Normal Quantile Plots

This page is devoted to presenting the Normal Quantile Plot as a method for getting a feel for the normality of a data set.

To demonstrate these steps we need to start with some data. We will use the GNRND4 program to generate that data. One option for that program asks it to generate data that is approximately normal. As we will see below, given the KEY values that we use, the program generates the data set:

Figure 1 We start with the run of the GNRND4 program, giving it the appropriate KEY values.

Figure 2 Figure 2 gives a view of the start and of the end of the list of generated va;ues. We see that we have exactly the desired values now stored in L₁.

Figure 3 There are four steps to creating "by hand" a normal quantile plot. The first step is to sort the values. We use the key to open the STAT menu, shown in Figure 3. Then use the key to move the highlight to the second item, SortA(. Press to select that option and paste the command onto the main screen.

Figure 4 Here, on the main screen, we complete the command via to form SortA(L₁). Press to perform the command. Note that this sorts the values in the specified list. We use to see the result.

Figure 5 The second step in the process is to form a parallel list with ascending values equally spaced and all within the open interval from 0 to 1. To do this for a list of n values, we use the fractions of the form p/q where q=2*n and p is an odd number from 1 to 2*n-1. In our case, we want to create a lies of 6 such values, namely, 1/12, 3/12, 5/12, 7/12, 9/12, and 11/12.
We could move to the Stat Editor and enter these values by hand. Alternatively, we can use the built-in function seq( to create such a list. In Figure 5 we used to open the LIST menu, and we moved the highlight to OPS. That submenu contains the desired seq( command at position 5. In this Figure we have moved the highlight to that spot. Then press to paste the command onto the main screen.

Figure 6 Now we need to complete the command. The seq( command has four arguments. The first is an expression to evaluate to produce a typical item in the list. The second is the base variable whose values will change to produce different items in the list. The third is the starting value that is to be used. The fourth is the final value to be used. For our case, we want to let a variable, X, take on values from 1 to 6, and each such value we want to calculate (2*X-1)/12. Thus the desired command is seq((2*X-1)/12,X,1,6) which will generate the list of values that we can then assign to L₂.

Figure 7 Of course, the keystrokes to finish the command on the calculator are . Then press to perform the command.

Figure 8 Figure 8 shows the start of the list of values that we have generated.

Figure 9 Use to move to the STAT menu. Use to select the Edit option, thus moving to the Stat Editor shown in Figure 10.

Figure 10 Now we can see the values that have been placed into L₂. We use the key to move over to enter the first value into L₃.
The third step in generating the normal quantile plot is to produce another parallel list of values, this time finding the appropriate z-score that has the area under the normal curve to the left of that z-score equal to the corresponding value in L₂. Thus, the value in L₂(1) is 1/12 so we want the corresponding value in L₃ to be the z-score that has an area equal to 1/12 under the normal curve and to the left of that point. From the home screen we would find this via the command invNorm(1/12). We do the same in the editor.
As shown in Figure 10 we are ready to enter the first item in L₃. Press to open the DIST menu.

Figure 11 Now we use the key to move the highlight down to the invNorm( option.

Figure 12 From Figure 11 we press to paste invNorm( at the bottom of the screen.

Figure 13 Now, we need to complete the command. In truth, we could just enter the required value 1/12, but we will take the slightly longer, but more general, approach of using the value in L₂(1). To do this we complete the command via , changing the last line of the screen to that shown in Figure 13. We complete the command by pressing the key.

Figure 14 In Figure 14 we see that the appropriate value has been entered as the first item in L₃. Furthermore, the calculator is now ready for us to enter a value for L₃(2).

Figure 15 For Figure 15 we have repeated the process to generate invNorm(L₂(2)) at the bottom of the screen. Again we use to perform the command.

Figure 16 Here we see the new value added to L₃.

Figure 17 Figure 17 shows the result of completing this process for the remaining values in L₃. At this point we are done creating the parallel lists that we desired. Press to leave the editor and return to the main screen in Figure 18.

Figure 18 Having accomplished the first three steps in the process:
Sort the values (in L₁)
Create a parallel list of evenly spaced values between 0 and 1 (in L₂)
For each of those evenly spaced values get its invNorm( value (in L₃)
We are left with the final, fourth step:
Create a Scatter Plot using the sorted values (in L₁) and the invNorm( values in L₃).
We use the keys to open the main Stat Plots menu shown in Figure 19.

Figure 19 For the calculator being used here we see that all of the plots are Off. At the moment the highlight is on the first plot, the one we will use here. Therefore, we press to move to the screen controling the settings for that plot.

Figure 20 Figure 20 was captured without changing anything on the screen. The blinking cursor is sitting over the On option (in fact it is covering that option in this screen capture) but we see that the Off option is the highlighted one, corresponding to the fact that, at the moment, Plot1 is Off.
To set Plot1 to the On status, we just press the key. The immediate result is shown in Figure 21.

Figure 21 Now, not only is the setting changed to On but also the blinking cursor is still on that option.
We now turn our attention to the Type of plot. At the moment, in Figure 21, the plot is set to the histogram option, . However, we want a Scatter Plot, currently displayed as . To do this we use to move the highlight to the Type selections. Once the highlight is on the Scatter Plot option we press to change the setting to .

Figure 22 Figure 22 was captured once the change has been made, although the blinking cursor is still on that choice and is obscuring it at the moment.
Looking at the reest of the settings we see that they have changed from what they were back in Figures 20 and 21. The value of Xlist is correctly set as L₁ but, for this calculator, the value of Ylist needs to be changed to L₃. To do this we use the key to move the highlight to the Ylist field and we press to change that value to L₃. That is the condition that we find in Figure 23.

Figure 23 Note that the change to the Ylist field has been made. We will take one extra step just to be sure that the values are set, namely we will use the key to move the highlight to the last field, the one where we can set the Mark value. This produces Figure 24.

Figure 24 Now that the values have been set all that we need to do is to use the keys to open the Zoom menu and then to choose the ZoomStat option. Doing so will move the calculator to display the Scatter Plot in a window that has had its parameters set appropriately for the given data.

Figure 25 We have gone through all of this to produce this graph. A true normal distribution should produce the set of points in approximately a diagonal straight line from the lower left to the upper right corner. The plot in Figure 25 does not conform to that. We would have to conclude that the original data was not really a Normal distribution. [On the other hand, the data set is really small. Taking such a small sample will often lead to such non-conforming plots.]

The data that we used above came from the GNRND4 program when we specifically asked that program to generate data that is approximately normally distributed. Given what we found one might think that there is an error in the program. However, as just noted, having a small sample of data often leads to conclusions of non-normality. Let us use the program to generate a larger sample to see what happens in that case. We will generate a list of data on the calculator using GNRND4 with Key 1=357843504 and Key 2=700053. That list will be the same numbers that appear in the following table: Thus, our problem will be to generate a normal quantile plot for the data in the list above.

Figure 26 First we generate the data.

Figure 27 The values in Figure 27 correspond to the values that we saw in the table above. These values are stored in L₁.
We could go through the four step process shown above to sort the data, then generate the parallel lists and then produce the scatter plot. However, the third step of that process, finding the invNorm( value of each of the 36 evenly spaced values between 0 and 1 that we would generated in the second step looks daunting at best. This looks like too much work. There must be an easier way! In fact we will see two easier ways.

Figure 28 Remember that L₁ holds our values. The calculator has a special plot option that automatically does the normal quantile plot without our doing any of the work. We return to the General Plot menu via . Then we press to move to the Plot1 screen shown in Figure 29.

Figure 29 In Figure 29 we have moved the cursor to the last of the Type options and selected it. Note that the option is selected. We confirm that the other values are correct. Note that the Data List is set to L₁. Once everything is set we use to open the ZOOM menu and select ZoomStat.
You might notice that it takes the calculator a bit of time to do all of the computations that it needs to generate the requested normal quantile plot shown in Figure 30.

Figure 30 Here is the desired plot. In this case we can see that the points do line up in an approximate diagonal. Therefore, this plot supports the conclusion that the original data points, our L₁ values, are normally distributed.
Because the calculator generates this plot without our doing any of the work shown in Figures 5 through 25, it would seem that we have no benefit from doing that work ourselves. However, note that the calculator did the work but that it does not directly supply the values in the two parallel lists, nor does it give us the sorted values of the original data. It is possible that we may need these values.

Figure 31 We could extract the sorted values and their associated invNorm( values from the plot. We move into TRACE mode via the key. The calculator then displays, at the bottom of the screen the first sorted point, in this case 36, and the associated invNorm( value, in this case -2.200411.

Figure 32 We can use the key to highlight the second point on the plot to get its value, 41, and its associated invNorm( value, -1.731664. We could continue to use the cursor key to move through the rest of the points to read out the sorted values and their associated invNorm( values. This would be tedious, but it could be done. However, the graph does not provide us with that list of evenly spaced values between 0 and 1 that we had generated when we did all of this in Figures 5 through 25.

Figure 33 Back in the explanation tied to Figure 33 we noted that there would be two easier ways to generate our normal quantile plot. The first was to use the built-in Plot Type that we just demonstrated. The second is to have a program that does all of our steps from Figures 5 through 25, and more, for us.
We leave Figure 32 via the sequence to return to our main screen. Then we open the program list menu via the key. We find and highlight the program named NRMLQNTL. Then press to paste the command to run that program onto our screen. This is shown in Figure 33. Press to run the program.

Figure 34 The program asks for the location of the original data. We respond with to tell it that the data is in L₁. Then press to continue the program.

Figure 35 The program does all of the computations that we had done in our original process, along with finding the linear regression equations for the original data points and their associated invNorm( values. It then displays both the scatter plot and the regression equation. Having the graph of the regression equation makes it even easier to get a feeling for the linearity of the plotted points in the scatter graph.

Figure 36 Just as before, we could move into TRACE mode, via the key, and see the individual values behind each of the points on the plot.

Figure 37 However, we could just get out of the plot, via , and open the StatEditor via and we see that indeed the program did do what we had done earlier, namely, it
sorted the values and put them into L₁
created the evenly spaced values between 0 and 1 and put them into L₂
created a list of associated invNorm( values in L₃

Figure 1	We start with the run of the GNRND4 program, giving it the appropriate KEY values.
Figure 2	Figure 2 gives a view of the start and of the end of the list of generated va;ues. We see that we have exactly the desired values now stored in L₁.
Figure 3	There are four steps to creating "by hand" a normal quantile plot. The first step is to sort the values. We use the key to open the STAT menu, shown in Figure 3. Then use the key to move the highlight to the second item, SortA(. Press to select that option and paste the command onto the main screen.
Figure 4	Here, on the main screen, we complete the command via to form SortA(L₁). Press to perform the command. Note that this sorts the values in the specified list. We use to see the result.
Figure 5	The second step in the process is to form a parallel list with ascending values equally spaced and all within the open interval from 0 to 1. To do this for a list of n values, we use the fractions of the form p/q where q=2n and p is an odd number from 1 to 2n-1. In our case, we want to create a lies of 6 such values, namely, 1/12, 3/12, 5/12, 7/12, 9/12, and 11/12. We could move to the Stat Editor and enter these values by hand. Alternatively, we can use the built-in function seq( to create such a list. In Figure 5 we used to open the LIST menu, and we moved the highlight to OPS. That submenu contains the desired seq( command at position 5. In this Figure we have moved the highlight to that spot. Then press to paste the command onto the main screen.
Figure 6	Now we need to complete the command. The seq( command has four arguments. The first is an expression to evaluate to produce a typical item in the list. The second is the base variable whose values will change to produce different items in the list. The third is the starting value that is to be used. The fourth is the final value to be used. For our case, we want to let a variable, X, take on values from 1 to 6, and each such value we want to calculate *(2X-1)/12. Thus the desired command is seq((2X-1)/12,X,1,6)* which will generate the list of values that we can then assign to L₂.
Figure 7	Of course, the keystrokes to finish the command on the calculator are . Then press to perform the command.
Figure 8	Figure 8 shows the start of the list of values that we have generated.
Figure 9	Use to move to the STAT menu. Use to select the Edit option, thus moving to the Stat Editor shown in Figure 10.
Figure 10	Now we can see the values that have been placed into L₂. We use the key to move over to enter the first value into L₃. The third step in generating the normal quantile plot is to produce another parallel list of values, this time finding the appropriate z-score that has the area under the normal curve to the left of that z-score equal to the corresponding value in L₂. Thus, the value in L₂(1) is 1/12 so we want the corresponding value in L₃ to be the z-score that has an area equal to 1/12 under the normal curve and to the left of that point. From the home screen we would find this via the command invNorm(1/12). We do the same in the editor. As shown in Figure 10 we are ready to enter the first item in L₃. Press to open the DIST menu.
Figure 11	Now we use the key to move the highlight down to the invNorm( option.
Figure 12	From Figure 11 we press to paste invNorm( at the bottom of the screen.
Figure 13	Now, we need to complete the command. In truth, we could just enter the required value 1/12, but we will take the slightly longer, but more general, approach of using the value in L₂(1). To do this we complete the command via , changing the last line of the screen to that shown in Figure 13. We complete the command by pressing the key.
Figure 14	In Figure 14 we see that the appropriate value has been entered as the first item in L₃. Furthermore, the calculator is now ready for us to enter a value for L₃(2).
Figure 15	For Figure 15 we have repeated the process to generate invNorm(L₂(2)) at the bottom of the screen. Again we use to perform the command.
Figure 16	Here we see the new value added to L₃.
Figure 17	Figure 17 shows the result of completing this process for the remaining values in L₃. At this point we are done creating the parallel lists that we desired. Press to leave the editor and return to the main screen in Figure 18.
Figure 18	Having accomplished the first three steps in the process: Sort the values (in L₁) Create a parallel list of evenly spaced values between 0 and 1 (in L₂) For each of those evenly spaced values get its invNorm( value (in L₃) We are left with the final, fourth step: Create a Scatter Plot using the sorted values (in L₁) and the invNorm( values in L₃). We use the keys to open the main Stat Plots menu shown in Figure 19.
Figure 19	For the calculator being used here we see that all of the plots are Off. At the moment the highlight is on the first plot, the one we will use here. Therefore, we press to move to the screen controling the settings for that plot.
Figure 20	Figure 20 was captured without changing anything on the screen. The blinking cursor is sitting over the On option (in fact it is covering that option in this screen capture) but we see that the Off option is the highlighted one, corresponding to the fact that, at the moment, Plot1 is Off. To set Plot1 to the On status, we just press the key. The immediate result is shown in Figure 21.
Figure 21	Now, not only is the setting changed to On but also the blinking cursor is still on that option. We now turn our attention to the Type of plot. At the moment, in Figure 21, the plot is set to the histogram option, . However, we want a Scatter Plot, currently displayed as . To do this we use to move the highlight to the Type selections. Once the highlight is on the Scatter Plot option we press to change the setting to .
Figure 22	Figure 22 was captured once the change has been made, although the blinking cursor is still on that choice and is obscuring it at the moment. Looking at the reest of the settings we see that they have changed from what they were back in Figures 20 and 21. The value of Xlist is correctly set as L₁ but, for this calculator, the value of Ylist needs to be changed to L₃. To do this we use the key to move the highlight to the Ylist field and we press to change that value to L₃. That is the condition that we find in Figure 23.
Figure 23	Note that the change to the Ylist field has been made. We will take one extra step just to be sure that the values are set, namely we will use the key to move the highlight to the last field, the one where we can set the Mark value. This produces Figure 24.
Figure 24	Now that the values have been set all that we need to do is to use the keys to open the Zoom menu and then to choose the ZoomStat option. Doing so will move the calculator to display the Scatter Plot in a window that has had its parameters set appropriately for the given data.
Figure 25	We have gone through all of this to produce this graph. A true normal distribution should produce the set of points in approximately a diagonal straight line from the lower left to the upper right corner. The plot in Figure 25 does not conform to that. We would have to conclude that the original data was not really a Normal distribution. [On the other hand, the data set is really small. Taking such a small sample will often lead to such non-conforming plots.]

Figure 26	First we generate the data.
Figure 27	The values in Figure 27 correspond to the values that we saw in the table above. These values are stored in L₁. We could go through the four step process shown above to sort the data, then generate the parallel lists and then produce the scatter plot. However, the third step of that process, finding the invNorm( value of each of the 36 evenly spaced values between 0 and 1 that we would generated in the second step looks daunting at best. This looks like too much work. There must be an easier way! In fact we will see two easier ways.
Figure 28	Remember that L₁ holds our values. The calculator has a special plot option that automatically does the normal quantile plot without our doing any of the work. We return to the General Plot menu via . Then we press to move to the Plot1 screen shown in Figure 29.
Figure 29	In Figure 29 we have moved the cursor to the last of the Type options and selected it. Note that the option is selected. We confirm that the other values are correct. Note that the Data List is set to L₁. Once everything is set we use to open the ZOOM menu and select ZoomStat. You might notice that it takes the calculator a bit of time to do all of the computations that it needs to generate the requested normal quantile plot shown in Figure 30.
Figure 30	Here is the desired plot. In this case we can see that the points do line up in an approximate diagonal. Therefore, this plot supports the conclusion that the original data points, our L₁ values, are normally distributed. Because the calculator generates this plot without our doing any of the work shown in Figures 5 through 25, it would seem that we have no benefit from doing that work ourselves. However, note that the calculator did the work but that it does not directly supply the values in the two parallel lists, nor does it give us the sorted values of the original data. It is possible that we may need these values.
Figure 31	We could extract the sorted values and their associated invNorm( values from the plot. We move into TRACE mode via the key. The calculator then displays, at the bottom of the screen the first sorted point, in this case 36, and the associated invNorm( value, in this case -2.200411.
Figure 32	We can use the key to highlight the second point on the plot to get its value, 41, and its associated invNorm( value, -1.731664. We could continue to use the cursor key to move through the rest of the points to read out the sorted values and their associated invNorm( values. This would be tedious, but it could be done. However, the graph does not provide us with that list of evenly spaced values between 0 and 1 that we had generated when we did all of this in Figures 5 through 25.
Figure 33	Back in the explanation tied to Figure 33 we noted that there would be two easier ways to generate our normal quantile plot. The first was to use the built-in Plot Type that we just demonstrated. The second is to have a program that does all of our steps from Figures 5 through 25, and more, for us. We leave Figure 32 via the sequence to return to our main screen. Then we open the program list menu via the key. We find and highlight the program named NRMLQNTL. Then press to paste the command to run that program onto our screen. This is shown in Figure 33. Press to run the program.
Figure 34	The program asks for the location of the original data. We respond with to tell it that the data is in L₁. Then press to continue the program.
Figure 35	The program does all of the computations that we had done in our original process, along with finding the linear regression equations for the original data points and their associated invNorm( values. It then displays both the scatter plot and the regression equation. Having the graph of the regression equation makes it even easier to get a feeling for the linearity of the plotted points in the scatter graph.
Figure 36	Just as before, we could move into TRACE mode, via the key, and see the individual values behind each of the points on the plot.
Figure 37	However, we could just get out of the plot, via , and open the StatEditor via and we see that indeed the program did do what we had done earlier, namely, it sorted the values and put them into L₁ created the evenly spaced values between 0 and 1 and put them into L₂ created a list of associated invNorm( values in L₃