes the calculator to display the 19^th throught the 24^th values. Also, we note that ^–116 joins the previous most frequently found values with a frequency of 3.

Figure 65 We go on with another key.

Figure 66 Pressing shows another set of six pairs of values.

Figure 67 And, pressing gives us the final five values and their associated frequencies. In this final set of values we see that ^–91 appears 3 times in the original list of values. Thus, we have a 4-way tie for the most frequently occurring value. THe list is multi-modal with mode values of ^–122, ^–120, ^–116, and ^–91 . The program is now ready to try to produce a histogram of the original data.

Figure 68 We press to move to the next phase of the program. The program determines that the width of a histogram column will have to be greater than 1, in fact it must be greater than 1.525, in order to be displayed nicely. We respond by pressing to set the width at 2.

Figure 69 Now we need to press to get the calculator to accept our value of 2, from Figure 68, and to create the histogram, shown in Figure 69.

Figure 70 As before, we can return to the HOME screen by pressing the key.

Between the built-in statistical features of the TI-89 and the COLLATE3 program we have been able to find the mean, median, and mode of the data values in our list. In addition, we have found the range, the quartile points, and the population standard deviation. The COLLATE3 program also gives us a frequency distribution table, and a histogram.

The collate3 program has some additional by-products. The output for the frequency distribution table is taken from two lists that COLLATE3 creates. The list OCL holds each of the different values found in the original list. The list ONL holds the frequency of occurrence for each of the corresponding values in OCL. If we look back to Figure 9, we will recall that the OneVar command required one list, but it had some optional parameters as well. The syntax for the OneVar command allows us to follow the command with a single list of values. That is what we have demonstrated above. Alternatively, the OneVar command allows us to follow the command by two lists, the first holding the different values, and the second list holding the frequency for each corresponding value. That is, the results that we saw in Figure 52 and 53 were the result of the OneVar command in Figure 48. After we have run the collate3 program, that is, after we have generated OCL and ONL, we could use this alternative syntax to issue the command "OneVar ocl,onl" and follow that by the "ShowStat" command. This would produce exactly the same results that we saw in Figures 52 and 53. There are additional forms of the OneVar command, but we will not cover them here.

Also, to do its work, COLLATE3 always produces a list called OSL, a copy of the original list but sorted into ascending order. We could have used that list to find values such as the median or the quartile points. There were 59 items in the original list. Therefore, in OSL, which holds the same 59 items but sorted, the 30^th item will be the median value.

Figure 71 To look at an item in OSL we can create its name followed by the position number in brackets, as a command. We press to lock in alphabetic mode. Then we continue with to generate osl.

Figure 72 We press to leave alphabetic mode, and we continue to form the psoition number in brackets, [30], by pressing .

Figure 73 Now, we press to give that command. The TI-89 responds, in the history area, with the command and the value of the list item in the 30^th position. This is indeed the median value.

Figure 74 As we look back at Figure 73, the entire command line is highlighted. If we want to look at the first quartile point, it will be in position 15 in that same list. We can use the text that was left in the command line and modify it to read osl[15]. To do this, we press to move the cursor to the end of the line.

Figure 75 We edit the contents of the command line by using the cursor control key to move one position to the left. Then we delete the 0 and the 3 by pressing and . Then we insert 15 by pressing and . These changes are reflected in Figure 75.

Figure 76 As usual, press to perform the command that was created in Figure 75. Then, to complete Figure 76, we edit the command to read osl[45] by pressing and . Finally, we have the TI-89 perform the command by pressing . The history area shows all of the results, namely, the first quartile is ^–123 and the third quartile is ^–103. These are the same values that we saw back in Figure 53.

Figure 1	Figure 1 represents a "clean" TI-89 home screen. That screen has a command entry line, almost at the bottom, and a history area for most of the screen (between the menu items at the top and the command entry line at the bottom). Pressing the key will clear the command line. It is not necessary to clear that line, but doing so does remove some of the clutter. In the same way, we can clear the history area by using the two key sequence . This will open a small window and select the Clear Home option. Again, this is not necessary.
Figure 2	We will start the problem by creating a list of values. We start a list with the { character. We press to generate that curly brace in the data line. Then we enter each of the numbers in the list, separating numbers with a comma. The keys to complete Figure 2 are and . This takes us to the right edge of the screen. We are not done entering the list, however.
Figure 3	In Figure 3 we continue entering the numbers in our list. Note that the command area display scrolls to the right so that we can see the new numbers and the commas. The remaining numbers are generated via and . We complete the list by entering the right curly brace, }, via the sequence . Finally, we want to store this list under the name l1 (note that this is a lowercase letter "l" and the digit "1"). The keys complete the input line from Figure 3.
Figure 4	We left Figure 3 with the complete command formulated in the command entry line. However, we still have to tell the TI-89 to perform the command. We do this by pressing the key. The TI-89 responds by first echoing the command in the history area and then displaying the result of the command in that same history area. We can see this in Figure 4 where we have two new lines in the history area. Note that the result of our creation of a list called l1 is an echo of that list. At the same time, the entire command input line has been highlighted.
Figure 5	We can move from the input line into the history area to inspect the information that is displayed there. To do this we press the key. Note in Figure 5 that the input area is no linger highlighted, and that the bottom line in the history area is now highlighted. The right-pointing arrow in that line indicates that we can shift the display to the right to see more of that information.
Figure 6	Figure 6 is the result of pressing the key 6 times, shifting the display for the highlighted line. First, note that the other lines are not affected. Second, we observe that the highlighted line now has both a left-pointed and right-pointed arrow at the respective edges of the list. These indicate that there are more values to be viewed at either end of the list.
Figure 7	We can move back to the input line area by pressing the key. Note, in Figure 7, that we have moved back to the input line, but that the shift in the bottom line of the history has not been changed. Now that the desired list has been created in the calculator, we can proceed to doing the statistical computations. As noted above, we want to use the OneVar command. We could type this command. However, that would require a number of keystrokes and it would require that we completely remember the command that we want to use.
Figure 8	A method for manufacturing the OneVar command is to find that command in the catalog, a list of commands on the calculator. We open that catalog by pressing the key. Figure 8 shows the catalog window. The contents of the catalog are arranged alphabetically. The pointer is at the first entry, abs(. We need to move that pointer down the list to find our desired command.
Figure 9	One way to jump down, or up, the list is to press the key associated with the first character of the command we want. In our case, we want the OneVar command. Therefore, we press the key that is associated with the letter "o", namely the key. Pressing that one key causes the calculator to jump down the catalog names to the o's. Fortunately for us, the first "o" command is the one we want, OneVar. Figure 9 shows that command at the top of the list with the selector arrow pointing to it. A careful examination of the screen in Figure 9 reveals that the calculator gives us some helpful hints about the appropriate use of the OneVar command. At the very bottom left of the screen we find the new message . This is information about the required and the optional parameters that are to follow the OneVar command. In this case we note that the command is to be followed by at least one list, and possibly by as many as four lists.
Figure 10	We press the key to actually select the OneVar command from Figure 9 and to paste it into the command line, replacing our entire former command, as is shown in Figure 10. As noted in the discussion following Figure 9, the OneVar command needs to be followed by the name of a list. Quite by accident, our list is named "l1" much like the "l1" suggested by the calculator in Figure 9. We will enter our name for our list in Figure 11.
Figure 11	We enter our list name by pressing the keys. Figure 11 shows our complete command.
Figure 12	Now we press to perform the OneVar command. Figure 12 illustrates the results of that command. It almost appears as if the command did nothing. The only thing that has changed is that we have the command added to the history area, at the bottom left, and the word "Done" has been added to the lower right of the history area. Unfortunately, the OneVar command instructs the calculator to do certain computations, but it does not cause the TI-89 to display those results. The "Done" is our only indicator that the command was successfully completed.
Figure 13	We still need to find some way to see the results of the OneVar command. The command to show those results is the ShowStat command. Again, we can find it in the catalog, which we re-open via the key. Note that the catlog appears just as we left it in Figure 9.
Figure 14	We are looking for the ShowStat command. To move down the command list to the "S's" we press the key. The calculator display shifts to that shown in Figure 14. Unfortunately, there are many commands that start with the letter S. Our desired command is not even on the screen yet.
Figure 15	We can press the key 11 times to move down to the ShowStat command, as has been done in Figure 15. Note that we merely want the selector arrow to be pointing to the ShowStat command.
Figure 16	Having selected the ShowStat command in Figure 15, we press to place that command into the command line in Figure 16.
Figure 17	Once the command is in place in Figure 16, we press again to perform the command. Figure 17 displays the window that the ShowStat command opens. That window indicates that the mean of the data is 3.333333, that the sum of the values in our "l1" list is 70, and that the sum of the squares of the values in the list is 300. The fourth value, gives an estimate of a standard deviation based on a sample. This statistic is beyond the scope of the material presented here. The next three lines in the window indicate that the number of items in the list is 21, that the lowest value in the list is 1, and that the first quartile point is at 1.5. The final line in the window, , gives us the median value of the list as being 3. Although, at first, there is no special indication of the fact, we might suppose that there is even more to see in this window. After all, we have seen the first quartile, but not the third, and we have seen the minimum value, but not the maximum value in the list. A closer examination shows that the equal sign in the line is really a tiny arrow pointing down. This seems to be meant as our only on-screen indication that more information is available.
Figure 18	We press the key 2 times to shift the display to that shown in Figure 18. Here we see the final two values: the third quartile is at 5 and the maximum value is 6. At the same time, note that the equal sign for the median, the medStat line, is now correctly formed, whereas, the equal sign in the top output line of the window now has its equal sign contorted to be an up-arrow.
Figure 19	We have seen all that the ShowStat command has to offer. We press the to close that window. We do need to recognize that ShowStat displays the results computed by the OneVar command issued earlier. The two commands need to be given to compute the values and then to display those values. We might have noticed that the ShowStat command does not display the population standard deviation for our values. This is a value that is computed by the OneVar command. It is just not displayed by ShowStat on the TI-89.
Figure 20	We would like to see the value of the population standard deviation. It is stored in a system variable called . We will need to generate that name. We can use the TI-89 menu system to do this. We press to open the CHAR window, shown in Figure 20.
Figure 21	In Figure 20, the Greek option was highlighted, and that is the option we want. We open the Greek sub-window by pressing the key. Figure 21 shows that window. Our desired Greek character, , is not in that window. However the down-arrow on the sixth line suggests that more characters follow.
Figure 22	We move down the sub-window by pressing the key until we find the desired character, as shown in Figure 22.
Figure 23	Now we paste that character into the command line by pressing the key. (Note: we have used the TI-89 menus to generate the Greek letter sigma, . An alternative method to generate the sigma character is to remember the sequence .)
Figure 24	Of course, we wanted not just , so we need to complete our command by pressing . This produces the command line shown in Figure 24.
Figure 25	Press to perform that command. The result is shown in Figure 25 where the history area holds the command and the associated value, 1.78174161275. This is the population standard deviation of the numbers in our original list.

Figure 26	In order to run the program COLLATE3 we need to manufacture that name in the command line. We could do this by typing the characters, or we could find the program in the variable listing on the TI-89. We will choose this second approach. We open the VAR-LINK window by pressing the keys. Note that the calculator used here has many programs and variables defined on it. If you are following along on a different calculator, then your display will be different, reflecting your different list of available programs and variables.
Figure 27	As before, we can move to the names that start with the letter "C" by pressing the key. On this calculator that action took our selection highlight directly to the collate3 program item.
Figure 28	Pressing copies collate3 to the command line, along with a left parenthesis. Programs on the TI-89 are followed by a possible list of parameters enclosed in parentheses. COLLATE3 does not have any parameters. nonetheless, the TI-89 will insist that the name be followed by a pair of parentheses enclosing whatever parameters there may be.
Figure 29	We complete our command by adding the required closing parenthesis by pressing the key.
Figure 30	In Figure 29 we completed forming the desired command. We press the key to start the program. Figure 30 shows us the input/output screen for the running program. Note how the TI-89 hazes out the menu area. Those menu items are no longer available. The program has prompted us for the name of the list of values that we wish it to process.
Figure 31	We respond to the prompt of Figure 30 by pressing the keys to generate our list name, l1.
Figure 32	Having typed the list name in Figure 31, we use to submit that name to the program. The collate3 program will process that list of values and it will produce many lines of output that just flow across the input-output screen. Finally, when the entire l1 list has been processed, the collate3 program pauses as shown in Figure 32. Note the indicator in the lower right corner of the screen.
Figure 33	We press to continue the program. The output gives us the details on the frequency distribution of the values in the original list. From Figure 33 we can see that the number 1 appears 5 times in l1. The numbers 2, 3, 4, and 6 each appear 3 times, while the value 5 appears 4 times. Therefore, the mode value is 1. Again, the calculator is paused, waiting for our command to go on to its next step.
Figure 34	Pressing the key moves us from Figure 33 to Figure 34. The collate3 program concludes by producing a histogram of the input data values. In a subtle change, the calculator has also moved from the input-output screen of Figure 33 to the graphics screen of Figure 34. Note that an entirely new set of menu options has appeared for Figure 34.
Figure 35	So far we have seen a three different kinds of screens. We started on the HOME screen. The programs accept input and produce output on their own screen. And, there is a graphics screen. To continue our work we will want to return to the HOME screen. We do this by pressing the key. For Figure 35 we have returned to the HOME screen.

Figure 36	We open the VAR-LINK window via th keys.
Figure 37	We want to find the gnrnd9 program in the list. We press to jump to the start of the "G" entries in the window. On the calculator used to make these images, the first such "G" entry is "game". This is the situation shown in Figure 37.
Figure 38	We move from Figure 37 to Figure 38 by pressing the key 3 times to move the highlight to the gnrnd9 program entry.
Figure 39	Presing the key will paste gnrnd9 and the left parenthesis onto the command line. Then we press to enter the closing parenthesis. Again, the gnrnd9 program does not use parameters.
Figure 40	Pressing the key again starts the program. The TI-89 shifts back to the input-output screen. Here, in Figure 40, we see the remnants of the earlier program, and the new output from our program. The GNRND5 is text produced by the program (it should be changed to GNRND9, but the original version of the program was written on a TI-85 and the text output was not changed in the conversion to TI-89 code). The program has prompted us for the first seed value.
Figure 41	In Figure 41 we have pressed the keys needed to generate that seed value, as given above, 54365448139.
Figure 42	Press to accept that value. The program then asks for the second seed value. Enter the number 5391885598 and press . The program asks for the checksum value. Enter the value 232 as given in the problem above.
Figure 43	Press to accept the checksum. Assuming that we entered the correct values, the program works on the task for a while. When it has completed the task of generating all of the numbers that were in the problem table, the program displays those values, and pauses while we inspect the values. Note the PAUSE.
Figure 44	We can use the key to shift the display so that we can see more of the generated list. For Figure 44 we have pressed that key 6 times.
Figure 45	Remembering that the GNRND9 program is still running, we press to allow the program to run to completion. The only noticeable change in Figure 45 is that the indicator has disappeared. Figure 45 continues to reflect the input-output program screen. If we are to do more work we will have to return to the main HOME screen.
Figure 46	We press to return to the HOME screen. It appears just as we left it, expect that we can see that gnrnd9 has run to completion. GNRND9 produced our desired list of values in a list called l1. Our next step is to do the OneVar analysis on the values stored in l1. We could go back to the catalog to find the OneVar command, and then we could append the name of our list. However, this is exactly the command that we gave earlier. In fact, we can see that earlier command at the top of our history area. Our approach will be to retrieve that command and to re-issue it to do a new analysis on the new data stored in l1.
Figure 47	We can move the highlight up to the OneVar command by using the key 10 times. Figure 47 shows the result of doing this.
Figure 48	We press to retrieve that command and to paste it into the command line, as shown in Figure 48.
Figure 49	We press again to perform the action in the command line. The result is the simple acknowledgement that the task was DONE. Recall that we need to issue the ShowStat command to see the results. That command is in the history area. Let us return there to retrieve it.
Figure 50	Press 10 times to move the highlight to the desired command..
Figure 51	Press to paste that command into the command line, as in Figure 51.
Figure 52	Then we press to perform the command. This opens the new window, seen in Figure 52. We can read the most of our desired values from this table.
Figure 53	To see the remaining values, the 3rd quartile point and the maximum value, we press the key 2 times. Figure 53 shows the change in the display window.
Figure 54	We press the key to leave the statistics display window and return to the HOME screen. We still want to find the population standard deviation. Again, the desired command, , is in the history area.
Figure 55	Press the key 10 times to move the highlight to the command in the history area.
Figure 56	Press to retrieve the command and to perform it. The result is shown in Figure 56. Continuing to repeat our earlier actions, we want to run the COLALTE3 program so that we can examine the frequency distribution of the values in the original list. That way we can identify the mode value in the list.
Figure 57	We return to the history area and select the old command by pressing 10 times.
Figure 58	Paste that command into the command line by pressing the key.
Figure 59	We start the program by pressing . The program asks for the name of our original list. We respond by pressing to indicate that the name of the list is l1.
Figure 60	We press to accept the name l1 from Figure 59. The program processes the values in that list, producing lines of output as it goes along. The program pauses, as shown in Figure 60, when it is ready to give us the results.
Figure 61	Pressing has the calculator display the first six values, and their associated frequencies.
Figure 62	Pressing causes the next group of values to be shown.
Figure 63	Another brings up another set of values and their associated frequencies. Note that the output on this screen indicates both ^–122 and ^–120 appear 3 times in the original list.
Figure 64	Yet another causes the calculator to display the 19^th throught the 24^th values. Also, we note that ^–116 joins the previous most frequently found values with a frequency of 3.
Figure 65	We go on with another key.
Figure 66	Pressing shows another set of six pairs of values.
Figure 67	And, pressing gives us the final five values and their associated frequencies. In this final set of values we see that ^–91 appears 3 times in the original list of values. Thus, we have a 4-way tie for the most frequently occurring value. THe list is multi-modal with mode values of ^–122, ^–120, ^–116, and ^–91 . The program is now ready to try to produce a histogram of the original data.
Figure 68	We press to move to the next phase of the program. The program determines that the width of a histogram column will have to be greater than 1, in fact it must be greater than 1.525, in order to be displayed nicely. We respond by pressing to set the width at 2.
Figure 69	Now we need to press to get the calculator to accept our value of 2, from Figure 68, and to create the histogram, shown in Figure 69.
Figure 70	As before, we can return to the HOME screen by pressing the key.

Figure 71	To look at an item in OSL we can create its name followed by the position number in brackets, as a command. We press to lock in alphabetic mode. Then we continue with to generate osl.
Figure 72	We press to leave alphabetic mode, and we continue to form the psoition number in brackets, [30], by pressing .
Figure 73	Now, we press to give that command. The TI-89 responds, in the history area, with the command and the value of the list item in the 30^th position. This is indeed the median value.
Figure 74	As we look back at Figure 73, the entire command line is highlighted. If we want to look at the first quartile point, it will be in position 15 in that same list. We can use the text that was left in the command line and modify it to read osl[15]. To do this, we press to move the cursor to the end of the line.
Figure 75	We edit the contents of the command line by using the cursor control key to move one position to the left. Then we delete the 0 and the 3 by pressing and . Then we insert 15 by pressing and . These changes are reflected in Figure 75.
Figure 76	As usual, press to perform the command that was created in Figure 75. Then, to complete Figure 76, we edit the command to read osl[45] by pressing and . Finally, we have the TI-89 perform the command by pressing . The history area shows all of the results, namely, the first quartile is ^–123 and the third quartile is ^–103. These are the same values that we saw back in Figure 53.