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Brief User Guide for TI-82 Statistics



INDEX:

To facilitate lookup, the instructions are divided into the following categories:

         I.   Data Manipulation - Entering data, sorting data, clearing lists, friendly values from graphs.
        II.  Single-Variable Statistics - Histogram by hand, simple histogram with the calculator, choosing 
             your own classes when using the calculator, frequency polygon, cumulative frequency (Ogive), percentile
             graph, relative frequency polygon, cumulative relative frequency graph, histogram from grouped data,
             frequency and cumulative frequency graphs from grouped data, box and whisker plot, discrete probability
            distribution, coefficient of  variation, finding standard deviation, finding standard deviation from grouped
            data, standard deviation with a computation formula.
       III.  Two Variable Statistics scatter plot, regression analysis, finding r, r2, a, and b in correlation using a
              calculator, finding r, r2, a, and b in correlation using a computation formula.
       IV.  Aids in doing statistics by hand
        V.  Permutations, combinations, factorials, random numbers.
        VI. Binomial Distribution, pdf, cdf.

       VII.  Probability with the Normal Distribution:

       Notice that the TI-82 does not have the capability to do distributions and inferential statistics as does the
      TI-83 Plus and TI-83.
      


RELEASE DATE:  Beta version - 11/5/08       DATE LAST REVISED:  11/5/08
Printer friendly page here.   Sorry, no printer friendly version. 
NOTE:  Copying  limitations and printing hints are at the end of this document.

IMPORTANT NOTICE:  This is a beta version, meaning that I have not checked it all out on my calculator.  Please
    use it with care. 

FORWARD:  It seems that at the ends of the spectrum of opinions about using calculators there are two polar
opposites:  Use a calculator to the maximum or don't use it for anything except arithmetic.  I have tried to take
into consideration the broad spectrum and include methods for both use of a calculator only and use of the
calculator to take some of the drudgery of arithmetic out of the use of the computation formulas.

NOW A WORD ABOUT MY USE OF LISTS:  Lists are a  powerful tool for doing statistics. In most computation
formulas, lists can be substituted for the variables in most applications.   When doing list arithmetic such as
multiplication, addition, and subtraction and storing the result in another list, the operation can be done from
either the list screen or the home screen.  Fro example L1*L2 will do the same thing at the list screen as
L1*L2
→L3 at the home screen.  (The arrow is a result of pressing STO.)  But when using a function such as
sum( , the operation must be done from the home screen.  So, I will be using both the home screen and the list
screen to do list operations in this document.


 
I.  Data Manipulation
     (NOTE:  In some instances you may want to clear a list or lists before you start entering data.  You  
     can overwrite data already in a list, but remember that if the old list was longer than the new one,
     you must delete the remaining old data an item at a time.  The easiest way to clear one of the tabular
     lists, L 1 -L 6 is to place the cursor on the name above the list and press CLEAR; then ENTER.  DO NOT
     press the DEL key or you will delete the list entry. You can also clear a number of lists or any list as
     follows:
      a) Press STAT, 4 (ClrList).  This will paste "ClrList"  to the home screen.  Press 2nd; then the button
          for the list number you want to clear, for example L1 ; then press ENTER.  If you want to clear more
          than one list, separate the lists by a comma.
      1)  Entering Data:
          a)  Press STAT; then ENTER.  Tables for entering data will appear.
          b)  To enter data, just place the cursor where you want to enter the data and press the correct
               numbers.  You don't have to erase old data if there is already data in the list, but if the old list
               is longer than the new list, you will need to delete the remaining old data items.  Just place
               the cursor over the data and press DEL.       

    2)  Putting Data in Order:
           a)  Press STAT, 2 (SortA).  This will paste SortA to the home screen. 
           b)  Press 2nd, L1 (or whatever list you want to sort); then press ENTER. Finally, return to your
                tables to view the sorted data. Note that you can also sort data in descending order with
                SortD.

   3)  Friendly Values on Graphs Using TRACE:
      Many times when you use the TRACE function, you will get an x-value such as 2.784532.  If you
      change the x-range in the WINDOW function to be a multiple of 4.7, the x-values will be "friendlier"
      values that can be more easily plotted by hand.  Usually the easiest way to do this is to press ZOOM,
      4, for ZDecimal and use  Zoom In or Zoom Out to adjust the window size if it's not satisfactory.  That's
      fine if you are satisfied with a symmetric window.  If you need an asymmetric window, you can get the friendly
      values by pressing WINDOW and setting the window parameters by hand.  Let's take a value and say
      that after a stat plot we get some "unfriendly" values and we press WINDOW and get X-min = .6 and
      X-max = 8.2.  If we change X-min to 0 and X-max to 2*4.7 = 9.4; then we will have friendlier values when
     using TRACE.

II.  Single-Variable Statistics  
    
1)  Doing a Frequency Distribution Histogram by Hand:
           a)  Use items 1 and 2 in Section I above to enter and sort your data.
           b)  Find the class width as follows:
                  (1)  Let S represent the smallest data number (The first number in your sorted list.), L be 
                        the largest number (The largest number in the sorted list.), and C be the number of
                        classes you've chosen. Find the class width, W, with the formula W = (L-S)/C.  Round
                        the number up to the nest higher whole number.

          
c)  Determine the limits of the classes by adding the class width to each successive class. 
                Don't forget that the lower class limit is counted as part of the class width.
           d)  Determine the number of data points in each class as follows:
                (1)  If your data is in L1, go to that list. Make sure your data is sorted in ascending order;  
                      then scroll down to the last number that falls within the upper limit of the first class.  At
                      the bottom of the list your will see L1(#), where # is the number of data items in your first
                      class. 
                (2)  Scroll down to the last item of the second class and subtract the number of items in the
                      first class from the number that appears in L1(#). Continue this until you come to the
                      end of the list.  Note that if you also want cumulative frequency, just write down the
                      numbers as you progress. 
            e)  Subtract 0.5 from each lower class limit of the first class to get the lower boundary of the
                 first class.  Add the class width to get successive boundaries.
             f)  Alternatively, you could do the histogram with the calculator as described below and use the data
                classes and values from that histogram.

      2)  Doing a Histogram with the TI-82:
          
This procedure describes how to do  a simple histogram for which the calculator selects the class 
           width and, therefore, the number of classes..  First you need to get your data into lists. 
           a)  Go to the graphing screen by pressing the Y= button and deselecting or clearing any  functions so 
                that they won't be displayed with your graph.
                Now, go to the list and enter data as follows:
           b)  Press [STAT], [ENTER]
           c)  Then enter the numbers in L1.  (Or whatever list you choose.)
           d)  Press [2nd], [STAT PLOT] and press [ENTER] , [ENTER] to turn Plot 1 on.
           e)  Cursor to the  icons opposite Type, select the fourth icon, histogram,  and press [ENTER] to 
                highlight the histogram icon.
           f)  Enter L1 (or whatever list your data is in) opposite Xlist, by pressing 2nd, L1.  Make sure that
               1 opposite Freq is highlighted if you have ungrouped data.
          g)  Press [ZOOM]; then 9 (ZoomStat)  and the histogram will appear on the screen.  You may need
               to press WINDOW and adjust the window size to get the best display.

          h)  To find the numbers for the limits of the classes and the number of items in the class, press
                [TRACE]; then use  the cursor to move across the tops of the bars in the histogram and read the  
                various numbers.
                                                                

    3)  Selecting Your Own Class Widths for the Histogram Generated by the Calculator.
          a)  Enter your data into List L1.  If your data is not in order, you can sort it by pressing STAT,  
               selecting SortA(, then entering the list name of the data (often L1).  As an example, you might
               have this displayed on your screen: SortA(L1.  Now, press ENTER, and your data will be sorted.
          b)  Now, from the sorted data, determine the class width and lower boundary of the lowest class as
               described under "Doing a Frequency Distribution by Hand" above. 
                Now do this:
          c)  Press [2nd], [STAT PLOT], highlight 1, and press [ENTER].
          d)  Highlight ON on the next screen; then highlight the histogram symbol, the fourth symbol..  Make sure
                L1 is highlighted for  the Xlist.  Note that if you do not have tables data ( where frequencies are given), 
               highlight 1 opposite Freq. 
           e)  Press [WINDOW],  and enter the lower boundary of the lowest class as Xmin  and your 
                 chosen  class width in Xscl.  Note that once the lower boundary  and class width are set, the 
                 upper limit is automatically determined.  Set Ymin at zero.
            f)  Press [GRAPH] and the histogram will appear.  You can use [TRACE] to display the value of the
                boundary limits and frequencies of a particular bar on the histogram.
           g)  If the graph extends above the top edge of the screen, press WINDOW and increase the Ymax
                value.  I also usually set Ymin to -1. 

  4.  Constructing a Frequency Polygon from Ungrouped Data:
          After graphing the histogram, you can use TRACE to get the data for the frequency polygon and a cumulative
          frequency graph if you wish. 
          a)  Press TRACE and use the arrow to move across the histogram bars.  Record the values for x-min, x-max, and "n"
                on a sheet of paper in tabular form.
          b)  Add one-half the class width to each x-min value and record those values.   Store these values in a list, for example
                L2 if you have your histogram data in L1.  Store the corresponding values of "n" in L3.
          c)  Press 2nd, STAT PLOT, ENTER.  If "On" is not highlighted; then select it and press ENTER.
          d)  Highlight the second icon; then highlight L2 opposite Xlist and L3 opposite Ylist.
          e)  Press ZOOM, 9 and the graph will appear on the screen.
          NOTE:  Some teachers or texts prefer return-to-zero graphs.  If your course requires that, do the following after step b)  
          above:
          A.  Calculate a midpoint of a new class preceding the first class and another midpoint after the last class.  These
               values will be entered into L2.  To do that place the cursor at the first item in L2, press INS and replace the zero that 
               appears with your the first midpoint you calculated. Go to the bottom of the L2 list and enter the second value you
               calculated.
         B.  Now you want to enter zero in L3 opposite each of these new midpoints.  Place the cursor at the top of L3 and press
              INS.  A zero will be added.  Now cursor to the bottom of the list and enter a zero opposite the last new midpoint
              that you entered in L2.
         C. Proceed with step c) above. 

5.  Constructing a Cumulative Frequency Chart (Ogive) Graph:
          a)  Enter the Xmax values that you recorded above in a list.  For example, Lif you still have data in the  other  lists.
          b) Next we want to store the cumulative frequencies in L 5, but the TI-82 has no cumSum function.  So, we have three
               options:  Either do them by hand, use an expression,  or use a program.  I'll first show a method that I have given
               students to do the cumulative frequencies by hand; then at the end of this procedure, I'll show give you a short
               calculator program and also how to do it using an expression.  Either of the latter  two methods will store
               the cumulative frequencies in a list.  First doing them by hand we'll assume that your frequencies
              are stored in L4 , and your want to store the cumulative frequencies in L5.
        c)   From the home screen press 2nd, L4 , STO, 2nd, L5 to store the list 4 data in list 5.  Then press 2nd, LIST, ENTER
               to go to the lists.  
       d)  Let's assume for simplicity that we have 1, 2, 5 , 6 stored in L5.  Place the cursor on 2 and enter 2+1 to get 3 in
             the second item.  Move down to 5 and enter 5+3 to get 8 as the third item.  Next move down to 6 and enter 6+8
             to get 14.  Continue with this method until you have completed the list.
         e)  Now that you have the cumulative frequencies in L5,  press 2nd, STAT PLOT; then press ENTER, ENTER to
              turn on the plot function. 
          e)  Highlight the second icon;  then highlight L4 opposite Xlist and press ENTER.  High L5 opposite Ylist and press
                ENTER. .
               NOTE:  If you did a return-to-zero graph for the frequency polygon, go to the list and delete the last
               midpoint and zero in L4 and L5 respectively.
          f)  Press ZOOM, 9 and the graph will appear on the screen.
          Now the expression:
             
(1)  First we want to store the first number in L4 in the first position of L5.  Do that by pressing 2nd, L4(1), STO, 2nd,
                     L5, (, 1, ).   You should now have L4(1) L5(1).  Press ENTER to store the number.   Alternatively, you cold just
                     go to the list and enter the number.
              (2)  Now,  we want to store 0 in the variable X.   From the home screen, press 0, STO, X,T,O, ENTER.
              (3)  Finally, enter the following expression:  X+1, STO, X, ALPHA, : L5(X) +L4(X+1), STO, L2(X+1).  You should have
                     the following expression on the home screen: X+1→X:L5(X) +L4(X+1), , L2(X+1).  Press ENTER continuously
                     until the an error message appears.  That error signals that you have reached the end of the list.  Note that you
                     can change the lists if your list data is not in the list I have used.  I used L4 and L5 only because I was continuing
                     the problem above.
               (4)  Now go back to item "e" to do the plotting.
           Now for the short program: 
            Note that I have written this for lists 1 and 2.  You can change it to anything you
                wish.
                CUMSUM  (This is the name I gave the program)
                dim L1 →N
                 N→dim L2
                 L1(1)→.L2(1)
                For (K, 1, N-1)
                L1(K+1) +L2((K)→L2(K+1)
                END
                Now go back to item "e" to do the plotting.

     6)   Relative Frequency polygon and Cumulative Relative Frequency (Ogive) Graphs: 
             Do these exactly  as in the frequency polygon and cumulative frequency graph above except that they require
             relative frequency data obtained as described in the Cumulative Frequency section.  

  7)  Histogram Using Grouped Data:
          
a)  Enter the midpoints of the classes into L1 and the corresponding frequencies into L2 .
           b)  Press 2nd, STAT PLOT, ENTER.
           c)  If "On" is not highlighted, select it and press ENTER.
           d)  Move the cursor to the histogram symbol and press ENTER; then enter L1 opposite Xlist and L2 opposite Ylist.
           e)  Press ZOOM, 9 and the histogram will be displayed. 
           Note:  If you want to select your own classes do this before pressing ZOOM 9 in step "e" above.
              1)  Press WINDOW and enter the lowest boundary value opposite Xmin
                    and  the class width opposite Xscl.  You may also want to change Ymin to something like zero or -1 so that
                    histogram will not be so far above the baseline.
              2)  Press GRAPH and the histogram will be displayed.

     8)  Frequency Polygon Using Grouped Data:
          
Do this exactly like the histogram, except select the line graph icon, the second icon.  If you've already done the   
           histogram,  just change the icon and press GRAPH.                

     9)  Cumulative Frequency (Ogive) Graph from Grouped Date:
           a)  Enter the upper class limits in a list, for example, L3 if you have data in the first two lists.
           b)  If you have the frequency in L2 , do the following: 
                Use either of the three methods described at the end of  item 5 above for the cumulative frequency.
          c)  Press 2nd, STAT PLOT, ENTER.   If "On" is not highlighted, select it and press ENTER.
          d)  Highlight the second icon, and enter L3 opposite Xlist and L4 opposite Ylist.
          e)  Press ZOOM, 9 and the graph will be displayed.

    10)  Relative Frequency and Cumulative Relative Frequency Graphs for Grouped Data:
           
Do these exactly as in the frequency polygon and cumulative frequency graph above except that  after storing
            the data for the frequency polygon do this step:  Press 2nd, L4 /N,  STO, 2nd, L4 .  This will convert the data
            in L4 to relative frequency. This assumes that the frequency data is stored in L4 .
            N is the total number of data points. 

    11)  Percentile Graphs:
           
This graph is fairly similar to the Ogive graph.  We will do this in two groups of steps:  Preparing data
            and plotting data.
            Preparing Data:
            a)  Enter upper boundaries in L1 and the corresponding frequencies in L2.  If you want the graph to start
                  at zero, enter the first lower boundary with zero for the frequency.
           b)  Press 2nd, QUIT to get out of the List.
           c)  You must now get the cumulative sum of each term and divide it by the sum of L2, multiply that by 100 and
                 store that result in L3.  It's probably easier to just calculate the sum of L2 than to try to enter that into the string.   
                
To simplify matters, let's just store that sum in the variable A. To do that, press 2nd, LIST, 5(sum), 2nd, L2, STO,
                 ALPHA, A, ENTER.
                 and record the number for future use. 
          Now the expression:
             
(1)  First we want to store the first number in L2/A in the first position of L3.  Do that by pressing 2nd, L2(1), 100,
                     STO, 2nd, L3.  You should now have L2(1)/100→L3(1).  Press ENTER and the number will be stored.
                     STO, 2nd, L3.
             (2)  Now,  we want to store 0 in the variable X.   From the home screen, press 0, STO, X,T,O, ENTER.
              (3)  Finally, enter the following expression:  X+1, STO, X, ALPHA, : L5(X) +L4(X+1), STO, L2(X+1).  You should have
                     the following expression on the home screen: X+1→X:(L2(X) +L3(X+1))/A*100, , L3(X+1).  Press ENTER continuously
                     until the an error message appears.  That error signals that you have reached the end of the list.  Note that you
                     can change the lists if your list data is not in the list I have used.  I used L4 and L5 only because I was continuing
                     the problem above.  
          Plotting the Data:
           i)  Press 2nd, STAT PLOT, ENTER
           j)  Select the second icon and enter L1 opposite Xlist and L3 opposite Ylist.
           k)  Press ZOOM, 9 and your graph will be displayed.
           l)  You can find the exact percentiles of the boundaries by using TRACE, and approximate percentiles of
                other x-values by using the cursor.

        
    12) Box and Whisker Plot
         a)  First go to the graphing screen by pressing the Y= button.  Deselect any  Y= functions so that 
               they won't be entered on your graph.  If you choose, clear the list as described at the beginning
              of this document.
         b)  Press [STAT], [ENTER] to go to the list tables.
         c)  Enter your  numbers in L1.  (Or whatever list you choose.)
         d)  Press [2nd], [STAT PLOT] and press [ENTER] to turn on  Plot 1.
         e)  Opposite the word Type, cursor  to the icon that represents a box-and-whisker plot, icon 3, and 
              press [ENTER] to highlight the box plot icon. 
         f)  Enter the list you put the data in, usually L1, in the Xlist, by pressing 2nd, L1. or whatever list
             you chose.  Highlight 1 for the Freq.
         g)  Press [ZOOM]; then 9 (ZoomStat)  and the box-and-whisker plot will appear on the screen.

         h)  To find the numbers for the limits of the quartiles, press [TRACE]; then use  the cursor to move
               across the diagram and obtain the values for quartiles or the beginning and ending values.

13) Box and Whisker Plot by Hand
       You can save yourself considerable calculation if you use the calculator to find Q1, Median, and Q3
       when doing a box-and-whisker plot by hand.  To find those values do the following:
       a)  Press STAT, cursor to CALC  and press ENTER.  "1-Var Stats" will be displayed on the home 
            screen.
       b)  If your data is in list L1 just press ENTER.  Otherwise press 2nd and the list name where your
            data is stored.
       c)  Cursor down and you will find Q 1 , Q3 , and Med listed.  "Med" is the median.
 
14)  Discrete Probability Distribution
       
Let's take a simple example to demonstrate this:  Suppose a word is flashed on a screen several
       times while people are  trying to recognize the word.  The list below indicates what percentage of the
       group required a given number of flashes to recognize the word.  
              No. of Flashes     1        2        3     4       5
              Percent                27      31     18     9      15
              P(x)                     .27     .31     .18    .09   .15
       
         In summary, the method is to enter the number of flashes into list L1 and the corresponding P(x)  
         values  into L2 as P(x).  The details are as follows:
         a)   Enter the number of flashes in list L1 and the corresponding P(x) values in L2 opposite the
                number of flashes.  (How to enter data in a list is covered at the beginning of this document.)
         b)  Now let's find the mean:  Remember that the mean is this:  mean = Σxp(x). 
         c) 
Press 2nd, LIST, move the cursor to MATH and press 5(sum).   Press (,  2nd, L1,*, 2nd, L2, ).  You
               should have this expression: sum( L1* L2)
         d)
 Press ENTER and the mean, 2.54, will be displayed.
         e)  Now let's do the variance.  Remember that the formula for the variance is this: Σx2 P(x) - 2
              or in terms of xP9x) this: Σx2 P(x) - (Σxp(x))2  .
          f)   Press 2nd, LIST, move the cursor to MATH and press 5(sum).   Press (,  2nd, L1, x,x, 2nd, L2, ) -
               (, 2nd, LIST, move the cursor to MATH and press 5.  You should now have this on the screen 
              sum(
L12*L2) - (sum( 
          g) 
Press 2nd, L1, x, 2nd, L2, ),) .  You should now have the following on the screen:
                sum(L12*L2) - (sum(L1 *L2))2
          h)  Press ENTER and the variance, 1.868..., will be displayed.
          i)  If you need the standard deviation do this:  Press 2nd, , 2nd, Ans, ENTER.  The value, 1.3668... will
              be displayed.  

  15)  Doing a Discrete Probability Distribution by Hand
         Many teachers still see value in cranking out the numbers for these statistics
, so here are methods
            to take some of the drudgery out of doing the arithmetic.
         The mean can be obtained by the following formula: mean = Σxp(x).
            To obtain the individual values and store them in list L3, do the following:  (The x-values should
             should be stored in L1  and the p(x) values in L2.)
             a)  Highlight the title for L3 and Press 2ND, L1, x, 2ND, L2.  You will now have L1*L2 in the left bottom of
                  the list screen.
             b)  Press ENTER and you will have the individual values stored in list L3.
             c)  To get the sum of these values,  do this.
                      (1) 
Press 2nd, LIST; cursor to MATH, and press 5.  The expression sum( will be pasted to
                              the home screen. 
                      (2)  Press 2ND, L1 ,x, 2ND, L2 , ), STO, 2ND, L3.  You will have sum(L1 *L1)→L3 pasted
                             to the home screen.
                      (3)  Press ENTER and the sum of those values will be displayed.  Obviously if you only
                             need the mean and not the details of the arithmetic, do only part c.
             You can obtain the variance and standard deviation by first solving for the variance using
             the
formula:  Σx2 P(x) - 2 where is the mean obtained as above.  To obtain the individual
             values of the first term,  x2 P(x). and store them in list L3, do the following:
              a)  From the home screen, press 2ND, L1, x2, ,x, 2ND, L2 , STO, 2ND, L3.  You will have L12*L2→L3
                   
pasted to the home screen.

              b)  Press ENTER and the individual values will be entered in list L3 and pasted to the home
                    screen.
              c)  To get the sum of these values do the following:
                      (1)  From the home screen,
press 2nd, LIST; cursor to MATH, and press 5.  The expression sum( will be pasted to
                              the home screen. 
                      (2)  Press 2ND, L1 ,x2 ,x, 2ND, L2 , ), STO, 2ND, L3.  You will have sum(L12*L2)→L3 pasted
                             to the home screen.
                      (3)  Press ENTER and the sum of those values will be displayed and stored in L3.  Obviously
                             if you only need the sum of the values in the first term  and not the details of the arithmetic,
                            do only part c.
            
 d)  Now subtract the value for 2 from the last value obtained and that will be the variance.
               e)  To obtain the standard deviation, take the square root of the variance as follows:
                        (1)  If you have just calculated the variance do press 2ND, √, 2nd, ANS, ENTER.  Otherwise,
                               insert the value for the variance in place of ANS.

            NOTE:   Obviously, if you only want to obtain the values for the  these three parameters,  you can
            use this method, but it is much easier to use method 14 above.   Just as information, the total
           expression for the variance using this method would the this:  sum(L12*L2) - (sum(L1 *L2))2 .

16)  Calculation of Coefficient of Variation from List Data:
         The coefficient of variation, CV=s/x-bar, is a simple arithmetic calculation if you have the mean
         and standard deviation.  But calculations from a list are a little more involved.  Here's an easy way
         to do it.
         a)  Store the data in a list, for example L1, and press 2nd, QUIT to leave the lists.
         b)  Press 2nd, STAT, move the cursor to CALC.  1-Var Stats will be displayed on the home screen. 
         c)  Enter the given value for x-bar (an x with a line above it.), press divide, enter the value for
              Sx or
σx whichever is appropriate.
         d) Press ENTER and the CV will be displayed.

17.  Finding the Standard Deviation and Variance of Ungrouped Data:

A.  Calculated by the Calculator Only    
     
a)  Entering Data:

          1)  Press STAT; then ENTER.  Tables for entering data will appear.  If you need to clear a    
               list, move the cursor up to highlight the list name; then press CLEAR, ENTER.
          2)  To enter data, just place the cursor where you want to enter the data and press the 
               correct numbers, then press ENTER.  You don't have to erase old data if there is  
               already data in the list, but if the old list  is longer than the new list, you will need to
               delete the remaining old data items.  Just place the cursor over the data and press
               DEL. 
      b)  Suppose that you have the sample of data listed immediately below and you want to find   
            the standard deviation and variance.
            Data:  22, 27, 15, 35, 30, 52, 35
      c) Press STAT, move the cursor to CALC, and press ENTER. The expression 1-Var Stats
            should be pasted to the home screen. If the data is in L1, just press ENTER, otherwise  
            press 2nd and the list number where the data is stored and press ENTER.  In either case,
            the standard deviation and several other statistics will be displayed.

B.  Calculating  Numbers to Plug into a Computation Formula::
    
The standard deviation can be found easily by using 1-Var Stats as described above, but  
      many teachers require that students do the calculations by hand to learn the details of the  
      process.  The following give a method for using the TI-82, TI-83 Plus, or TI-84 for doing much  
     of the arithmetic required and obtaining numbers to plug into the formulas.
     Suppose that students did sit-ups according the table shown below.    

Student

Sit-ups (x) (L1)

x2  (L2)

1

22

484

2

27

729

3

15

225

4

35

1225

5

30

900

6

52

2704

7

35

1225

 

 

 

 n=7

Σx=216

Σx=7492

The variance computation formula is as follows:   s2 = [(Σx -(Σx))/n)]/(n-1), where s2 is the variance .
So,  we will need  ∑x2 and ∑x to plug into the formula.
 
 a)  Enter the data in the table as indicated previously in this document
  
b) Press STAT, move the cursor to CALC, and press ENTER. The expression 1-Var Stats
      should be pasted to the home screen. If the data is in L1, just press ENTER, otherwise 
      press 2nd and the list number where the data is stored.
   c)  Copy n=7, ∑x = 216, and ∑x2 =7492.
  NOTE:  You now have enough data to plug into the formula and solve for the variance and standard deviation. 
 If you are not required to do the detailed calculations, ship to filling in the formula in step f.  Otherwise, continue
 with the next step.
  
d)  Now well need an x2 column.   Highlight the title on list L2, and press 2nd, L1, x2.   You should have
         L12 at the bottom left of the list screen.
    e)  Press ENTER and the numbers will be in list L2.
   
f)  Now, we want to use the number that we previously recorded to plug into the variance
        formula.   So, at the home screen enter
(7492-2162/7)/(6). 
     g)  Press ENTER and you should have 137.8, which is the variance.
     h)  To find  the standard deviation, press 2ND, √ , 2ND, Ans, ENTER, and you will have
           11.39...

18.  Finding the Variance and Standard Deviation of Grouped data.
    A.  Calculated by the Calculator Only:
     
a)  Entering Data:

          1)  Press STAT; then ENTER.  Tables for entering data will appear.  If you need to clear a    
               list, move the cursor up to highlight the list name; then press CLEAR, ENTER.
          2)  To enter data, just place the cursor where you want to enter the data and press the 
               correct numbers and press ENTER.  You don't have to erase old data if there is already  
               data in the list, but if the old list  is longer than the new list, you will need to delete the  
               remaining old data tems.  Just place the cursor over the data and press DEL. 
      b)  Suppose that you have the sample of data listed in the table below and you want to find
            the standard deviation and variance.
        

Classes

Class
Midpoint  x (L1)

Freq. (f) (L2)

35-45

40

2

45-55

50

2

55-65

60

7

65-75

70

13

75-85

80

11

685-95

90

11

95-105

100

4

       c)  Enter the class midpoints in list L1You can either do the midpoints by hand or calculate   
            and store them in list L1 as follows:
            (1) Store the lower boundaries in list L1 and the upper boundaries in L2
            (2) Press 2ND, QUIT to get out of the list editor and press (, 2ND, L1, + 2ND, L2,), divide
                  symbol, 2 STO, L1.  You should have (L1 + L2)/2
L1 on the home screen.  Press     
                  ENTER and the midpoints will be stored in L1.
      
d)  Enter the frequencies in L2 as described under Entering Data immediately above, then
            press  2nd , QUIT to leave the tables.
            Now well calculate the required statistics.
       e) Press STAT, move the cursor to CALC, and press ENTER. The expression 1-Var Stats
            should be pasted to the home screen. Press 2nd, L1 ; then press the comma and finally
            press 2nd, L2.  
        e)  Press ENTER, and the standard deviation along with several other statistics will be
             displayed.  The sample standard deviation is 14.868.
         f)  To find the variance, just square the standard deviation by entering the value, pressing
             the x2 button, and then ENTER. 
 

  B.  Calculating  from Grouped Data to Plug into a Computation Formula:
    
The standard deviation and variance for grouped are similar to ungrouped data except that the
     x-values are replaced by the midpoints of the classes.  Let's assume some sort of grouped  
     data as indicated by the first and third columns below.
    

Classes

Class
Midpoint  x (L1)

Freq. (f) (L2)

xf
(L3)

x2f
(L4)

35-45

40

2

80

3200

45-55

50

2

100

5000

55-65

60

7

420

25200

65-75

70

13

910

63700

75-85

80

11

880

70400

685-95

90

11

990

89100

95-105

100

4

400

40000

 

 

n=Σf=50

∑x=Σxf=3780

∑x2 =  Σxf=296600

The formula for the grouped data variance is this:
           s2 =(
Σx2  -(Σxf)2 /Σf)/(Σf-a) 
a) You can either do the midpoints by hand or calculate and store them in list L1 as follows:
     (1) Store the lower boundaries in list L1 and the upper boundaries in L2
     (2) Press 2ND, QUIT to get out of the list editor and press (, 2ND, L1, + 2ND, L2,), divide
          symbol, 2 STO, L1.  You should have (L1 + L2)/2
L1 on the home screen.  Press ENTER
          and the midpoints will be stored in L1.
b)  Press STAT, ENTER to go to the lists and store the frequencies in list L2.  After you have     
     finished entering the frequencies and midpoints, press 2nd, QUIT to leave the lists.
 Now lets calculate the required numbers.
c) Press STAT, move the cursor to CALC, and press ENTER. The expression 1-Var Stats
    should be pasted to the home screen. Press 2nd, L1 ; then press the comma and finally
    press 2nd, L2
d)  Press ENTER and several statistics along with the standard deviation will be displayed.   
     Record the standard deviation, Sx =14.868 for a reference.  Also record ∑x=∑xf=3780,
     ∑x2=∑x2f=296600, and n=50.  Youll need these values later. 
      Notice that the value for ∑f is listed as n in the calculator and ∑xf is listed as ∑x and ∑x2f is  
      listed as ∑x2
NOTE:   You now have enough numbers to plug into the formula and solve for the variance.  
If you are not required to do the detailed calculations to fill in the table, skip to item j below. 
Otherwise continue with the next step.    
e)  Calculate xf and store it in
L3.  Highlight the title for L3 and press  2ND, L1, *, 2ND, L2.. You should have
      L1*L2 at the bottom left of the list screen.   Press ENTER and the products will be stored in list L3.
f)  Calculate x2f and store it it L4.  Highlight the title for L4 and press 2ND, L1, x2 , * , 2ND,  L2.  You should
     now have L12 *L2 at the bottom left of the list screen. 
g)  Press ENTER and the results will be stored in list L4 .
 h)
 You dont need to calculate Σf.  That is the value for n that you previously recorded.  
 
i)   You dont need to calculate Σxf.  That is the value for ∑ x that you previously recorded.
 j)  Now, you want to plug the appropriate numbers into the formula for the variance. From the  
     home screen enter
(296600-3780/50)/(49)
 k)  Press ENTER and you should have 221.06, which is the variance.
 l)  If you want the standard deviation, press 2ND, √ , 2ND, Ans, ENTER, and you will have 14.868...

 III. Two-variable Statistics
 1)  
Scatter Plot
       First you need to get your data into lists. 
       a)  Go to the graphing screen by pressing the Y= button and deselecting any  functions so that 
             they won't be entered on your graph.  If you want to clear the lists before entering data, see the
             note at the beginning of this document.
       b)  Press [STAT], [ENTER] to go to the list tables.
       c)  Enter the data-point numbers ( the x-values)  in L1 and the corresponding values (y-
             values)   in L2. 
      d)  Press [2nd], [STAT PLOT] and press [ENTER] to turn Plot 1 on.
      e)  Cursor to the scatter diagram, the first icon opposite Type,  and press [ENTER] to highlight the
            scatter diagram icon.
      f)  Highlight  L1 opposite Xlist, and L2 opposite Ylist (do this by pressing 2nd and the appropriate list button);  
            then select the type marker you prefer.  (I like the + symbol. ).
      g)  Press [Zoom], 9 and the scatter plot will appear on the screen.

2)  Plotting  x-y line chart
     
Do that the same as the scatter plot in item 1 above except that when you select the type, choose the 
      second icon for the line symbol rather that the scatter-diagram  icon.

3)  Regression Analysis:
     
Assume that you have the following information on the heights and weights on a group of young
     women:

  1 2 3 4 5 6 7 8
Height x 65 65 62 67 69 65 61 67
Weight y 105 125 110 120 140 135 95 130

       First you need to get your data in lists.  You can do that from the home screen, but if you have any   
      significant amount of data, it's much easier to enter it into List tables.  See the note at the beginning of
      this document for instructions on clearing lists if you want to clear your lists before data entry.
       Here's how to enter data:
      a)  Press [STAT], [ENTER];  then enter the numbers for the independent variable, x-values,   in L1 and
            the corresponding values in L2.
      b)  After you have finished entering data, Press[STAT]. 
      c)   Cursor to CALC and press <9>, [ENTER] (Where <9> is just the number 9 from the keyboard.)
             LinReg (a+bx) will appear on the screen if you chose 9.   Note that if you want to use QuadReg
            or some other analysis, press the number to the left of that entry.
       d) If you want to graph the equation of the best-fit line, ship to item e below.  If you have your data  
           in the L1 and L2 as described above, just press ENTER.  If  you have your data in other lists, youll  
           need to  enter the lists by pressing 2nd, press the list number for x, comma, 2nd, press the list number
           for y; then press ENTER.  In either case a, b, and r will be displayed. 
            ANSWER:  If you pressed ENTER you should have these values:  a=-186.47.., b=4.705,
           and r=.7979, if you want r2 , just enter the value for r and press x2 , then ENTER. You will get
           r2 =.63366.
      e)  If you want to graph the equation,  press Y=, VARS, 5 (for Statistics), move the cursor to EQ, and press 7
           (for RegEQ.  The regression equation will be entered opposite Y1=.
       f)  Press ZOOM, 9 (for ZoomStat) and the graph will be displayed.
       g)  If you also want to graph the scatter plot on the same screen as the line graph, do the following:
            1)  Press 2nd, STAT PLOT;  then press ENTER, ENTER.
            2)  On the screen that appears, select the first icon and press ENTER.
            3)  Move the cursor to L1  opposite Xlist and press ENTER.  Next,  move the cursor to L2 opposite
                  Ylist and press ENTER. 
           4)  Select your favorite mark is you have one; then press ZOOM, 9 and the scatter diagram and the
                best-fit line will be displayed.

4)  Plotting a graph with the scatter plot and the regression equation on the same axis.
     First you need to do the regression graph as described above in item 3.  Now, you want to put the 
     scatter plot on the screen with the graph. To do this:
     a) Press [2nd], [STAT PLOT] and press [ENTER], ENTER to turn Plot 1 on.
     b) Cursor to the scatter diagram for Type (the first icon) and press [ENTER] to highlight the scatter  
         diagram.
     c) Move the cursor opposite  Xlist,  highlight L1 and press ENTER.  Move the cursor opposite Ylist, 
          highlight L2  and press ENTER.   Select the mark you choose if you wish.  (I like a + ).
     d) Press ZOOM, 9 (for ZoomStat) and the scatter plot and best-fit graph will appear on the screen.
     e)  You can press [TRACE] to display the x-y values of the data points, or press the down arrow to 
           jump  to points on the line.
Note that if your data has several decimal places and you'd rather have fewer, you can make the data
 friendlier by making the x-distance (xmax-xmin) a multiple or sub-multiple of 9.4. 

5)  Finding the Correlation Values r and r2 Using a Computation Formula: 
                  
Assume that you have the following information on the heights and weights on a group of young women:

  1 2 3 4 5 6 7 8
Height x 65 65 62 67 69 65 61 67
Weight y 105 125 110 120 140 135 95 130

       First you need to get your data in lists.  You can do that from the home screen, but if you have any   
      significant amount of data, it's much easier to enter it into List tables.  See the note at the beginning of
      this document for instructions on clearing lists if you want to clear your lists before data entry.
       Here's how to enter data:
      a)  Press [STAT], [ENTER];  then enter the numbers for the independent variable, x-values,   in L1 and
            the corresponding values in L2.
            NOTE:  The formula for r is this:  (n
Σxy ΣxΣy)/[(√nΣx2- (Σx)2)(√nΣy2- (Σy)2)].  So, you will  
            need  Σx, Σy, ΣxΣy, Σx2, Σy2,, and n.  You can get all of these by using the 2-Var Stats
            function.  Use that as follows:
      b)  With the data in lists L1 and L2 press STAT, move the cursor to CALC, and press 2.  The
           expression 2-Var Stats, should be displayed on the screen. 
      c) If the data are in L1 and L2, press ENTER and the necessary values will be displayed.  If the
          data are not in those lists, you will have to enter the list numbers where the data are stored.  
          Notice that you will need to scroll down to get some of the values on the screen.  Record the
          values for these parameters:  Σx=521,  Σx2=33979, n=8, Σy=960, Σy2=116900, Σxy=62750.
      NOTE:  Just a few words on entering the data in the calculator:  All denominators and
      numerators with  more than one term must be enclosed in parentheses.  On the TI-82,  a square root
      with more than one term  must be enclosed in parentheses.  Example:  √(nΣx2- (Σx)2).
        Now lets plug the numbers into the equation for r:
        d)  r=
(nΣxy ΣxΣy)/[(√(nΣx2- (Σx)2)(√(nΣy2- (Σy)2)]
                = (8*62750-521*960)/(√(8*33979-5212)(√(8*116900-9602))
               =.7979..
       e)  Some students seem to have difficulty accurately entering a long expression such as in item "d." 
            Those students can do the calculation without loss of accuracy by using the following method.
          1)  Enter the numerator in the calculator and store it in variable N.  In this manner: 
               8*62750-521*960, STO, ALPHA, N. 
          2)  Calculate the denominator and store it in two separate variables M and D. In this manner
               √(8*33979-5212 )  , STO, ALPHA, M; then √(8*116900-9602), STO, ALPHA, D.
          3)  N(M*D), ENTER.  You'll get the same answer as above.

6)  Finding the Values a and b for the Best-Fit Equation Using a Computation Formula:
     
Assume that you have the following information on the heights and weights on a group of young women:

  1 2 3 4 5 6 7 8
Height x 65 65 62 67 69 65 61 67
Weight y 105 125 110 120 140 135 95 130


      The formula for b is this:  (nΣxy ΣxΣy)/(nΣx2- (Σx)2).  So, you will need to record the values
      for .  x-bar, y-bar, Σx, Σy, ΣxΣy, Σx2, Σy2, and n.. You can get all of these by  using the 2-Var Stats function.  
     Use that as  follows:
      a)  With the data in lists L1 and L2 press STAT, move the cursor to CALC, and press 2.  The
           expression 2-Var Stats, should be displayed on the screen. 
      b)  Press ENTER and the necessary values will be displayed.  Notice that you will need to
           scroll down to get some of the values on the screen.  Record the values for these
          parameters: ).  So, you will need to record these values: x-bar=65.125, Σx=521,  Σx2=33979,
          n=8, Σy=960, y-bar=120, Σy2=116900, Σxy=62750
      c)  Plug these numbers into the formula and then enter the expression your calculator. 
           Just a few notes on entering the data in the calculator:  All denominators and numerators
          with more than one term must be enclosed in parentheses.  On the TI-82,  a
          square root expression must be enclosed in parentheses.  Example:  √(nΣx2- (Σx)2)
   d)  Enter the values in the calculator  for this formula:
         b=
(nΣxy ΣxΣy)/(nΣx2- (Σx)2).
           =(8*62750-521*960)/(8*33979-5212)
           =4.7058..
     e)  Now, calculate the value for "a" from the formula:
          a= y-bar b(x-bar)
            =120-4.7058 *65.125
           =-186.465
      f)  Some students seem to have difficulty accurately entering a long expression such as in item "d." 
          Those students can do the calculation without loss of accuracy by using the following method.
          1)  Enter the numerator in the calculator and store it in variable N.  In this manner: 
               8*62750-521*960, STO, ALPHA, N. 
          2)  Enter the denominator and store it in variable D.  8*33979-5212 , STO, ALPHA, D.
         
3)  Enter  ND and press ENTER.  You'll get the same answer as above.  
                                                                       
                                                  
IV.  Aids in doing statistics by hand.
       General: 
Often in book problems in school you'll need to do a lot of calculations by hand.  These  
         techniques will save you a lot of arithmetic.
 
1.  Arranging Data In Order.  (This is the same as item 2 in section I above, which I will repeat here.)
     a)  Enter the data in one of the lists as indicated in Section I.    
     b)  Press STAT, 2 (SortA).  This will paste SortA to the home screen. 
     c)  Press 2nd, L1 (or whatever list you want to sort); then press ENTER.  "Done" will be displayed
          on the home screen, indicating your data has been sorted. Note that you can also sort data in 
          descending order with SortD.

2.  Finding Mean (x-bar), ∑x, or ∑x2 , σ, Median, Q1, Q3 for Grouped or Ungrouped Data.
    For Ungrouped Data:

    a)  After entering your data in the list as described in item 1 of Section I, above, press STAT, and
         cursor over to CALC, and press ENTER. "1-Var Stats" will be pasted to the home screen.
    b)  Enter the list name you want to operate on by pressing 2nd; then the list number, for example L1.
    
c)  Press ENTER.
    d)  A number of results will be displayed on the home screen.
     NOTE:  You can also find these values for discrete random variable statistics by entering the values
                 of the variable in L1 , for example, and the corresponding data values in L2.
   For Grouped data:
    a)  Find the midpoints of each group and enter those values in L1; then enter the corresponding frequencies
        L2.  Entering data in a list is described in item 1 of Section I, above.
        
    b)  Press STAT, cursor over to CALC, and press ENTER. "1-Var Stats" will be pasted to the home screen.
    
c)  Press 2nd, L1, 2nd, L2; then press ENTER.
    d)  Various statistics will be displayed on the home screen.  Note that for grouped data, ∑xf is listed on the
          calculator as ∑x and ∑x2 f is listed as ∑x2 .

3.  Finding products such as xy or (x-y):
     a) Assume that your x-data is in L1 and your y-data is in L2.  Then obtain the product by pressing
        2nd, L1; x (multiply symbol), 2nd, L2, ENTER.
     b)  If you want the data stored in a list, L3 for example, before pressing ENTER in item a, press 2nd,
          L1, STO, 2nd, L3.  Then press ENTER.
     c)  Obviously, x-y can be obtained by merely substituting the subtraction symbol for the
          multiplication symbol in atep a) above.

4.  Squaring operations such as elements of lists.
     a)  To square the elements of a data set, first enter the data in a list, for example L1.
     b)  Press 2nd, L1; then the x2 symbol, ENTER.  The squared elements will be displayed.
     c)  If you want to store the squared data in a list, for example L3, then before pressing ENTER in
          item b above, press 2nd, STO, 2nd,  L3.  Then press ENTER.
     d)  If you want to multiply corresponding elements of two lists and square each result; then your
         expression should be like this:  (L1 * L2)2 .

5.  Find x-x (Sorry, I have no symbol for the mean, so I displaced the bar.) from the data in
     list  L
1.

     a)  Enter 2nd, L1, -, 2nd, LIST.  Note that" -" is a minus sign not a negative sign.
     b)  Cursor to MATH and press 3.  You should now have "L1-mean(" pasted to the home screen.
     c)  Press 2nd, L1, ENTER.  The result will be displayed on the home screen. 
     d)  If you want to store the results in a list, for example L3, then before ENTER in item "c" above, press
         STO, 2nd, L3; then ENTER

 6.  Finding (x-x )2  
      a)  Press (, 2nd, L1, -, 2nd, LIST.
      b)  Cursor to MATH and press 3.  You should now have "(L1-mean(" pasted to the home screen.
        c)  Press 2nd, L1,),),x2 .  The expression ((L1-mean(L1))2 should now be displayed on the screen.
          Press ENTER and the results will be displayed on the home screen.
      d)  If you want to store the results in a list, for example L3, before pressing ENTER in item "c"
           above, press STO, 2nd, L3; then ENTER.

7.  Finding (Σx)2 and Σx2
    
Some computation formulas for the standard deviation require (Σx)2 .  To find that, do the following:
      a)  Enter your data in a list as described at the beginning of this document.  Press 2nd, QUIT to get
          out of the list. Press ( to enter a parenthesis on the home screen.
      b)  Press 2nd, LIST, and cursor over to MATH.
      c)  Press 5.  "(sum(" should be entered on the home screen.
      d)  Press 2nd, L1 or whatever list your data is stored in.
      e)  Press ), ), x2 .  You now should have (sum(L1))2 on your home screen.
       f)  Press ENTER and the results will be displayed on the screen.
       g) Σx2 can be found by using the "1-Var Stats" function under STATS, CALC, but you can also
          find it by entering "sum L12 "

8.  Notice that you may also do several other operations by pressing 2nd, STAT; then moving the cursor to
    MATH and entering the list name that you wish to operate on.

V.  Permutations, combinations, factorials, random numbers:
 1. Finding Permutations.
    a)  Suppose we want the permutations (arrangements) of  8 things 3 at a time, enter 8 on the home 
         screen.
    b) Press MATH and cursor over to PRB and press 2, (nPr). You will have 8 nPr pasted to the screen.
    c)  Enter 3 and press ENTER.  You will get 336.

 2. Finding Combinations:.
     a)  Suppose we want the combinations (groups) of  8 things 3 at a time, enter 8 on the home screen.
     b) Press MATH and cursor over to PRB and press 3. (nCr). You will have 8 nCr pasted to the screen.
     c)  Enter 3 and press ENTER.  You will get 56.

 3. Finding Factorials.
     a)  Suppose we want 5 factorial (5!).  From the home screen press 5.
     b) Press MATH and cursor over to PRB and press 4 (!)). You will have 5! pasted to the screen.
     c)  Press ENTER and you answer, 120, will be displayed.

VI.  Binomial Distribution, pdf, cdf:
      The binomial with n items, p probability, and r trials can be calculated for a specific value using a calculator or
      looked up in a table.  Let's suppose, however, that you have 12 items, each with a probability of 0.3, and you
      are asked to find the probability that 6
≥ r≤9.  You can always look the values in the table, but it's much easier
        if you use the little program below.
      

Prgrm:BINOMIAL
:INPUT "INPUT N", N
:INPUT "INPUT P", P
:INPUT "BEGIN  R", K
:INPUT "END R", R
:{0}
→L1
:R→dim (L1)

:For(V, K, R)
:N nCr V*P^V*(1-P)^(N-V)
→L1(V+1)
:End
:sum L1→A
:Disp "P=", A
:Stop
To do the problem described above, just start the program, enter values for N and P at the prompt.  After that, enter
6 at the "BEGIN R" prompt and 9 at the "END R" prompt.  If you want just one value, put that value in for both BEGIN
 and END.  Of course, if you want Cdf throug, say 5, just enter 0 (zero) for BEGIN and 5 for END.  If by chance you
need the individual values, just look in list L1.  You can get there by pressing 2ND, LIST, ENTER.

VII.  Probability with the Normal Distribution:
   Preliminaries:   Although students usually buy a TI-83 Plus, Casio CFX-9750, or newer calculator when they take
   statistics, some students may find themselves without funds to buy a new calculator.  For that reason, I am
   including how to do probabilities with a normal distribution.
   The method we will follow is to graph the normal curve using the graphing function  and integrate the curve within
   limits to find the area, which can be interpreted as the probability.
   First we must set up the equation for the normal curve on the Y= screen.  We'll store it in the Yo= so that it will not
   be in our way when we are graphing functions and doing plots.
   Important: We will use M for mean and S for standard deviation if the formula.  We will store new values for these
   variables when we have a new problem, but the equation will remain the same.
   Storing the equation:
   a)  Press Y= and move the cursor down to yo=.  Enter the following:  -.5((x-M)/S)2 )/(S√(2π)).  If you are not going
         to use the equation immediately, move the cursor to the equal sign opposite Yo and press ENTER to deselect
         that entry.
   b)  Now let's set the WINDOW.  In general, set the window Xmin at mean+ 5 standard deviations and Xmax at
         mean + 5 standard deviations.  Set Ymin at 0 and Ymax at .4/S.  That is .4 divided by the standard deviation.
    Now for examples:
    Ex 1: Let's say we have a distribution with a mean of 70 and a standard deviation of 4.  We want to find the
       probability that a random selection is less that 66.
     a)  Store 70, the mean,  and the standard deviation, 4, in the variable M and S respectively by entering 70 and
            pressing, STO, ALPHA, M, ALPHA, :,4, STO, ALPHA, S.  You should now have this:  70→M:4→S.  Press
            ENTER and the variable values will be stored.  If you are ever in doubt about whether a variable is
            stored, just press ALPHA the variable letter and ENTER.   The stored value will be displayed.
     b)  Press WINDOW and set enter 70-5*4 opposite Xmin and 70+5*4 opposite Xmax.  Set Xscl to 5 if you don't
           like the thick x-axis.  Set Ymin at 0 and Ymax at .4/4 = .1
     c)  Press GRAPH and a standard curve should be displayed.
     d)  Press 2nd, CALC, 7(∫f(x)dx), enter 50, the number for Xmin, and press ENTER.
     e)  Move the cursor a few x-units to the right and enter 66, ENTER.  The area will be shaded and the value for
           the area, 0.1586... will be displayed on the screen. 
    Ex 2: Suppose that again we have a mean of 70 but a standard deviation of 6.  We want to find the
       probability that a random selection will be between  66 and 76.
     a)  Store the standard deviation, 6, from the home screen  in the variable S by pressing 6, STO, ALPHA, S. 
          You should now have this: 6→S.  Press ENTER and the variable values will be stored.  If you are in doubt
          about whether you have the correct variable for M,  just press ALPHA, M,  ENTER and the stored value will
          be displayed.
     b)  Press WINDOW and set enter 70-5*6 opposite Xmin and 70+5*6 opposite Xmax.  Set Xscl to 5 if you don't
           like the thick x-axis.  Set Ymin at 0 and Ymax at .4/7 = .07.  Actually, Xmax and Xmin only need be wide
           enough to include the values 66 and 76 for this particular problem.
     c)  Press GRAPH and a standard curve should be displayed.
     d)  Press 2nd, CALC, 7(∫f(x)dx), enter 40, the number for Xmin, and press ENTER.
     e)  Move the cursor a few x-units to the right and enter 100, ENTER.  The area will be shaded and the value for
           the area, .5888... will be displayed on the screen. 
   Ex 3: Finally let's suppose that again we have the same problem as in Ex 2, but we want to find the probability
             that a random selection is greater than 76.
     a)  Assume the current mean, 70, and the standard deviation, 6, are still stored and the WINDOW is set as in
           Ex 2.  
     b)  Press 2nd, CALC, 7(∫f(x)dx), enter 76, the number for Xmin, and press ENTER.
     c)  Move the cursor a few x-units to the right and enter 100, ENTER.  The area will be shaded and the value for
           the area, .15865.. will be displayed on the screen. 

A FINAL WORD:   There are other things that could be done with the TI-83, but it is such an old calculator that
     I think it is more useful to spend my time with more modern calculators.  If you have question, I'll try to find
     time to answer them.


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