Random Numbers

One basic concept in statistics is the ability to select a Simple Random Sample, an SRS. One technique for doing this is to use random numbers to select items from a numbered list. Of course this just begs the question of how do we get random numbers. The TI–3/84 calculators have a facility for generating pseudo-random numbers. This page steps through a discussion of obtaining and using such numbers.

The random numbers generated by the calculator are called pseudo random numbers because although they look random they in fact are the result of some strange computations. Furthermore, they are completely reproducible. Once we accept this we can just drop the pseudo from the term pseudo random.

Figure 1 The random number generator in the calculator uses a seed value to both start and to continue the process of generating random numbers. In general there is no need for a user to alter that seed value. However, if you do change it then you will alter the sequence of random values produced on you calculator.
For this web page we want your calculator to be able to produce exactly the sequence that the calculator used here produced. To do this we will set the seed value.
In Figure 1 we have entered the value 25717 and then used the key. We are ready to store the value in the seed variable. We have to find that variable.

Figure 2 Press to open the MATH menu shown in Figure 2.

Figure 3 Use the key to move the highlight to "PRB". This will display the "Probability" sub-menu. The first item in that sub-menu is rand. That is the item we want. It is already selected. Therefore, press to use that item, in this case, to paste that item onto the calculator screen, as is shown in Figure 4.

Figure 4 The command is complete. This command will place the value 25717 into the random number seed. Press to perform the command.

Figure 5 The only visible response from the calculator is to echo the value 25717.

Figure 6 Use the key to open the MATH menu and then use the key to move to the PRB submenu. Again, we want to paste the rand item onto the main screen. Press to do this.

Figure 7 Figure 7 has the rand item on the main screen. When used in an expression, the rand item acts as a function and it returns a random value between 0 and 1. We will press to perform the command.

Figure 8 The calculator response is .5863844503, which certainly seems like a random value between 0 and 1.

Figure 9 We want to perform the rand function again. We could return to the MATH menu, go to the PRB sub menu, and select it again, or we could press to recall the last command. Either way, Figure 9 shows that we are ready to perform the rand command again.
Press to perform the command.

Figure 10 Figure 10 shows that the second time we performed the rand command the result was .194143297, another seemingly random value between 0 and 1.
Figure 10 goes on to show yet a third use of the rand command, this time producing the value .5827919869. We could continue to perform the rand command and we would find that it continues to produce what seem to be random values between 0 and 1.
Before we do anything like that let us demonstrate the effect of changing the seed value.

Figure 11 The top line of Figure 11 is the same as the bottom line of Figure10. We have just executed more commands and the screen has rolled up as we have done this. The new commands were first to reset the seed value to 25717 and then to perform the rand function two times. What we see is that the rand function produces the same set of values that it did in Figures 8 and 10. In fact, at this point, we know the exact value that rand will produce the next time we use it, namely the value .5827919869, shown at the bottom of Figure 10 and the top of Figure 11.

Figure 12 For us to understand the next command you need to recall that the TI–83/84 calculators "understand" implied multiplication. Typing 3π on these calculators is as good as typing 3*π. Thus, in Figure 12, when we recall the previous command via and then press the resulting command is rand20 which the calculator interprets as rand*20. We know that the next value of rand will be .5827919869, and we could compute 20 times this to arrive at the value 11.65583974, or we could just press and let the calculator do that work.

Figure 13 In Figure 13 we do see the expected result. We also see a new command, rand*20, and the result of performing that command, 3.30643276, a value that we could not have predicted from earlier screens because it involves using the next random number generated by the function rand.

Figure 14 We have seen that rand will produce a random value between 0 and 1. We have seen that we can form the expression rand*20 and that expression will produce a random value between 0 and 20. What can we do to produce a random integer between 0 and 19 inclusive? The immediate answer is that we can take the integer value of the random number between 0 and 20. To do that we return to the MATH menu, via the key, and then we use the key to move to the NUM submenu shown in Figure 14. Then we use the key to move the highlight to point to the fourth item, int(. The int( function returns the largest integer less than or equal to the value it is given. If we give the int( function a random value between 0 and 20, then the only results of that will be integers etween 0 and 19 inclusive. We press to past the int( function onto the main screen as shown in Figure 15.

Figure 15 The int( function is in place. Now we will complete the expression.

Figure 16 Return to the MATH menu, via the key, use teh key to move to the PRB submenu. To paste rand to the main screen we can just press .

Figure 17 We finish our expression via and then press to have the calculator perfom the command. In this case the result is 12.

Before we go any further, it is about time that we look at the TI documentation related to the use of rand. Here is the material from the TI Guidebook:
The syntax of rand is given as rand[(numtrials)]. The items inside the square brackets are optional. Thus, it is perfectly legal to form the command, as we have been doing, rand. If the command is formed in that fashion, then the documentation states that the default for numtrials will be the value 1. That is to say, if we just use the command rand it will return just one random value between 0 and 1. The "Tip" goes on to say, as we have seen, that you can produce a wider range of values by multiplying rand by the desired width. Their example is to form the expression rand5 which is equivalent to the expression rand*5.
The documentation goes on to point out how you can set a new seed value for the random number generator. Finally, the documentation includes a screen shot that is probably an error since there is no explanation of the source of the third line on that screen.
The intriguing part of the presentation is that you can generate more than one random value by specifying the value numtrials. Let us see if we can do this.

Figure 18 Figure 18 picks up where we left things in Figure 17, but now we have created and had the calculator perform the command rand(5). The result is that we have produced a list of 5 values, the first one being .9923155961 and the second value is shown as starting with .9. The three dots after that indicate that there is more to see. We can use the key to scroll to the right on that last line of the display. We have done this a number of times to generate Figure 19.

Figure 19 In Figure 19, on the last line of the display, we see the end of the first random value from Figure 18, namely, ...61 and that is followed by most of the second value in the list, namely, .9813998199..., where the three dots mean that there is more to the display.

Figure 20 For Figure 20 we have combined the things we have learned to form the command int(rand(5)*20). To perform this command the calculator will produce 5 random values, each between 0 and 1. It will multiply each of those random values by 20. Then, for each of the five values, it will find the largest integer less than or equal to that value. The result is a list of 5 integer values, in the first case this was {3 7 6 4 10}. Then we recalled the command and executed it to get a new list. Then we recalled the command again and performed it to get the third set of five values.

The task of producing random integers within a given range happens so often that the TI–83/84 family of calculators has a special command that does just that. The command is randInt(. The TI documentation for randint( is:
That documentation gives the syntax of the command as randInt(lower,upper[,numtrials]) meaning that we have a required lower and upper specification but an optional numtrials value. We will explore this command.

Figure 21 For Figure 21 we have returned to the MATH menu and moved to the PRB submenu. Then we moved the highlight down to the fifth item, randInt(. We press to paste that item to the main screen.

Figure 22 In Figure 22 we completed the randInt( command by using 25 as the lower value and 35 as the higher value. Adding a closing right parenthesis, the command was complete and we press to have the calculator perform it. The result was 26. Then we recalled and performed the command two more times and the results were 31 and 30, all results being in the range of 25 to 35.

Figure 23 For Figure 30 we have built the same command, but this time we have specified the numtrials as 5. As usual, press to have the calculator perform the command and move to Figure 24.

Figure 24 Our first execution of the command randInt(25,35,5) produced the list {35 35 30 26 33}. We note that all the values are within the desired range and that we actually have a repeated value, namely 35. There is nothing to say that this cannot happen.
We then recalled the command and performed it again, this time generating the values {25 33 34 30 28}. We were lucky enough this time to not have any repeated values.

Figure 25 If the range of values is large enough, that is if higher-lower is a large enough value, then picking even a moderate number of random values in the range will likely not result in having any duplicate values. For example, if there are 12,500 credit students at the college this term and we want to select a random grup of 30 students then we could just number the students 1 to 12,500 and ask the calculator to choose 30 random values in the range from 1 to 12,500. The command to do this, randInt(1,12500,30) is on Figure 25, along with the start of the resulting list of values.
We could just scroll over that list to see all the values, but we might as well take advantage of some of the features of the calculator to ease this task.

Figure 26 To generate Figure 26 we press . The calculator recognizes that by starting with the key we are implying that we want to store the previous answer. Therefore, the calculator generates the Ans in that line. We are storing that answer in a list called L₁. Naturally, we perform the command via the key. The calculator displays the start of the list of values again. However, now that the answer is in a standard list we can actually work with it.

Figure 27 For example, we can sort that list. Press to open the STAT menu. Then move the highlight down to item 2, SortA(, and press .

Figure 28 We have the start of the command specified. The command is to sort a list into ascending values. We just need to tell the calculator which list to sort.

Figure 29 We complete the command via to say sort items in list L₁ into ascending order. Press to perform the command and the calculator sorts the values and displays Done.

Figure 30 The key sequence will display the contents of L₁. Of course they still do not fit on the screen but we see the start of the list and we could use the key to scroll across the list. Because the list is sorted it will be easy to see if there are any duplicates.

Figure 31 For Figure 31 we have indeed scrolled over some of the list do display more items.

Figure 32 Another way to display the items in the list is to use teh List Editor on the calculator. We get to that list editor via the STAT menu. We return to the STAT menu via the key. Figure 32 shows that menu. The option that we want is the first option. It is already highlighted so we just hit the key to perform that option.

Figure 33 Figure 32 shows us the start of the List Editor screen. Here we see, in the first column, the first seven values in L₁. The highlight, in Figure 33 is on the first item. As such, the bottom of the screen tells us that item #1 in the list has the value 25. If we use to move the highlight the information displayed at the bottom will change appropriately. If we move the highlight far enough down the list new items appear as the list scrolls up.

Figure 34 For Figure 34 we have scrolled down to display item 13 which has the value 4519.

Figure 35 For Figure 35 we have scrolled down to display item 19 which has the value 8089.

Figure 36 For Figure 36 we have scrolled down to display item 25 which has the value 9256.

Figure 37 For Figure 37 we have scrolled down to display item 31 which has the no value at this time. Remember that we generated just 30 values via the command in Figure 25. In Figures 33 through 37 we have seen the list of the 30 students that we have randomly selected from the 12,500 credit students this term.

Figure 1	The random number generator in the calculator uses a seed value to both start and to continue the process of generating random numbers. In general there is no need for a user to alter that seed value. However, if you do change it then you will alter the sequence of random values produced on you calculator. For this web page we want your calculator to be able to produce exactly the sequence that the calculator used here produced. To do this we will set the seed value. In Figure 1 we have entered the value 25717 and then used the key. We are ready to store the value in the seed variable. We have to find that variable.
Figure 2	Press to open the MATH menu shown in Figure 2.
Figure 3	Use the key to move the highlight to "PRB". This will display the "Probability" sub-menu. The first item in that sub-menu is rand. That is the item we want. It is already selected. Therefore, press to use that item, in this case, to paste that item onto the calculator screen, as is shown in Figure 4.
Figure 4	The command is complete. This command will place the value 25717 into the random number seed. Press to perform the command.
Figure 5	The only visible response from the calculator is to echo the value 25717.
Figure 6	Use the key to open the MATH menu and then use the key to move to the PRB submenu. Again, we want to paste the rand item onto the main screen. Press to do this.
Figure 7	Figure 7 has the rand item on the main screen. When used in an expression, the rand item acts as a function and it returns a random value between 0 and 1. We will press to perform the command.
Figure 8	The calculator response is .5863844503, which certainly seems like a random value between 0 and 1.
Figure 9	We want to perform the rand function again. We could return to the MATH menu, go to the PRB sub menu, and select it again, or we could press to recall the last command. Either way, Figure 9 shows that we are ready to perform the rand command again. Press to perform the command.
Figure 10	Figure 10 shows that the second time we performed the rand command the result was .194143297, another seemingly random value between 0 and 1. Figure 10 goes on to show yet a third use of the rand command, this time producing the value .5827919869. We could continue to perform the rand command and we would find that it continues to produce what seem to be random values between 0 and 1. Before we do anything like that let us demonstrate the effect of changing the seed value.
Figure 11	The top line of Figure 11 is the same as the bottom line of Figure10. We have just executed more commands and the screen has rolled up as we have done this. The new commands were first to reset the seed value to 25717 and then to perform the rand function two times. What we see is that the rand function produces the same set of values that it did in Figures 8 and 10. In fact, at this point, we know the exact value that rand will produce the next time we use it, namely the value .5827919869, shown at the bottom of Figure 10 and the top of Figure 11.
Figure 12	For us to understand the next command you need to recall that the TI–83/84 calculators "understand" implied multiplication. Typing 3π on these calculators is as good as typing 3π. Thus, in Figure 12, when we recall the previous command via and then press the resulting command is rand20* which the calculator interprets as *rand20. We know that the next value of rand will be .5827919869, and we could compute 20 times this to arrive at the value 11.65583974**, or we could just press and let the calculator do that work.
Figure 13	In Figure 13 we do see the expected result. We also see a new command, *rand20, and the result of performing that command, 3.30643276, a value that we could not have predicted from earlier screens because it involves using the next random number generated by the function rand**.
Figure 14	We have seen that rand will produce a random value between 0 and 1. We have seen that we can form the expression *rand20 and that expression will produce a random value between 0 and 20. What can we do to produce a random integer between 0 and 19 inclusive? The immediate answer is that we can take the integer value of the random number between 0 and 20. To do that we return to the MATH menu, via the key, and then we use the key to move to the NUM submenu shown in Figure 14. Then we use the key to move the highlight to point to the fourth item, int(. The int( function returns the largest integer less than or equal to the value it is given. If we give the int( function a random value between 0 and 20, then the only results of that will be integers etween 0 and 19 inclusive. We press to past the int(** function onto the main screen as shown in Figure 15.
Figure 15	The int( function is in place. Now we will complete the expression.
Figure 16	Return to the MATH menu, via the key, use teh key to move to the PRB submenu. To paste rand to the main screen we can just press .
Figure 17	We finish our expression via and then press to have the calculator perfom the command. In this case the result is 12.

Figure 18	Figure 18 picks up where we left things in Figure 17, but now we have created and had the calculator perform the command rand(5). The result is that we have produced a list of 5 values, the first one being .9923155961 and the second value is shown as starting with .9. The three dots after that indicate that there is more to see. We can use the key to scroll to the right on that last line of the display. We have done this a number of times to generate Figure 19.
Figure 19	In Figure 19, on the last line of the display, we see the end of the first random value from Figure 18, namely, ...61 and that is followed by most of the second value in the list, namely, .9813998199..., where the three dots mean that there is more to the display.
Figure 20	For Figure 20 we have combined the things we have learned to form the command *int(rand(5)20). To perform this command the calculator will produce 5 random values, each between 0 and 1. It will multiply each of those random values by 20. Then, for each of the five values, it will find the largest integer less than or equal to that value. The result is a list of 5 integer values, in the first case this was {3 7 6 4 10}**. Then we recalled the command and executed it to get a new list. Then we recalled the command again and performed it to get the third set of five values.

Figure 21	For Figure 21 we have returned to the MATH menu and moved to the PRB submenu. Then we moved the highlight down to the fifth item, randInt(. We press to paste that item to the main screen.
Figure 22	In Figure 22 we completed the randInt( command by using 25 as the lower value and 35 as the higher value. Adding a closing right parenthesis, the command was complete and we press to have the calculator perform it. The result was 26. Then we recalled and performed the command two more times and the results were 31 and 30, all results being in the range of 25 to 35.
Figure 23	For Figure 30 we have built the same command, but this time we have specified the numtrials as 5. As usual, press to have the calculator perform the command and move to Figure 24.
Figure 24	Our first execution of the command randInt(25,35,5) produced the list {35 35 30 26 33}. We note that all the values are within the desired range and that we actually have a repeated value, namely 35. There is nothing to say that this cannot happen. We then recalled the command and performed it again, this time generating the values {25 33 34 30 28}. We were lucky enough this time to not have any repeated values.
Figure 25	If the range of values is large enough, that is if higher-lower is a large enough value, then picking even a moderate number of random values in the range will likely not result in having any duplicate values. For example, if there are 12,500 credit students at the college this term and we want to select a random grup of 30 students then we could just number the students 1 to 12,500 and ask the calculator to choose 30 random values in the range from 1 to 12,500. The command to do this, randInt(1,12500,30) is on Figure 25, along with the start of the resulting list of values. We could just scroll over that list to see all the values, but we might as well take advantage of some of the features of the calculator to ease this task.
Figure 26	To generate Figure 26 we press . The calculator recognizes that by starting with the key we are implying that we want to store the previous answer. Therefore, the calculator generates the Ans in that line. We are storing that answer in a list called L₁. Naturally, we perform the command via the key. The calculator displays the start of the list of values again. However, now that the answer is in a standard list we can actually work with it.
Figure 27	For example, we can sort that list. Press to open the STAT menu. Then move the highlight down to item 2, SortA(, and press .
Figure 28	We have the start of the command specified. The command is to sort a list into ascending values. We just need to tell the calculator which list to sort.
Figure 29	We complete the command via to say sort items in list L₁ into ascending order. Press to perform the command and the calculator sorts the values and displays Done.
Figure 30	The key sequence will display the contents of L₁. Of course they still do not fit on the screen but we see the start of the list and we could use the key to scroll across the list. Because the list is sorted it will be easy to see if there are any duplicates.
Figure 31	For Figure 31 we have indeed scrolled over some of the list do display more items.
Figure 32	Another way to display the items in the list is to use teh List Editor on the calculator. We get to that list editor via the STAT menu. We return to the STAT menu via the key. Figure 32 shows that menu. The option that we want is the first option. It is already highlighted so we just hit the key to perform that option.
Figure 33	Figure 32 shows us the start of the List Editor screen. Here we see, in the first column, the first seven values in L₁. The highlight, in Figure 33 is on the first item. As such, the bottom of the screen tells us that item #1 in the list has the value 25. If we use to move the highlight the information displayed at the bottom will change appropriately. If we move the highlight far enough down the list new items appear as the list scrolls up.
Figure 34	For Figure 34 we have scrolled down to display item 13 which has the value 4519.
Figure 35	For Figure 35 we have scrolled down to display item 19 which has the value 8089.
Figure 36	For Figure 36 we have scrolled down to display item 25 which has the value 9256.
Figure 37	For Figure 37 we have scrolled down to display item 31 which has the no value at this time. Remember that we generated just 30 values via the command in Figure 25. In Figures 33 through 37 we have seen the list of the 30 students that we have randomly selected from the 12,500 credit students this term.