To demonstrate these steps we need to start with some data. We are given the following table of sample data:

Our first step is to get the midpoint of each class in Table 1. Fortunately for us, the class width given to us is a nice even value, namely 10, and the class lower limits are also nice integer values. It should not be too much for us to modify the original data to produce the following:

If we were to follow the directions in the textbook, our next steps would be to add a row to the table to hold, at the bottom of the Number in Class column, the sum of the frequencies, that is, to hold the value 79. Then we would add a column to hold the product of the frequencies and the class midpoint values. Would use the last row of that column to hold the total of the frequencies times the class midpoints. Then, from that information we could compute the mean of the data (the sum from this last column divided by the sum of the Freq column). For the numbers in our table that would be 25041/79=316.9746835. We would use that mean value as an approximation to the mean of the data that first generated Table 1. The table now appears as

Thereafter, we add another column to hold the difference between the midpoint and the calculated mean. Then we add a another column to hold the square of the calculalted difference between the midpoint and the calculated mean. Finally, we add a column that is the frequency times the square of the differences between the midpint and the calculated mean. We get the sum of the values in this last column. The table now appears as

The quotient, 40275.9/(79-1)=516.35, is the variance of the grouped data, assuming this is a set of sample data. The square root of that is 22.7235, which is then the standard deviation of the grouped data. We would use that value as an approximation to the standard deviation of the original data that generated table 1.

That is a great deal of work. The calculator can do this for us in just a few steps. First, we need to get the required values into the calculator. Those required values are just the class midpoint values and the class frequencies. We can store those two lists in L₁ and L₂.

Figure 1	We will start the process by making sure that we have a "clean" and "set up" calcuulator. We open the Memory menu via the keys. From here we will select the fourth option, ClrAllLists, by pressing the . This will paste that command onto our main screen.
Figure 2	Once the command is pasted here we press to have the calculator perform the command. This has made sure tht there is no data in any lists that are on the calculator. Our next step is to make sure tht we set up the stat editor. To do this we press to open the STAT menu, then press to select and paste teh SetUpEditor command, and finally, on the main screen, press to perform the command. This has made sure that the StatEditor si set to display the six built-in lists, starting with L1.
Figure 3	In fact, our next step will be to move to use the StatEditor so that we can put the list of midpoint values into L1. Press to open the STAT menu, then press to start the Edit... process. Figure 3 shows the editor set up just as we have arranged. We proceed by entering the nine numbers for the midpoint values (279, 289, 299, 309, 319, 329, 339, 349, and 359).
Figure 4	We have captured Figure 4 just before we press to accept 329 as the sixth value. The first five are shown on the screen.
Figure 5	Figure 5 shows the screen after all 9 values have been entered. Note that the calculator is in position to enter a tenth. We do not have a tenth value to enter. Instead we want to start entering the frequencies into L2. Therefore, we press the to move to the next column.
Figure 6	Now, positioned in the L2 column, we can enter the frequencies (5, 7, 18, 9, 9, 7, 11, 11, and 2).
Figure 7	Figure 7 shows the list after we have entered six values and as we are about to enter the seventh.
Figure 8	All of the values have been entered. We can safely exit the editor via the sequence.
Figure 9	The command that we want to use is 1-Var Stats L1,L2. That command will instruct the calculator to perform the 1-Var Stats using the values found in L1 along with the associated frequencies found in L2. We start constructing that command by going to the STAT menu via , then using to move to the CALC sub-menu, shown in Figure 9, and then pressing to select the desiored command and have it pasted onto the main screen.
Figure 10	Now that the start of the command is here we contnue constructing the command with L1 L2.
Figure 11	Note how the calculator wraps the command onto a second line. If we were using a TI-84 Plus with MathPrint turned ON then this wrapping of commands would be replaced by a scrolling left and right feature. Once the command is created we press to have the calculator perform it.
Figure 12	Here we have all the answers, and more, that we worked so hard to get in the various tables at the start of this problem. The mean is given as 316.974835, and, since this is sample data, the standard deviation is 22.72351921. The calculator even gives us the sum of all the values as 25041. It is interesting to note that the calculator does not provide us with the other total we used above, namely, 40275.9. This is because there is another, equivalent, formula for finding the standard deviation, and it uses the sum of the squares of the values which is displayed here as 7977639.
Figure 13	We can also use the key to see the rest of the descriptive values.

The calculator method requires little of us. From the original problem statement in Table 1 we had to

• find the midpoint values (Table 1 only had the Low and HIGH values)
• enter the midpoint values into the calculator
• enter the frequency values into the calculator
• create and run the 1-Var Stats L₁,L₂ command

Figure 14	We move to the list of programs via the key, and then we move down, using the key, to find the desired program. On the calculator used here the program was so far down the list that the only way to select it is to move the highlight to that program, as shown in Figure 14, and then press the key.
Figure 15	This pastes the command prgmGRPAPPX onto the screen. We press to actually start running the program.
Figure 16	The program first asks for the number of classes. We had Table 1 as our starting point in the problem, so we know that there are 9 classes. Therefore, we respond with 9. Then the program asks for the lower limit of the first class. Again, from Table 1 we know this is 274.
Figure 17	The program moves on to ask for the class width. We can see that the width is 10 and we enter that value. (It is worth noting that we could have let the calculator determine this value by entering 284–274.)
Figure 18	The calculator then prompts us for the frequencies in each class. Figure 18 shows this after we have entered the frequencies of the first five classes and before we enter the frequency of the sixth class. Two important point should be made here. First, in an attempt to make the program pretty, the programmer got GRPAPPX to construct each of the prompts with text and class number. This takes time. Therefore, the program is not as fast as is the usual user. It is easy to get ahead of the program, trying to enter a frequency value here before the calculator has asked for it. In that case, the calcualtor will not record the key you have pressed and you will have an incorrect entry. Second, unlike the editor, you cannot go back to correct an entry in this program. Once a value is entered and you have moved to the next value, you cannot change what has been entered. If that happens, all you can do is to stop the program and start all over again.
Figure 19	Figure 19 shows the program as we are about to enter the final frequency.
Figure 20	The program accepts tht last frequency value, does all of the calculations, and starts to produce the output. The 79 at the top of the screen was the number of items found. The 25041 is supplied for reference. The mean of the data, in this case 316.9746835, is given. It is the exact mean of the data we have, but it is noted as an approximation because we are using this as an approximation to the mean of the data that originally created Table 1. The program is in a paused condition, press to continue.
Figure 21	In case this represents a population the program displays the variance and the standard deviation values. THis is probably overkill since there is little reason to believe that we should ever consider this to be a population. After all, the values in Table 1 are not the original 79 values. We are not using the original 79 values, we are using the midpoint of the classes to do our computations. Nonetheless, the program can do the calculations and present the results. It is up to the user to know which resutls to use. It is more likely that we should use the sample statistics. To get those we press to allow the program to continue.
Figure 22	Here we have the variance and standard deviation treating the data as coming from a sample. The standard deviation shown here is the same value that we found above, in Figure 12. However, in this case, the program also tells us the variance. Usually we do not worry about how a program actually computes the valeus that it does. We are satisfied if the program just produces the right results. We do know that the results of Figrue 12 came from the calculator's 1-Var Stats command. THe results shown in Figures 21 and 22 came from the GRPAPPX program, and the results match the earleir results. Still, it will be slightly instructive to see the methodology of the GRPAPPX program.
Figure 23	We return to the StatEditor via . Here we see that the first three columns are given the names LOW to hold the lower limit of each class MDPT to hold the midpoint of each class FREQ to hold the frequency for each class. The valeus in the lists are appropriate for our problem.
Figure 24	Using to move to the right so that we can see the next three lists we find MTF to hold the midpoint times the frequency MMM to hold the midpoint minus the mean MMM2 to hold the square of the midpiont minus the mean.
Figure 25	Again, moving to the right we see the final column MMM2F to hold the frequency times the square of the midpoint minus the mean These are the columns that we used in Table 4. It was from these values that we found the sum of the final column and from that sum we found the variance. Now, rather than have us actually add the values in that final column, we can get the calculator to do taht for us.
Figure 26	We left Figure 25 via the sequence. We then move to the LIST menu via the key sequence, and from there, using , we move to the MATH sub-menu, shown in Figure 26. We want the fifth option, sum(. To get that we presss the key.
Figure 27	That pastes the start of the command onto the screen. We want the calculator to find the sum of the valeus in MMM2F. We need to find that name.
Figure 28	Return to the LIST menu. Move down the list of names until we find MMM2F. Then we can select the highlighted name by pressing the key.
Figure 29	Complete the command by adding the closing parenthesis via the key. (Note that the actual name pasted by the calcualtor is slightly different from the item we selected. That difference is the leading small L. This is the character that the calculator uses to denote a list name at various times.) We can press to hav ethe calculator perform the command.
Figure 30	The result, shown in Figure 30, is the same value that we got back in Table 4. From that we could calculate the sample variance by dividing that value by (79-1), that is, by 78. We could get the standard deviation by taking the square root of that answer.

A quick review of our progress thus far reveals that before we even started the problem there was some set of 79 values. We have not seen those values. We start the problem with a frequency table that has equal width classes aand that gives us the lower limit of each class and the frequency of each class. From this we can determine approximate values for the mean, variance, and standard deviation statistics of the original 79 values.

Having done this work, we are just about ready to turn in our results when the person who gave us the original table returns to tell us that, on second thought, it would be better if the classes had "nicer" boundaries. Therefore, we now have a new table and we need to redo our work. The new table is

Figure 31	We might as well use the program. Return to the list of programs and again select GRPAPPX. This pastes the name onto the main screen. Use to start the program.
Figure 32	Enter the number of classes, 9, and the first class lower limit, 270.
Figure 33	Then enter the width of the classes, 11, followed by the frequency for each class.
Figure 34	Finish entering the frequencies.
Figure 35	The new approximate mean is 317.2721519. This is different from the mean that we got from Table 3 (although it was presented just above the display of that table), Figure 12, and Figure 20. This approximate mean is based on a different organization of the original data. Both answers are good. Both are good approximations. Had we been given the original 79 values then we could have actully found their mean. This is the best we can do with the data that we have.
Figure 36	We continue to find the population variance and standard deviation.
Figure 37	And then to find the sample variance and standard deviation. These are new values. They are close to but not identical to the values that we found before. They remain good approximations to the true value.

Since it did not take much time to get our approximtions, we decide to provide a historgram of the data in Table 5.

Figure 38	In preparation for this we will return to the StatEditor just to verify the values there. We use the keys to get to Figure 38. We will be happy if we can get a histogram using the values in MDPT and FREQ. We could safely exit Figure 38 via .
Figure 39	Use to open the STAT PLOT menu. This particualr claculator has all of the plots turned Off. Plot1 is highlighted. Pressing will open the settings for Plot1.
Figure 40	Figure 40 shows the initial settings for Plot1 on this calculator. We want to change these to On, the Type: needs to be , Xlist: needs to be our MDPT list, and Freq: needs to be our FREQ list.
Figure 41	For Figure 41 we have made these changes. Before we leave this Figure, note that we just typed the names of the two lists for this. That is, the calculator had already put itself into alpha mode (note the blinking cursor). It was sufficient to press to generate the FREQ. An alternative approach would have been to go to the LIST menu and to find and select the names that we needed here. Had we done tht then the names here would have had the initial L in front of them. However we achieve the changes, once the settings have been changed we do a ZoomStat, via . This will take us to the graph in Figure 42.
Figure 42	This is the histogram that the ZoomStat command determined by computing new values for the WINDOW settings. This is an accurate histogram, but not the one we want. For one thing, this hs only 8 columns and we want a different column for each of our 9 classes. For another, we know that the list of frequencies is 1, 10, 18, 10, 9, 10, 11, 8, and 2. That is a bit different that the image here.
Figure 43	If we move into TRACE mode, via , the calculator telss us the limits on the first "classs", as well as the fact that the first class has 11 values in it. We see, from the limits on the first class of 275.5 to 288.07143 that the first class deterined here contains both of our first two classes, one with a midpoint of 275.5 and the other with a midpoint of 286.5. We can solve this issue by returning to the WINDOW settings and putting in our own values.
Figure 44	We get to the WINDOW settings via the key. In Figure 44 we ahve already changed the Xmin to 270, the Xmax to 369, and a new Xscl set to 11. With these values in place we can return directly to the graph via .
Figure 45	Here is the revised histogrm and it looks just the way we would expect.
Figure 46	Returning to TRACE mode, via , we see the limits of the first class and the fact that there is just 1 value in that class.
Figure 47	Moving to the right via we see the limits of the second class and the fact that there are 10 items in that class. We could continue to scan across the columns to see the values in each class.

As it turns out, the backgound data, the 79 values, that went into the making of both Table 1 and Table 5 came from a run of the GNRND4 program. It will be instructive for us to look at the real set of values and to determine the actual mean and standard deviation of those values. That way we can compare the approximations that we found above to the real statistics (parameters) of the original sample (population). Here is a table of the original vlaues:

Figure 48	We start by running the GNRND4 program and giving it the necessary key values.
Figure 49	The program completes. The values are now stored in L1. We get the 1-Var Stats command. Left alone this will do the analysis of the data in L1. That is what we want. Press to start the actual processing.
Figure 50	The result gives us the true mean, 316.5189873, and the standard deviation if this is a sample of some data, 22.86687267, and the standard deviation if this is a population, 22.72168471. We can see that our earlier approximations were really quite good.
Figure 51	We move down to see the rest of the information from 1-Var Stats.
Figure 52	Just to confirm that the data of Table 6 would be grouped as in Table 1, we run the COLLATE3 program giving it L1 as the source for the data.
Figure 53	We set the lower limit of the first class to be 274 and we set the class width as 10.
Figure 54	The program displays the values shown. Remember that these values have been rounded to 4 decimal places.
Figure 55	The program concludes. In order to really seen the values that we need to confirm Table 1 we will go tot he StatEditor.
Figure 56	In Figure 56 we see exactly the same lower limits and frequencies that we had in Figure 1, at least through the first 7 rows of the table.
Figure 57	We conclude by moving down the editor screen to see and verify the last two rows of the table.