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Brief User Guide for TI-83 Plus Statistics Contents: This document covers single- and two-variable statistics, scatter plot, regression analysis, normalpdf, Student's Distribution,
geometric mean, and much more.
Last Revised: 9/25/15

INDEX:

To facilitate lookup, the instructions are divided into the following categories:
PART I: BASIC STATISTICS:

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,  sorting data into classes,
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, weighted average, median of grouped data, geometric mean.
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 and matrices, testing the correlation coefficient,
IV.  Aids in doing statistics by hand
V.  Permutations, combinations, factorials, random numbers.
VI.  Normal Distribution - Area under a normal curve, Finding Z values, Graphing a curve,  WINDOW
settings for graphing a curve, Probability Distribution Function using normalpdf(, Graphing the
Normal Distribution Using normalpdf(, normalcdf(,  normalcdf( using lists, and ZInterval,
VII. Other Distributions -  Finding a TInterval, Student's t Distribution, Using invT to Find a t-value given
α
and df,
Chi-squared Distribution, binomialpdf, binomialcdf.
VIII.  Hypothesis testing - mean and z-test (data), mean and z-test (statistics), mean and t-test (data),
mean and t-test (statistics).

IX.  Statistics of two Populations - confidence interval for two dependent population, confidence interval for two

independent populations (Data and Stats),
X.  Other Tests and Inferences -  one-way ANOVA, Chi-Square test for independence
, X2 Goodness of Fit,
XI. Special Procedures - Covariance, Variance-covariance matrix, a smidgen of mean-variance optimization

APPENDIX:  Simple program for calculating inverseT with at TI-83 Plus RELEASE DATE:  10/1/06         DATE LAST REVISED:  9/21/15

© 2003 Frank Kizer     NOTE:  Copying  restrictions and printing hints are at the end of this document.

PART I: BASIC STATISTICS

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 that use only a calculator  and those that use 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, subtraction, and raising to a power and storing the result in another list, the operation
can be done from either the list screen or the home screen.  For example L1*L2  with the L3 title highlighted 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.)
There may be occasions when a function such as sum( must be done from the home screen, but it usually is more
convenient to do arithmetic operations from the lists screenBut when using a function such as sum( , the operation
must be done from the home screen.

I.  Data Manipulation
1) Clearing Lists:
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.  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.
2)  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.

3)  Putting Data in Order:
In some of the procedures below, you may need to put the data in order of value.  You can do
do that as follows:
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.

4)  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)  Graphing or Drawing a Frequency Distribution Histogram (Ungrouped Data):
I have included two methods for graphing a histogram or getting data to construct the histogram by hand.  The first
method allows the calculator to calculate the class limits and boundaries.  The second method requires some involvement
in the procedure by the calculator user, although the calculator does all of the arithmetic.  Also I give a procedure
for getting the data from these graphs to draw a histogram by hand in the event that an instructor requires that be
done.  Finally, I have included a calculator program to sort the data into classes.

A)  Entering Data:
a)  Press STAT; then ENTER.  Tables for entering data will appear.  You may want to completely clear the list
you are planning to use by moving the cursor to the title, for example L1, and pressing CLEAR, ENTER.
DO NOT press DEL while you have the title highlighted or your delete that list from the tables.
b)  To enter data, just place the cursor where you want to enter the data and press the correct
numbers.  If you have not cleared the list and 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 you want to delete and press DEL.

B)  Doing the Histogram by Letting the Calculator Choose the Class Limits:

a)  Go to the lists and enter data.  First, press [STAT], [ENTER] to display the list tables.
b)  Enter  the numbers in L1.  (Or whatever list you choose.)
c)  Press [2nd], [STAT PLOT] and press [ENTER] to turn Plot 1 on.
d)  Cursor to the  icons opposite Type, select the third icon, histogram,  and press [ENTER] to
highlight the histogram icon.
e)  Enter L1 (or whatever list your data is in) opposite Xlist, by pressing 2nd, L1.  Make sure there's a
1 opposite Freq if you have ungrouped data.
f)  Press [ZOOM]; then 9 (ZoomStat)  and the histogram will appear on the screen.
g)  If you want to know the class limits and
the number of data points in each class, press TRACE and
move the cursor across the tops of the bars.

C)  Doing the Histogram by Choosing Your Own Class Widths:

a)  Go to the lists and enter data.  First, press [STAT], [ENTER] to display the list tables.
b)  Enter  the numbers in L1.  (Or whatever list you choose.)

c)  Find the class width as follows:
First you will need to sort the data so that you can determine the smallest and largest number.
(1) Press STAT, 2 (for SortA) to paste SortA( to the home screen.
(2) Press STAT, ENTER and record the smallest number (the first one) and the largest number (the last one).
(3) Let S represent the smallest data number, L be the largest number, 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 next higher whole number.
NOTE:  It may also be that you are given the number of classes by a textbook problem.  In that case simply enter
that number is place of "C" in the above formula.

d)  Press [2nd], [STAT PLOT] and press [ENTER] to turn Plot 1 on.
e)  Cursor to the  icons opposite Type, select the third 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 there's a
1 opposite Freq if you have ungrouped data.
g)  Press [ZOOM]; then 9 (ZoomStat)  and the histogram will appear on the screen.

h)  To get the number of classes you want, press WINDOW and change Xscl to the class width.
i)  Press GRAPH and the new graph with the correct number of classes will be displayed.

D)  Using Either of These Methods to Get Data to Plot a Histogram by Hand:

First make a suitable table for Class Limits, Class Boundaries, and Frequency to record the numbers
that will be found in the procedure that follows.

Determine the class limits and the number of data points in each class from whichever of the graphs above that you choose.
a)  Press TRACE and move the cursor across the tops bars of the graph using the cursor controls. The class limits will be
displayed. Write these down in your table.  If you have a display such as min=1, max<9, the class limits of that class would
be 1 and 8. Don't  forget that the lower class limit is counted as part of the class width.  The number of data points in the
class being viewed will be indicated by n= some number.
b)  Determine the class boundaries as follows:
First, you must set the lowest class boundary. If, for example,  the lower class limit is 1, press WINDOW and change x-min
from 1 to 0.5. Press GRAPH to redraw the histogram. Now press TRACE and move the cursor across the top of the histogram
bars.  The numbers displayed are the class boundaries.  Consider the < symbol as an equal sign for the upper boundary.
Record these number in the table that you made.
c)  You now have all of the data you will need to draw the histogram by hand.

E)  Using my program NOSTOCLS to sort the data into classes:

There are times when it would be useful to check data for two or three different classes.  This might be useful for grading
papers or for saving class time for something more important than tallying.  With my program, this can be done in about
one minute after entering the data in  list L1. Suppose we have a certain set of  60 numbers with values from 1 to 47.  We
want to check the distribution for six, seven, and eight classes.  The program will quickly give the distributions, 9,14, 17, 6,
5, 3, 3, 3 for a class width of 6; 11,18, 14, 7, 3, 4, 3 for class width of 7; and 14,21,11,6,4,4 for class width of eight.  The program
is included at the end of this document.

2.  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 on the first row; then enter 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.

3.  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)  Now, store the cumulative frequency data in L 5 as follows:  Place the cursor over the list title, L1.  Press 2nd, LIST,
cursor to OPS, and press 6.  The expression cumSum( will be displayed at the bottom of the screen.
c)  With the cursor after the parenthesis, press 2nd, L3, ), , ENTER.  You will now have cumSum(L3)
at the bottom of  the lists screen.
d)  Press 2nd, STAT PLOT, highlight "On" if necessary and press ENTER
e)  Highlight the second icon on the first row; then enter L4 opposite Xlist and L5 opposite Ylist.
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.

4)   Relative Frequency polygon and Cumulative Relative Frequency (Ogive) Graphs:
These are done similarly to the as frequency polygon.  After storing the data for the x-values and frequencies, do the
steps listed for each type graph.
Relative Frequency:

Assume that we want to store the relative frequencies in list L3, the frequencies are in L2,and the x-values  are in L1 .

a)  First place the cursor to highlight the list title, L3.  Press 2ND, L2, ÷, 2nd, LIST, move the cursor to MATH and press 5.  You
should now have L2/sum( displayed on the bottom of the list screen.
b)  Press 2ND, L2, ), ENTER and the relative frequencies will be stored in list L3.

c)  To plot a graph of the relative frequency, press 2nd, STAT PLOT, ENTER.   If "On" is not highlighted, select it and press ENTER.
d)  Highlight the second icon, and enter L1 opposite Xlist and L3 opposite Ylist.
e)  Press ZOOM, 9 and the graph will be displayed.
Cumulative Relative Frequency:
Assume that we want to store the cumulative relative frequencies in list L4 and that the relative frequency
is still stored in L3 from the above relative frequency operation above, and that the x-values are in L1.

a)  First place the cursor to highlight the list title, L4.  Press 2ND, LIST, move the cursor to OPS and press 6.  You should now
have cumsum( at the bottom of the screen.
b)  Press 2ND, L3, ), ENTER. The cumulative relative frequencies will now be stored in L4.

c)  Press 2nd, STAT PLOT, ENTER.   If "On" is not highlighted, select it and press ENTER.
d)  Highlight the second icon, and enter L1 opposite Xlist and L4 opposite Ylist.
e)  Press ZOOM, 9 and the graph will be displayed.

5)  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 Freq.
e)  Press ZOOM, 9 and the histogram will be displayed.
Note:  If you want to select your own classes do the following 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.  Further, you may want to set Xmax to a value slightly above the
last class boundary.
2)  Press GRAPH and the histogram will be displayed.

6)  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.

7)  Cumulative Frequency (Ogive) Graph from Grouped Date:
a)  Enter the  class upper boundaries in a list, for example, L3 if you have data in the first two lists.
b)  If you have the frequency in L2 , place the cursor over the list title, L4, and  do the following:
A)   Press 2nd, LIST, cursor to OPS, and press 6.  cumSum(  will be displayed at the bottom of the list screen.
B)  With the cursor after the parenthesis, press 2nd, L2, ) .  You will now have
cumSum(L2) at the bottom of the lists screen.  Press ENTER.
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.

8)  Relative Frequency and Cumulative Relative Frequency Graphs for Grouped Data:

These are done similarly to the as frequency polygon.  After storing the data for the midpoints and frequencies, do the
steps listed for each type graph.
Relative Frequency:

Assume that we want to store the relative frequencies in list L5 and the upper limits on the classes are in L3 .

a)  First place the cursor to highlight the list title, L5.  Press 2ND, L3, ÷, 2nd, LIST, move the cursor to MATH and press 5.  You
should now have L3/sum( displayed on the bottom of the list screen.
b)  Press 2ND, L3, ), ENTER.

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 L5 opposite Ylist.
e)  Press ZOOM, 9 and the graph will be displayed.
Cumulative Relative Frequency:
Assume that we want to store the cumulative relative frequencies in list L6 and that the relative frequency
is still stored in L5 from the above relative frequency operation above, and that the class upper boundaries are in L3.

a)  First place the cursor to highlight the list title, L6.  Press 2ND, LIST, move the cursor to OPS and press 6.  You should now
have cumsum( at the bottom of the screen.
b)  Press 2ND, L5, ), ENTER. The cumulative relative frequencies will now be stored in L6.

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 L6 opposite Ylist.
e)  Press ZOOM, 9 and the graph will be displayed.

9)  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)  Highlight L3 at the top of the lists on the LIST screen.
c)  Press 2nd, LIST, move the cursor to OPS, and press 6 to paste cumSum( to the bottom of the list screen.
d)  Press 2nd, L2 , ), ÷ .  You now should have cumSum(L2)/ at the bottom of the list screen.
e)  Press 2nd, LIST, cursor to MATH and press 5 to paste sum( to the list screen.
f)  Press 2nd, L2, ).  You now should have cumSum(L2)/Sum(L2) at the bottom of the list screen.
g)  Press x (the multiply symbol), 100.  You now should have cumSum(L2)/Sum(L2) *100 at the bottom
of the list screen.
h)  Press ENTER and the data will be stored in L3 .
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.

10) Box and Whisker Plot
NB:  Users should be aware that there is no standard way of finding quartiles.  The TI uses the Moore and
McCabe (M-and-M) method.  Minitab and Excel use different methods. If you get different answers
from those in your book, check to see if your book is using a method other than M-and-M.
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 5, and
press [ENTER] to highlight the box plot icon.  (See the note at the end of this topic for when to
use icon 4.)
f)  Enter the list you put the data in, usually L1, in the Xlist, by pressing 2nd, L1. or whatever list
you chose.  Make sure the number 1 is opposite 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.
NOTE:  If you have one or two outliers (numbers much larger than the rest) you may want to use
icon 4.  This will not include the outliers in the last whisker, but will plot them as separate points
after the end of the last whisker.

11) 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.

12)  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 the frequency.  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)  Press STAT, cursor to CALC and press ENTER.  1-Var Stats will be displayed on the home
screen.
c)  Press 2nd, L1, press the comma,  then 2nd, L2 .  You should now have 1-Var Stats L1, L2 on the
home screen.
d)  Press ENTER and the values for the mean (expected value), standard deviation and other
statistics will be displayed.
e)  If you need the variance, merely re-enter the value for the  standard deviation, σx  , and square it
Note:  If you round off the standard deviation, you may have a slightly different answer than you would if
you had calculated the variance separately by hand.  To avoid that, enter all decimal places for
σx and
square that value.  If you don't like entering long numbers, you can do this:  Press VARS, 5, 3, ENTER, x2 ,
ENTER.

13)  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)  From the list screen, highlight the title of L3 and press 2ND, L1, x, 2ND, L2..  You will now have L1*L2
at the bottom left 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.
CAUTION:  DO NOT store sums in the lists if the particular list is going to be used in a succeeding arithmetic
operation.  Instead, do these calculations from the home screen.
(1)  Move the cursor down to the first blank space in L3.
Press 2nd, LIST; cursor to MATH, and press 5.
The expression sum( will be displayed at the bottom of  the list screen.
(2)  Press 2ND, L3 , ).You will have sum(L3) at the bottom of the list screen.
(3)  Press ENTER and the sum of those values will be displayed as the last item of L3
You can obtain the variance and standard deviation by first solving for the variance using the
formula:  Σx2 P(x) - µ2
where µ is the mean obtained as above.  To obtain the individual values of the first term,  x2 P(x), and store them in list L4,
do the following:
a)  From the list screen, place the cursor on the title for list L4 , press 2ND, L1, x2, ,x, 2ND, L2.  You will
have L12*L2 at the bottom left of the lists.

b)  Press ENTER and the individual values will be entered in list L4.
c)  To get the sum of these values do the following:
(1)  Caution:  Do not store sums in lists if you plan to use this list in another arithmetic operation.  Place the cursor
in the first blank space in L4, then p
ress 2nd, LIST; cursor to MATH, and press 5.  The expression
sum( will be displayed at the bottom left of the LIST screen.
(2)  Press 2ND, L4, )..  You will have sum(L4) at the bottom of the list screen.
(3)  Press ENTER and the sum of those values in L4 will be displayed as the last entry in L4.

d)  Now we want to subtract the value for µ2 from the last value obtained and that will be the variance.  You can always do that
by hand but if you want to be a little more creative, do it this way.  First press 2ND, QUIT to go to the home screen.
Suppose that your sum for L3, µ, and L4, Σx2 P(x), are in rows 6.   Press 2ND, L4, (, 6, ), -, 2ND, L3, (, 6, ).  You should now have this:
L4,(6 ) -L3,( 6).  Press ENTER and the variance will be displayed.
e)  To calculate the standard deviation from the variance in the list assuming that the variance is in L3(7),  move the cursor down one
space to L3(8)and
press 2ND, √, 2nd, L3,(,7,) and press ENTER.  The standard deviation will be displayed in L3(8).
f)  Of course if you calculated the standard deviation from the home screen, if you have just calculated the variance, press 2ND, √,
2nd, ANS, ENTER.

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 let the calculator do it all.   Just as information, the total
expression for the variance using this method would be this:  sum(L12*L2) - (sum(L1 *L2))2 .

14)  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 move the cursor to the first blank space at the end of the data.
b)  Press 2nd, LIST and move the cursor to MATH.
c)  Press 7 to paste StdDev( to the bottom of the list  screen.
d) Press 2nd, L1, ), and then press the divide symbol.
e)  Press 2nd, LIST, move the cursor to MATH, and press 3 for mean.
f)  Press 2nd, L1 , close the parentheses and then.  You should now have StdDev(L1)/mean(L1).  Press ENTER to display the
CV as the last number in L1.  If you want CV in percent, multiply this number by 100.
NOTE: If you're going to use this list for other calculations, be sure to delete the CV value before performing any operations.

15)  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)  Enter the data in list L1 as described under Entering Data immediately above.
d) 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 then press ENTER.  The standard
deviation and several other statistics will be displayed.

e)  To calculate  the variance, merely re-enter the value for the  standard deviation, Sx  ,
and square it Note:  If you round off the standard deviation, you may have a slightly different answer than
you would if you had calculated the variance separately by hand.  To avoid that, enter all decimal places for Sx and
square that value.  If you don't like entering long numbers, you can do this:  Press VARS, 5, 3, ENTER, x2 ,
ENTER.

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 themselves to learn the details of the
process.  The following gives 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)  in (L1) x2 in    (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 Σx²=7492

The variance computation formula is as follows:   s2 = [(Σx² -(Σx)²)/n)]/(n-1), where s2 is the variance .
So,  we will need x2 ,  ∑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 and Sx = 11.73923.
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 show the detailed calculations, skip to filling in the formula in step “f.”  Otherwise, continue
with the next step.

d)  Now we’ll need an x2 column.   Place the cursor on the title L2, press 2nd, L1, x2, ENTER.  The squares of the
numbers in L1 will be displayed in L2.  You can enter into your table  the numbers that you found for n, ∑x, and
∑x² from the 1-Var Stats function.

e)  Now, we want to use the numbers that we previously recorded to plug into the variance
formula.   So, from the home screen enter
(7492-2162/7)/(6).  You can either merely enter these numbers or your
worksheet or test sheet and square the standard deviation you found above and enter for the answer, or you can
do more time and work to enter the numbers in your calculator and find the variance.
f)  If you entered the numbers in the calculator, press ENTER and you should have 137.8…, which is the variance.
g)  To find  the standard deviation, press 2ND, √ , 2ND, Ans, ENTER,  or you can just record the standard deviation that
and you recorded above.  In either case, you will have 11.73...

16.  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 terms.  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 85-95 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) Place the cursor on the title of L1; then press (, 2ND, L1, + 2ND, L2,), ÷,
2 .  You should have (L1 + L2)/2
at the bottom left of the tables.  Press
ENTER and the midpoints will be stored in L1.

d)  Enter the frequencies in L2 as described under Entering Data immediately above.
Now we’ll 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.
f)  Press ENTER, and the standard deviation along with several other statistics will be
displayed.  The sample standard deviation is 14.868….
g)  To find the variance, just square the standard deviation by entering the number, 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 data 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 =  Σx²f=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  and store them in L1 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) Place the cursor on the title of L1; then press (, 2ND, L1, + 2ND, L2,), ÷,
2 .  You should have (L1 + L2)/2
at the bottom left of the tables.  Press
ENTER and the midpoints will be stored in L1.
Now let’s calculate the required numbers.
b) 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
c)  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.  You’ll 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 from the Lists screen. .
d)  Calculate xf and store it in
L3 by placing the cursor over the title for L3 pressing  2ND, L1, *, 2ND, L2..
You should have L1*L3 at the bottom left of the tables.  Press ENTER and the products will be stored
in list L3.
e)  Calculate x2f by placing the cursor on the title for L4  and  pressing 2ND, L1, x2 , * , 2ND,  L2
You should now have L12 *L2  at the bottom left of the tables.
f)  Press ENTER and the results will be stored in list L4.
g)
You don’t need to calculate Σf.  That is the value for “n” that you previously recorded.

h)   You don’t need to calculate Σxf.  That is the value for ∑ x that you previously recorded.
i)  At this point you can either just record the formula with the numbers plugged in on your work sheet
or test sheet or you can do the extra work to do the calculation with your calculator.  To get the
answer without putting the numbers in your calculator, merely square the standard deviation, which
you previously recorded. If you're going to do it with the calculator, do the next steps.
Now, you want to plug the appropriate numbers into the formula for the variance. From the
home screen enter
(296600-3780²/50)/(49)
j)  Press ENTER and you should have 221.06, which is the variance.
k)  If you want the standard deviation, you can just use the one you previously recorded or you can calculate
calculate by pressing 2ND, √ , 2ND, Ans, ENTER, and you will have 14.868...
Note that if you calculated the standard deviation first, just square that value to get the variance.

17)  Weighted Average:

Suppose you have some scores with the weights indicated:
Score      Weight

83              .3
85              .3
85              .5
89              .3
90              .7
a)  Press STAT, ENTER and enter the scores in list L1 and the weights in L2.
b) Press STAT, move the cursor to CALC, and press ENTER to paste "1-Var Stats"  to the home screen.
c)  If your scores and weights are in lists as indicated above, press ENTER and the weighted average will be
given as x-bar (x with a bar over it.).  If your data are in other lists, enter those lists separated by a comma and
press ENTER.

18)  Median of Grouped data:

Consider the following table.

 Age 5-14 15-24 25-34 35-44 45-54 Midpoint 9.5 19.5 29.5 39.5 49.5 Freq (People) 750 2005 1950 195 100

a)  To find the median class, enter the midpoints in L1 and the frequencies in L2 .
b)  Press STAT, move the cursor to highlight CALC and press ENTER.  1-Var Stats will be displayed on the home screen.
c)  If the data are in lists L1 and L2 just press ENTER.  If they are in other lists, you must enter the lists.  For example,
press 2ND, L2, comma, 2ND, L3 and then press ENTER
d)  Press ENTER and scroll down to Med=19.5.  That is the median of the class 15-
24. So 15-24 is the median class.
e)  Enter the appropriate data into the following formula:
Median = L + I *(N/2 - F)/f
Where
L = lower boundary of the interval containing the median.
I = width of the interval containing the median.
N = total number of respondents.
F = cumulative frequency of those below the median class.
f = number of cases in the median class.
f)  When you are finished entering, you should have this:
14.5+10(5000/2-750)/2005
g)  Press ENTER and you should get 23.228… Notice that the answer is different form
the value of 19.5 given by the calculator. That value of 19.5 was chosen by merely finding the  midpoint
of the median class.

19) 3. Geometric Mean:
Let's do the geometric mean of the S&P 500 as listed in the table at the beginning below.

 FUND OR BENCHMARK YEARLY RETUNS S& P 500 10.88, 4.91, 15.79, 5.49, -37.00, 26.46, 15.06, 2.11, 16.00, 32.39,

a) Press STAT, ENTER and enter the returns data in list L1 or whatever list is convenient.
b) Place the cursor on the list name of list L2 or whatever list you choose, and enter 2nd, L1/100 +1 and press ENTER.
Now, we're going to use the formula (products of L2)1/k , where "k" is the number of return values. In out case that's
10.
c) Press 2nd, STAT, move the cursor to MATH on the menu that appears and press 6 for prod(.
d) Press 2nd, L2 so that you have prod(L2 on the display. Press ENTER.
e) Now enter the following: (Ans)1/10 -1 . You may need to enter the entire exponent in parentheses if you have
an older calculator. If you did the S&P returns, you should get 7.4.

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.  It is not absolutely necessary to sort your data, but the TRACE will sometimes operate
in a confusing manner without sorting.  So, I recommend sorting.  To sort, press[STAT], select 2,
SortA( for ascending order.  SortA(  will be posted to the home screen.  Press
[2nd], L1, 2nd, L2, [ENTER].
d)  Press [2nd], [STAT PLOT] and press [ENTER] to turn Plot 1 on.
e)  Move the cursor to the scatter diagram, the first icon opposite Type,  and press [ENTER] to highlight the
scatter diagram icon.
f)  Enter L1 in the Xlist, and L2 in the 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.  You can use TRACE and the arrows
to move along and read the data pairs.

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 <8>, [ENTER] (Where <8> is just the number 8 from the keyboard.)
Note that if you want to use QuadReg or some other analysis, press the number to the left of that
entry. LinReg (a+bx) will appear on the screen if you chose 8.
d) If you want to graph the equation of the best-fit line, ship to item “e” below. If you don't want to graph,
continue with these instructions.   If you have your data in the L1 and L2 as described above, just press
ENTER.  If  you have your data in other lists, you’ll  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, r2, and r will
be displayed.  Note that if r and r2  are not displayed, press 2nd , CATALOG, D, ; then scroll to DiagnosticON
and press ENTER. ANSWER:  If you pressed ENTER you should have these values:  a=-186.47.., b=4.705…,
r2 =.63366…, and r=.7979…
e)  If you want to graph the equation,  then immediately after LinReg(a+bx),  enter the lists separated by
commas if the lists are not in L1 and L2.  If the numbers are in
L1 and L2, you need not enter the list names.
f)  Now, you want to store this as a Y-variable, say, Y1. So, do it this way:  Press [VARS], Cursor to
Y-VARS, [ENTER], [ENTER].  You should now have this on your screen.  LinReg Y1. (If the numbers
are in other lists, the lists followed by commas are also included
g)  Press [ENTER].   After a few seconds a long equation with coefficients having several decimal
places will appear on the screen.
h)  To graph that, you could just press GRAPH.  Depending on your data values, you may need to
adjust the WINDOW.  Pressing ZOOM, 0 (zero), for ZoomFit will get you a preliminary window setting.
i)  Note that if you have already done the regression equation without storing it in a Y-variable, you
can do that as follows:
1)  Press Y=; then VARS; then 5 (Statistics).
2)  Cursor over to EQ and press 1 (or ENTER). The regression equation will be stored in the Y1=
position.  You can then  graph as indicated previously.

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) Enter L1 in the Xlist, and L2 in the Ylist; then select the type marker you prefer. (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:

We will use calculator functions to reduce the arithmetic necessary for these formulas.  First we will
use "2-Var Stats" to obtain the values for such expressions as
ΣxΣy and Σx² to enter on our worksheet
or test sheet.   Then we will use LinReg(a+bx
to find the values for "r" and "r²."
This last procedure will eliminate the necessity for entering the numbers for the formulas into our calculators

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-83 Plus or
TI-84,  a square root expression must be enclosed in parentheses.  Example:  √(nΣx2- (Σx)2).
Now let’s plug the numbers into the equation for r:
At this point you will save yourself a lot of time if you calculate r and r² with the calculator.  To do that,
press STAT, move the cursor to CALC and press 8 for LinReg(a+bx.  Press ENTER if the lists are in L1 and
L2.  If they are not in those lists, you will need to enter the lists separated by a comma.  Press ENTER and
r and r² will be displayed along with other statistics.  If you are required to show your work, you will need
to write the numbers and on your paper.   Record the following:
d)  r=
(nΣxy –ΣxΣy)/[(√(nΣx2- (Σx)2)(√(nΣy2- (Σy)2)]
= (8*62750-521*960)/(√(8*33979-5212)(√(8*116900-9602))
=.7979…..
If you chose to put the numbers in the calculator, you might want to read the following:
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.
g)  Of course, r² is just the square of Ans, or you can just copy if from the stats calculation.

6)  Finding the Values a and b for the Best-Fit Equation Using a Computation Formula:

We will use calculator functions to reduce the arithmetic necessary for these formulas.  First we will
use "2-Var Stats" to obtain the values for such expressions as
ΣxΣy and Σx² which we can enter in the
formula on our worksheet or test sheet.  Then
, we will use the calculator function LinReg(a+bx to find
the values for "a" and "b" without doing the arithmetic on our calculators.  Finally, for those who are
allowed to use a simper method than the arithmetic intensive formula, I will suggest the use of matrices
and the calculator function rref( for finding the final answers. intensive substitution method used in many textbooks.

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 13

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 the following
parameters:  x¯=65.125, Σx=521,  Σx2=33979,
n=8, Σy=960, y-bar=120, Σy2=116900, Σxy=62750

At this point you will save yourself a lot of time if you calculate "a" and" b" with the calculator.  To do that,
press STAT, move the cursor to CALC and press 8 for LinReg(a+bx.  Press ENTER if the lists are in L1 and
L2.  If they are not in those lists, you will need to enter the lists separated by a comma.  Press ENTER and
"a" and "b" will be displayed along with other statistics.  If you are required to show your work, you will need
to write the numbers and on your paper.

c)  Plug these numbers into the formula and then enter the expression in 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-83 Plus or TI-84,  a
square root expression must be enclosed in parentheses.  Example:  √(nΣx2- (Σx)2)
d)  Enter the following  formula on your worksheet,  or in the calculator  if you're going to calculate the values.:
b=
(nΣxy –ΣxΣy)/(nΣx2- (Σx)2).
Now record these numbers on your worksheet
or in the calculator  if you're going to calculate the values.:
=(8*62750-521*960)/(8*33979-5212)
=4.7058…..
e)  Now, record this formula on your worksheet:
a= y-bar –b(x-bar)
Now just enter these numbers on your worksheet or enter them in your calculator if you're going to
calculate the value.
=120-4.7058 *65.125
=-186.465…
Using a Matrix to Solve for "a" and "b."

a. After you have set up the matrix as described in Method I, go to the home screen by pressing 2nd, QUIT.
b. To enter the value 6, for “n,” in the first element of the matrix press 6, STO, 2nd, MATRIX, ENTER, (1, 1). You
should have 6
à[A](1,1) on the screen. Press ENTER to transfer the value to the matrix.
c. Now, we want to do the summations. First press 2nd, ENTER to display the expression above once more. Place the
cursor on the value 6 and press VARS, 5. Move the cursor to ∑ and press ENTER.. Move the cursor to the element designation and change it to 1,2. You should now have ∑x
à[A](1,2). Press ENTER to display the value and transfer it to the matrix.
d. Press 2nd, ENTER to display the expression again and change the matrix element to 2,1. You should now have ∑x
à[A](2,1). Press ENTER to display the value and transfer it to the matrix.
e. Now, we want to transfer ∑y in matrix element 1,3. First press 2nd, ENTER to display the expression above once more.
Edit the expression using VARS, 5; move the cursor to ∑and press 3. Change the matrix element so that you have ∑y
à[A](1,3). Press ENTER to transfer the number to the
matrix.
f. Now, we want to transfer ∑x2 into matrix element 2,2. Display the expression once again and edit it using the listings under VARS, Statistics to get ∑x2
à[A](2,2). Press ENTER to transfer the information.  Using the same technique, First press 2nd, ENTER to display the expression above once more.
g. In a similar manner, transfer the ∑xy to element 2,3.  You should have ∑xy
à[A](2,3).
h. Now we want to solve the matrix. Press 2nd, MATRIX, move the cursor, to
MATH and scroll down to rref(.
That’s usually item “B.” Press ENTER to display rref( on the home screen.
i. Now you want to tell the calculator which matrix you want to solve. To do that, press 2nd, MATRIX, ENTER if
you have the data in [A].
j. Finally, press ENTER to display the answer matrix. It should be the following

┌ 1 0 .4ךּ
0 1 1 ﻠ
Sorry, my effort at making matrix symbols leaves a lot to be desired.

7.  Testing the Correlation Coefficient:
Suppose that we have the data given in the table below and we want to test the correlation coefficient at a significance level
of 1%.  Further suppose we believe that the correlation coefficient is positive. The null hypothesis is that the correlation is 0

 1 2 3 4 5 6 x 9.9 11.4 8.1 14.7 8.5 12.6 y 37.1 43 33.4 47.1 26.5 40.2

a)  If the data are not already in the lists, press STAT, ENTER and enter the x-values in list L1 and the
y-values in list L2.
b)  Press STAT, move the cursor to TESTS , and scroll down to LinRegTTest.  Press ENTER.
c)  On the screen that appears, move the cursor down to >0 and press ENTER.  If REGEQ has an entry, delete that.
d)  Move the cursor to Calculate and press ENTER.
e)  Among the items displayed is P=.009218...which is less that 0.01.  So, we reject the null hypothesis
and conclude that the correlation coefficient is positive.

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.  If you want to sort data in an independent and dependent list, L1 and L2,

for example, use SortA(L1,L2).

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, first press STATS, ENTER and highlight the list name L3.
Now, press 2nd,  L1, x (Multiply symbol),  2nd, L2.  Then press ENTER.
c)  Obviously, x-y can be obtained by merely substituting the subtraction symbol for the
multiplication symbol in the steps 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, first press STATS, ENTER and highlight the list name L3.
Now,  press 2nd,  L1, x² (the square symbol; 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, highlight the list name where you
want the data stored; then enter the operation as described above.  Finally, press 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, highlight the list name where you
want the data stored; then enter the operation as described above.  Finally, press 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.

4.  Randomly generated data sets:
Sometimes problems use a randomly generated set of data. Suppose we want to generate 10
random numbers between 1 and 50 and store them in List 1.  The proper syntax is randint(lower,
upper, how many).
That can be obtained as follows:
a)  Press MATH, cursor over to PRB and press the number 5. randint( will appear on the screen.
b)  Enter 1, 50, 10, so that your screen displays randint(1,50,10).  Press ENTER
c)  Now, if you want to cause these numbers to be stored in L1, before pressing ENTER in item b,
press STO;2nd, L1. The  entries, randint(1,50,10)->L1, will appear on the screen.
d)  Press ENTER and the numbers generated will appear on the screen and will be stored in list L1.
e)  Alternately, you can go to the lists, hightlight the name where you want the numbers stored and then enter
the randint(1,50,10) as described above.

VI.  Normal Distribution:
Note:
In this section, a general method will be outlined; then a specific example will be worked.  The same
problem will be used in several of the examples.

General, normalcdf(:  This function returns the value of the area between two values of the random variable
"x."  This can be interpreted as the  probability that a randomly selected variable will fall within those two
values of "x," or as a percentage of the x-values that will lie within that range.  The syntax for this function is
normalcdf( lower bound, upper bound, μ, σ.  If the mean and standard deviation are not given, then the
calculation defaults to the standard normal curve with a mean of 0 and a standard deviation of 1. I use the
values -1E9 and 1E9 for left or right tails.  The E in obtained by pressing 2nd, EE.  This can be used to solve
such problems as the following: P(x<90), P(x>100), or P(90<x<120).  If
µ and σ are omitted, the default
distribution allows the solution of the following: P(z<a), P(z>a), or
P(a<z<b).

1.  normalcdf(: Area under a curve between two points with μ (mean) and σ (std. dev.) given.

a)  Press 2nd, DISTR, 2.  The term "normalcdf(" will appear on the home screen.
b)  Enter the number for the left boundary, right boundary, μ, and σ in that order.  You do not need
to close the parentheses, but it's okay if you do.
c)  Press ENTER and the value of the area between the two points will be displayed. Notice that
you do not explicitly convert the points to z-values as in the hand method.
Ex. 1:
Assume a normal distribution of values for which the mean is 70 and the std. dev. is 4.5.
Find the probability that a value is between 65 and 80, inclusive.
a)  Complete item a) above.
b)   Enter numbers so that your display is the following:  normalcdf(65,80,70,4.5.
c)  Press ENTER and you'll get 0.85361 which is, of course, 85.361 percent.
1A. normalcdf from Lists:

Frist calculate the mean and standard deviation as follows:
a) Enter the data in lists, say L1 and the frequence in L2.

b) Press STAT, move the cursor to 1-Var Stats and press ENTER.
c) The following will be displayed. (Note that the applicable lists have already been entered.)
List: L1
Freq List: L2

Calculate
d) If the lists haven't already been entered, enter them by pressing 2nd and the proper list key. Now,
move the cursor to calculate and press ENTER. The statistics will be displayed.
e) Begin the calculation of the probability by pressing 2nd, DISTR, 2, ENTER to go to the format screen.
That screen with data already entered will be as follows:
lower:160.5
upper: 175.5
mean: x-- (This is x-bar)

σ: σx
Paste
Note: To enter the mean and standard deviation, position the cursor opposite the appropriate item
and press VARS, 5. In the first column, select the mean and press ENTER, Next go through the same
steps to select either the population or sample standard deviation as appropriate.
f) Now move the cursor to highlight paste and press ENTER. The expression normalcdf(160.5, 175.5, x-- , σx)
will be displayed. Note that this is for the TI-84. If you are doing the TI-83 Plus, just enter the expression
normalcdf(160.5, 175.5, x-- , σx). In either case, press ENTER and  the probability will be displayed.

2.  normalcdf(: Area under a curve to the left of a point with μ (mean) and σ (std. dev.) given.
Ex. 2:  In the above problem, determine the probability that the value is less than 62.
a)  Complete item a) in the general method above.
b)   Enter numbers so that your display is the following:  normalcdf(-1E9, 62,70,4.5.  Notice that
the "-" is a negative sign, not a minus sign.  Enter "E" by pressing 2nd, EE (The comma
key.)
c)  Press ENTER and you'll get 0.03772 which is, of course, 3.772 per percent.

3.  normalcdf(: Area under a curve to the right of a point with μ (mean) and σ (std. dev.) given.
Ex. 3:  In the above problem, determine the probability that a value is greater than or equal to 75.
a)  Complete item a) in the general method above.
b)   Enter numbers so that your display is the following:  normalcdf(75, 1E9,70,4.5.
Enter "E" by pressing 2nd, EE (The comma key.)
c)  Press ENTER and you'll get 0.13326 which is, of course, 13.326 per percent.

4.  normalcdf(:  Sample taken from a normal distrubution:

Suppose a sample of 35 is taken from the population above (μ=70 and σ=4.5).  What is the probability
that the mean is greater than 72?
a)  Complete item a) in the general method above.
b)   Enter numbers so that your display is the following:  normalcdf(72, 1E9,70,4.5/
√(35)
Enter "E" by pressing 2nd, EE (The comma key.)
c)  Press ENTER and you'll get .00427... which is, of course, 0.427 per percent.

5.   ShadeNorm(:  Displaying a graph of the area under the normal curve.
General:
This function draws the normal density function specified by
µ and σ and shades the area
between the upper and lower bounds.  This is essentially a graph of normalcdf(.  It will display the
area and upper and lower bounds.  Not including
µ and σ defaults to a normal curve.  The following
instructions, "a" through "c," are general instruction  to follow.

a)  First turn off any Y= functions that may be active.  Do this by moving the cursor to a
highlighted = sign and pressing ENTER.
b)  Press 2nd, DISTR and cursor over to DRAW.  Press 1 and ShadeNorm( will appear on the
home screen.  Enter the correct parameters depending on whether the problem is like 1, 2,
or 3 above.
c)  Press ENTER, and the graph may be visible on the screen.  You will almost certainly need
to reset the Window parameters by pressing WINDOW and changing Xmin, Xmax, Ymin, and
Ymax settings to get a decent display. As a first approximation, set Xmin at 5 standard
deviations below the mean and Xmax at 5 above the mean. (See the following example.)  Start out with
a Ymax about   0.3 and go from there.  You can set the Ymin at 0, or if you wish, set it at about
negative one-fiftieth of Ymax.  You may need to fine tune from there.
Ex 1:  Draw the graph of example 2 above.
a)  Press WINDOW and set Xmin=50, Xmax=90, ymin=-.01, Ymax = 0.1.  You can reset the
scales as you choose to eliminate the broad baseline.
b)  Press 2nd, DISTR and cursor over to DRAW.  Press 1 and ShadeNorm( will appear on the
home screen.
c)  Enter parameters so that your display looks like this:  ShadeNorm(-1E9, 62, 70, 4.5.
d)  Press ENTER and a reasonable looking graph should appear on the screen.
Note that if you reset the window, you may need to activate the expression again.  To do that, press
2nd, ENTER, ENTER.

6.  invNorm(:  Inverse Probability Calculation:
Find the number x, in a normal distribution such that a number is less than x with a given
probability. The syntax for this is invNorm(area, [μ, σ]).  The part in brackets indicates that there
is a default for those values.  The default is the standard curve with mean=0 and standard deviation is  1.
Ex. 1:   In Ex. 1 immediately above, find the number x, such that a randomly selected number will be below
that number with a 90% probability.
a)  Press 2nd, DISTR, 3 to select invNorm(.
b)  Enter parameters so that your display looks like this: invNormal(.90,70,4.5.
Ex. 2:  Given a normal distribution with a mean of 100 and standard deviation of 20.  Find a value Xo such
that the given x-value is below Xo is .6523.  That is P(X<Xo) = .6523.
a)  Press 2nd, DISTR, 3 to place "invNORM(" on the home screen.
b)  Enter information so that the entry looks like the following:  invNORM(.6523,100, 20.
Ex. 3:  What is the lowest score possible to be in the upper 10% of the class if the mean is 70 and the
standard deviation is 12?
a)  Press 2nd, DISTR, 3. to place "invNORM(" on the home screen.
b)  Enter information so that the entry looks like the following:  invNORM(1-.1,70, 12.  Your answer will
be 85.38 or 86 rounded off.

General:
If you are accustomed to graphing using the standard WINDOW settings called by
ZOOM, 6, then you're in for a big surprise if you use those settings for graphing the normal
curve.  So, before you display the ShadeNorm( function, press WINDOW and set the values
as follows:

a)  Xmin =
μ - 4σ. Round of to the next integer.
b)  Add the same number to the mean that you subtracted from the Xmin to get Xmax.
c)  Xscl= Set at the standard deviation.
d)  Ymin=0.  Some people like to set this at a small negative number, but if you have
problems with a wide range of std. devs. you'll have to keep changing it.  I set it at 0; then
I'm done with it.
e)  Ymax= As a first approximation, set this at 0.4/σ.
f)  Yscl= Most of the time the y-axis is not displayed, so I usually just set it at 0.01 and
leave it there.

Ex. 1:
Obviously if you wanted to graph the example immediately above, you could use the
ShadeNorm( using the lower bound of -1E9 and the upper bound of 75.766.  You would do that
as follows:

a)  Press WINDOW and set Xmin=50, Xmax=90, ymin=-.005, Ymax = 0.1.  You can reset the
scales as you choose to eliminate the broad baseline.
b)  Press 2nd, DISTR and cursor over to DRAW.  Press 1 and ShadeNorm( will appear on the
home screen.
c)  Enter parameters so that your display looks like this:  ShadeNorm(-1E9, 75.766, 70, 4.5.
d)  Press ENTER and a reasonable looking graph should appear on the screen.
Note that if you wanted to shade the region where the probability would be greater than 90%,
you would choose 75.766 for the lower boundary and 1E9 as the upper bound.

Ex. 2:
Suppose you wanted to graph a distribution and shade the area between the points 40 and 54,
with a mean of  46 and a std. dev. of 8.5

a)  Press WINDOW and set Xmin=12, Xmax=80, Ymin=-.005, Ymax = 0.06.  You can reset the
scales as you choose to eliminate the broad baseline.
b)  Press 2nd, DISTR and cursor over to DRAW.  Press 1 and ShadeNorm( will appear on the
home screen.
c)  Enter parameters so that your display looks like this:  ShadeNorm(40, 54, 46, 8.5.
d)  Press ENTER and a reasonable looking graph should appear on the screen.  The area
under the curve, 0.56562, will be displayed on the screen along with the upper and lower
bounds.

9.   normalpdf(:  Probability Distribution Function using normalpdf( :
General:
This function is used to find the fraction, and therefore also the percentage, of the
distribution that corresponds to a particular value of x.  The syntax of this function is
normalpdf(X, μ, σ
A) Finding the Percentage of a Single Value:
Ex. 1:
Suppose that the mean of a certain distribution is 60 and the standard deviation is 12.
What percentage of the population will have the value 50?
a)  Press 2nd, DISTR, 1 to paste normalpdf( to the home screen.
b)  Enter data so that your display is as follows:  normalpdf(50,60,12.

B)  Graphing the distribution:

Ex. 1:  Suppose that the mean of a certain distribution is 60 and the standard deviation is 12.
Investigate percentages for several x-values.
a)  First press WINDOW and set Xmin 12 (mean minus 4 std. dev.).  Set Xmax at the same
number of units above the mean, i.e., 108.
b)  Press Y= and select the Y1= position; then press 2nd, DISTR, 1 to paste normalpdf( to
the Y1= position.
c)  Enter data so that the entry after Y1= looks line this:  normalpdf(X, 60,12.
d)  Press ZOOM, 0 to select ZoomFit and the curve should appear on the screen.
e)  Press TRACE and you can move along the curve and read the values for different x-
values.  If you want a specific value, perhaps to get rid of the x-value decimals, just enter
that number and press ENTER.

10. ZInterval:  This gives the range within which the population mean is expected to fall with a desired
confidence level.  The sample size should be > 30 if the population standard devation is not
known.
Ex. 1:  Suppose we have a sample of 90 with sample mean x¯  = 15.58 and
σ = 4.61.  What is the 95%
confidence level interval?
a)  Press STAT, cursor to TESTS, and press 7.
b)  On the screen that appears, cursor to "Stats" on the ZInterval screen and press ENTER.
c)  Enter data opposite positions as follows:
σ: 4.61, x¯ :15.58, n:90, and C-Level: .95.
d)  Cursor down to Calculate, press ENTER, and the interval (14.628, 16.532) will appear along with
the values for "n" and the mean.
Ex. 2:  Suppose that you have a set of 35 temperature measurements with population
σ = .5 and you want
to know with a 95% confidence level within what limits the population mean of temperature measurement
will fall.
a)  First you need to enter the data in a list, say L1, by pressing STAT, ENTER, and entering your data
in the list that appears.  Just enter a data point and press either ENTER or the down arrow.
b)  Press STAT, move the cursor to TEST and press 7 to get the ZInterval screen.
c)  Cursor to "Data" and press ENTER.
d)  First enter the value for
σ opposite σ: on the calculator.
e)  Next you need to have the sample mean. That is obtained by entering the list designation where the data
is stored opposite List. Press 2nd, L1 , or whatever list you have your data in.
f)  Enter additional information as follows:  Freq: 1, C-Level: .95.
g)
Move the cursor to Calculate and press ENTER.  The same type data will be displayed as in Ex. 1 above.

VII. Other Distributions and Calculations:

1. TInterval:  If the sample size is <30, then the sample mean cannot be used for the population mean,  and
the ZInterval cannot be used.  However, if the distribution is essentially normal, i.e., known to be normal
form other sources or has only one mode and is essentially symmetrical, then the Student t Distribution
can be used.
Ex. 1:  Suppose you take ten temperature measurements with sample mean x¯  = 98.44 and s = .3.
What is the 95% confidence level interval?
a)  Press STAT, cursor to TESTS, and press 8.
b)  On the screen that appears, cursor to "Stats" and press ENTER.
c)  Enter data opposite positions as follows:  x¯ :98.44, S x : .3   n:10, and C-Level: .95.
d)  Cursor down to "Calculate", press ENTER, and, after a few seconds, the interval (98.225, 98.655)
will appear along with the values for  "n" and the mean.
Ex. 2:  Suppose that you have a set of 10 temperature measurements and you want to know with a 95%
confidence level what limits the population mean of temperature measurement will fall within.
a)  First you need to enter the data in a list, say L1, by pressing STAT, ENTER, and entering your data
in the list that appears.  Just enter a data point and press either ENTER or the down arrow.
b)  Press STAT, cursor to "TEST" and press 8 to get the TInterval screen.
c)  Cursor to "Data" on the TInterval screen and press ENTER.
d)  Enter information as follows:  List: Press 2nd, L1, Freq: 1, C-Level: .95.
e)  Cursor to "Calculate" and press ENTER.  After a few seconds, the interval (xx.xxx, xx.xx)
will appear along with the values for  "n," the mean, and sample standard deviation.

2.  Student's t Distribution:  The Student's t Distribution is applied similar to the normal probability function, but it
can be applied when there are less than 30 data points, for example: P(t> 1.4|df = 19).  The last part means
that the number of degrees of freedom ( one less that the number of data points) is 19.
Ex. 1:  Find the probability that t> 1.4 give that you have 20 data points.
a)  Press 2nd, DISTR,  5, (6 on a TI-84) to paste tcdf( to the home screen.
b)  Enter data so that your display is as follows:  tcdf(1.4, 1E9,19.

3.  invT: Finding a t-value Given α and df:
If you are working a problem using the t-value, there are different options depending on your needs and whether
you're using a TI-83 Plus or a TI-84 Silver Edition.
TI-84 Plus Silver Edition:  This calculator has an invT, so do the following:
(1)  Press 2nd, DISTR, 4, and invT( will be pasted to the screen.
(2)  Enter
α or 1-α, depending on whether you have a left or right tail; then enter the degrees of freedom, df.
(3)  Press ENTER and the value for "t" will be displayed.  Note that you may need to divide
α by 2 if you

TI-83 Plus:
This calculator does not have an invT, so you can do either of two procedures:
(1)  Look up the t-value in your book.  This is by far the easier.
(2)  If you have an α that's not in the table or don't have a table, you can do this:
Suppose you want the t-value for α=.1 for a left-tailed test.
(a)  Press MATH, 0, and the solver will be pasted to the screen.
(b)  Press the UP arrow so that the equation is displayed.
(c)  Press 2nd, DISTR, 5 and tcdf( will be pasted in as a formula.
(d)  Enter data so that your entry will look like this: tcdf(-1E9, X, 10) - .100 and press ENTER.
(e)  Press the UP arrow and enter -1 opposite X.
(f)   Press ALPHA, SOLVE, and the value for "t" will be displayed opposite X after about 20 seconds.
Suppose you want the t-value for α=.1 for a right-tailed test.
The steps are exactly the same except for these.
(d)  Enter data so that your entry will look like this: tcdf(-1E9, X, 10) - .900 and press ENTER.
(e)  Press the UP arrow and enter 1 opposite X.
Use a Calculator Program:

There are several program posted on the Web, for example, at www.ticalc.org .  I included a program of
my own at the end of this document.  It may not be the greatest, but it's simple and it works.

4.  The Chi-squared Distribution:  The χ2 Distribution is implemented similar to the Student's t
Distribution.
Ex. 1:  Assume that you want to find P(χ2 > 24|df=20) the same as in the above Student's t Distribution.
a )  Press 2nd, DISTR,  7 (8 for TI-84), to paste χ2cdf( to the home screen.
b)  Enter data so that your display is as follows:  χ2cdf(24, 1E9,19.

5.  Binomial Distribution, binonpdf(:

Suppose that you know that 5% of the bolts coming out of a factory are defective.  You take a sample of 12.
Determine the probability that 4 of them are defective.
a)  Press 2ND, DISTR, move the cursor down to A:binompdf( and press ENTER.
b)  Enter numbers so that your entry is like this: binompdf(12, .05, 4.
c)  Press ENTER and 0.00205 will be displayed.

6.  Binomial Distribution, binoncdf(:

Suppose that you know that 5% of the bolts coming out of a factory are defective.  You take a sample of 12.
Determine the probability that 4 or more of them are defective.
First I'll show a very easy way that gives only the answer; then I'll show a method that takes more time, but
provides much more intermediate results.
Short Way:
a)  Press 1, and then - , the subtraction sign.
b)  Press 2ND, DISTR, move the cursor down to B:binomcdf( ( or alternately press ALPHA, B) and press ENTER.
c)  Enter numbers so that the display looks like this:  binomcdf(12, .05, 3.
d)  Press ENTER and the answer, .0022364 will be displayed.
Longer Way:
a) Press 2ND, DISTR; then move the cursor to A:binompdf( (or press ALPHA, A) and press ENTER.
b)  Enter information so that your display looks like this:  binompdf(12, .05, {4, 5,6,7,8,9,10,11,12}).  Be sure
to use braces rather than parentheses.
c)  Press STO, 2ND, L1 to tell the calculator which list to store the individual values in.
Now, we want to also get the sum of all of these.  Do that as follows:
d)  Press ALPHA, : (the decimal point key); then 2ND, LIST, move the cursor to MATH, and press 5.  The expression
binompdf(12, .05, {4, 5,6,7,8,9,10,11,12}) : sum( should now  be displayed on the home screen.
e)  Press 2ND, L1,.  You should now have this expression:  binompdf(12, .05, {4, 5,6,7,8,9,10,11,12}) : sum( L1).
f)  Press ENTER,  and the answer, .0022364, will be displayed.  If you need the individual numbers,
they are in list L1.  Just press STAT, ENTER to see them.
Ex 2:  Suppose in the above example you want to know the probability of 3 and fewer.
a)  Press 2ND, DISTR, move the cursor down to B:binomcdf( and press ENTER.
b)  Enter numbers so that the display looks like this:  binomcdf(12, .05, 3.
c)  Press ENTER and the answer, .997763... will be displayed.
Ex 3:  Suppose that, on average, one out of ten apples in a fruit stand is unacceptable.  What is the probability that
8, 9, or 10 of a set of 11 such apples are acceptable?
a)  Press 2ND, LIST; move the cursor to MATH and press 5 to paste sum( to the home screen.
b)  Press 2ND, DISTR, ALPHA, A.  You will now have sum(binomialPdf( posted to the home screen.
c)  Enter data so that you have sum(binomialPdf(11, .9, {8,9,10}))  on the home screen.  Be sure to use braces
rather than parentheses enclosing the numbers 8, 9, 10.
d)  Press ENTER and .667...will be displayed.

VIII.  Hypothesis Testing:
1.  Testing for Mean  and z Distribution with Data:
a)  Enter the data into L1 or whatever list you choose.
b)  Press STAT and move the cursor over to TESTS.
c)  Press 1 or ENTER for Z-Test.
d)  Move the cursor to Data and press ENTER.
e)  Opposite
µo, enter the mean for the null hypothesis.
f)  Opposite σ, if you are using the sample standard deviation and it is not given, do the following: Press 2nd,
LIST, move the cursor to MATH and press 7.  stdDev(, will now be displayed opposite σ.  Now, enter you
list number where the dats is stored by pressing 2nd, and the list number, for example L1 .
g)  Enter L1 opposite List and 1 opposite Freq.
h)  Select the proper condition for the alternative hypothesis.
i)  Move the cursor to Calculate and press ENTER.
j)  If you want to use the calculator to find the z-value or critical value, see those procedures below.

2.  Testing for Mean and z Distribution with Statistics:
a)  Press STAT and move the cursor over to TESTS.
b)  Press 1 or ENTER for Z-Test.
c)  Move the cursor to Stats and press ENTER.
d)  Opposite
µo, enter the mean for the null hypothesis.
e)  Enter the given values for σ, x-bar, and n.
f)  Select the proper condition for the alternative hypothesis.
g)  Move the cursor to Calculate and press ENTER.  The z-value, p-value and some other statistics will
be displayed.

3)  Finding a z-vlaue for a particular confidence level:
Suppose you want the z-value for a particular
α, e.g., 5%. Do this:
a)  Press 2nd, DISTR, 3 for invNorm(.
b)  Enter
α for a left-tailed or 1-α for a right-tailed and press ENTER.
c)  The z-value will be displayed.

4)  Finding critical values of x.
Suppose you have a mean of 5.25, standard deviation of .6 and you want the critical number for an
α
of 5%.
a)  Press 2nd, DISTR, 3, and invNorm( will be pasted to the home screen.
b)  Enter numbers so that your entry looks like this:  invNorm(.05, 5.25, .6.  For a left tail, enter the value
for
α and for a right tail enter 1-α..
c) Press ENTER and the inverse will be displayed.

5.  Testing for Mean  and t Distribution with Data:
a)  Enter the data into L1 or whatever list you choose.
b)  Press STAT and move the cursor over to TESTS.
c)  Press 2 for T-Test.
d)  Move the cursor to Data and press ENTER.
e)  Opposite
µo, enter the mean for the null hypothesis.
f)  Enter L1 opposite List and 1 opposite Freq.
g)  Select the proper condition for the alternative hypothesis.
h)  Move the cursor to Calculate and press ENTER.
i)  If you are working a problem using the p-value test, read the p-value and compare it with
α or α-1 as appropriate.
j) If you are working a problem using the t-value test, you will need to know the critical values for the level of
significance,
α, that you have chosen.  There are different options depending on your needs and whether
you're using a TI-83 Plus or a TI-84 Silver Edition. See "
invT: Finding a t-value Given α and df:" in section VII of
this document for the details of these options.

6.  Testing for Mean and T Distribution with Statistics:

a)  Press STAT and move the cursor over to TESTS.
b)  Press 2 or ENTER for T-Test.
c)  Move the cursor to Stat and press ENTER.
d)  Opposite
µo, enter the mean for the null hypothesis.
e)  Enter the given values for σ, x-bar, and n. If you don't know x-bar you can enter it by placing the cursor opposite
the symbol for mean; then press 2nd, LIST, cursor to MATH, and press 3; then press ENTER. Enter L1 and
press ENTER.
h)  Select the proper condition for the alternative hypothesis.
i)  Move the cursor to Calculate and press ENTER.
j)  If you are working a problem using the p-value test,  read the p-value and compare it with
α or α-1 as appropriate.
k) If you are working a problem using the t-value test, you will need to know the critical values for the level of
significance,
α, that you have chosen.  There are different options depending on your needs and whether
you're using a TI-83 Plus or a TI-84 Silver Edition. See "
invT: Finding a t-value Given α and df:" in section VII of
this document
for the details of these options.

Simple Program for Calculating  InverseT:

I have written   a simple program for those who want to find t-values with a calculator.   Because  the TI-83Plus has a fairly slow clock speed,
a solution may take 20 seconds or so.  When you enter the program, you can add more letters to the menu items if you prefer.  I have
abbreviated them  to save memory space in my calculator.  The Program is in the Appendix at the end of this document.

IX.  Statistics of two Populations:

1.  Confidence Interval for Two Dependent Populations:

Enter the data from population 1 into L1 and the data from population 2 into L2.  Do this as follows:

a)  Press STAT, ENTER, and enter the data in the displayed lists.
b) After entering the data, highlight the list name, L3 example, where you want to store the data.
Now, store the paired differences in list L3 as follows:
c)  Press 2nd, L1, minus sign, 2nd, L2.
d)  You should now have L1 - L2 at the bottom  on the lists screen.  Press ENTER and the differences
will be stored in list L3.
Now, find the confidence level as follows:
e)  Press STAT, move the cursor to TESTS, and press 8 for TInterval.
f)   On the screen that appears, move the cursor to "Data" and press ENTER; then enter 1 opposite Freq
and press ENTER.
g)   Enter the confidence level you want opposite C-Level, for example .95.
h)  Move the cursor down to “Calculate” and press ENTER.  The confidence interval and other statistics will be
displayed.

2.  Confidence Interval for Two Dependent Populations (Stats):
If you do not have data, but have the mean, standard deviation, and n, use this procedure.
a)  Press STAT, move the cursor to TESTS, and press 8 for TInterval.
b)   On the screen that appears, move the cursor to "Stats" and press ENTER.
c)  Enter the sample mean, standard deviation, and the number of data points opposite "n.".
d)   Enter the confidence level you want opposite C-Level, for example .95.
f)  Move the cursor down to “Calculate” and press ENTER.  The confidence interval and other statistics will be
displayed.

3.  Confidence Interval for Two Independent Populations (Stats):

a)  Press STAT, move the cursor to TESTS, and press 0 (zero).
b)  On the screen that appears, move the cursor to Stats and press ENTER.
c)  Enter the sample means, standard deviations, and number of data points, n, for each sample.
d)  Set the confidence level you choose opposite "C-Level."
e) Highlight "No" opposite "Pooled" if there are no assumptions about the variations.
f)  Move the cursor to "Calculate" and press ENTER.  The confidence interval along with other statistics will be
displayed.

4.  Confidence Interval for Two Independent Populations (Data):

Enter the data from population 1 into L1 and the data from population 2 into L2.  Do this as follows:

a)  Press STAT, ENTER, and enter the data in the displayed lists.

To go to the confidence interval screen do this:
b)  Press STAT, move the cursor to TESTS, and press 0 (zero).
c)  On the screen that appears, move the cursor to Data and press ENTER.
d)  Opposite "List 1," press 2nd, L1 and opposite "List2," press 2nd, L2.
e)  Set the confidence level you choose opposite "C-Level."
f)   Highlight "No" opposite "Pooled" if  there are no assumptions about the variations.
g)   Move the cursor to "Calculate" and press ENTER.  The confidence interval along with other statistics will be
displayed.

X.  Other Tests and Inferences:

1.  One-way ANOVA:

Suppose that you are trying to determine a better way to motivate learning and come up with the following scores in response
to different types of motivations:

 Group 1 Group 2 Group 3 Group 4 x1 x2 x3 x4 9 2 3 5 8 4 7 3 9 3 9 7 8 5 8 7 7 5 6 6

Determine if  one of the methods is better that the others..
The null hypothesis,  Ho, will be that all of the means are equal.  Suppose that we want to determine if they are by
99 % confidence level.
a)  Press STAT, ENTER, and enter the each group of data in lists L1 through L4. To clear old data from a list, place the cursor on
the list title, press CLEAR; then ENTER.  DO NOT press DEL to clear a list.
b)  The syntax for ANOVA is ANOVA(List 1, List 2, List 3, List 4, ...List n).  So,  press STAT, move the cursor to TESTS, and move
the cursor down that list to ANOVA(.  That's item H: on the TI-84.  Press ENTER and ANOVA( will be displayed on the home
screen.
c)  Inter list numbers so that you have the following display: ANOVA(L1, L2, L3, L4.  You can either close the parentheses or not.
d)  Press ENTER and various results will be displayed.  One of these is P=.004764....  So, Ho is rejected.

2. Chi-Square Test for Independence:

Suppose that we have the observed (those indicated by O= ) values in the following table and we want to know if independence
is indicated at the α=0.01 level.

 Consumer Product Group #1 Group #2 Row Totals Brand #1 1 O=33: E=36 2 O=57: E=54 90 Brand #2 3 O=30: E=20 4 O=20 50 Brand #3 5 O=5: E=16 6 O=35 40 Brand #4 7 O=12 : E=8 8 O=8 :E=12 20 Column Totals 80 120 200

First we need to calculate the expected values, i.e., those already entered as E= in the table.  The expected value is this:
P(cell) * Sample Size.
The probability, P, of a cell is calculated using the rows and column totals.  For example the probability cell 5 is as follows:
P(5) = P(brand 3) and P(Group 1)
=40/200 * 80/200
Now, we multiply that by the sample size of 200:
E=40/200 * 80/200*200
=16
This can be simplified to the following:
E=[(Row Total)(Column Total)]/(Sample Size)
Now we want to test for independence, and we will first enter the observed values in Matrix [A] and the expected values in Matrix [B].
a)  Press 2ND, MATRIX, move the cursor to EDIT and press ENTER to edit matrix [A].
b)  Enter 4 x 2 for the matrix configuration and then enter the observed values.  Press 2ND, QUIT to leave this matrix.
c)  Press 2ND, MATRIX and move the cursor to EDIT.  Then press 2 to edit matrix [B].
d)  Enter 4 x 2 for the matrix configuration and then enter the  expected values in the matrix.  Press 2ND, QUIT to end the
matrix editing.
e)  Press STAT,  move the cursor to TEST, and select Χ2 -Test from that list and press ENTER.
f)  The Calculate screen will be displayed with the matrices indicated for storing observed and expected.
g)  Move the cursor to Calculate  and press ENTER.  Χ2 =24.64 and p=1.79467 will be displayed.
h)  Since P is smaller than the
α-value of 0.01, we reject the null hypothesis of independence and conclude that the preferences are dependent.
i)  Alternatively, we could have compared the statistic of Χ=24.64 with the critical value for α=0.01.  Using a df = (4-1)(2-1) and consulting a
table, we would have found Χ2 =11.34.  Since the test statistic is larger, we would reach the same conclusion as above.

3. Х2 Goodness of Fit:

The TI-83 Plus does not have a GOF function, so I will first give the procedure for the  TI-84; then I will give a procedure for the
TI-83 Plus.

Suppose that a cell phone vendor wants to test the colors of the cases of cell phones to see if customers have a color preference.
A sample is taken and the data in the following table is collected.  The vendor wants a confidence level of 95%.  That is,  α =.05.

 Frequency Red Blue Green Grape Lime Observed 30 28 20 15 12 Expected 21 21 21 21 21

The expected values are calculated by adding all of the observed values and dividing by 5, the number of categories.
HO: Customers show no color preference.
H1:  Customers show a color preference.
a) Press STAT and enter the observed values in L1 and the expected values in L2.
b) Press STAT, move the cursor to TESTS, and cursor down to D: Х2 GOF – TEST and press ENTER.
Alternately, you can press ALPHA, D to activate that procedure.
c)  On the screen that appears, make sure that L1 is opposite Observed and L2 is opposite Expected.
d)  Enter 4 opposite df.  The value of df is one less than the number of categories.
e)  Move the cursor to Calculate and press ENTER.
f)  The value P= .01882... will be displayed.  Since this value is less than α=.05, we reject the null
hypothesis.
Calculation for TI-83 Plus:
What we are actually going to do is first find the sum of the values listed opposite CONTRB on the TI-84.
This sum will give us the value for X2.  We will then use the X2 cdf to find the value for p.
a) Press STAT and enter the observed values in L1 and the expected values in L2.
b)  Press 2ND, QUIT to go to the home screen.
c)  Press 2ND, LIST and move the cursor to MATH.  Press 5 to paste sum( to the home screen.
d)  Enter information so that you have the following:  sum((L1-L2)2/L2). Press ENTER and you should get
the answer of 11.809 for X2.
Calculate the p-value:
f)  Press 2ND, DISTR, move th cursor to X2cdf( and press ENTER.
g)  Enter information so that you have the following: X2cdf(11.809, 1E9, 4).  The “E” is made by pressing
2ND; then the comma key.
h)  Press ENTER and the value .018829…, the same value as with the TI-84, should be displayed.

XI: SPECIAL PROCEDURES:
1. Covariance:

2. Variance-Covariance Matrix

3. Mean Variance Optimization

I'm going to do a very limited mean-variance optimization. I realize that Excel or a portfolio optimizer is the preferred method
of doing optimizations, but I'm thinking that maybe doing it with a calculator will give some insight as to what is being done
in Excel. Caveat: I'm aware that mean-variance optimization assumes that returns are normally distributed, that it often uses
historical rather than future data, and other shortcomings. I'm not going to deal with those issues. This section if for helping
students better understand what mean-variance optimization is all about, not evaluation of the effectiveness of procedures.
We will be using data from this table for our calculations.

 FUND OR BENCHMARK YEARLY RETUNS S& P 500 10.88, 4.91, 15.79, 5.49, -37.00, 26.46, 15.06, 2.11, 16.00, 32.39, Fund A 13.89,16.27,19.26,13.43, -48.02, 52.20, 14.48,-12.33, 18.72, 14.27 Fund B 20.84, 15.57, 26.64, 15.52, -44.10, 36.73, 11.04, -14.52,18.21. 15.14

So, with that, let's write down the formulas for the portfolio return and standard deviation.
Portfolio Expected Return:
E(Rp) = wT µ  Where wT is the transpose of the vector of the weights, w.
Standard Deviation of Portfolio:
σp =(wT V w)^(1/2) Where wT is as explained immediately above, V is the variance-covariance matrix and w is just the
vector of weights.
So, let's take a highly unrealistic portfolio of  S&P 500  and Fund A with the returns as given above and a weighting of .25 for
the S&P and .75 for Fund A. So, that would give us a weighting vector as follows:
[[.25]
[.75]]
Let's put that in matrix [H] for heavy
The terms used for the return matrix would be as follows:
So, µ =[[7.4]
[6.79]]

These values were obtained by taking the geometric means of the values in the tables above where r, r2 etc. are listed.
Let's put them in matrix [I]. Item 3, above, is a procedure for doing geometric mean.
So, the formula for the return is as follows:
E(Rp) = wT µ
So, in terms of our matrices, that would be as follows:
E(Rp) =[H]T [I]

a) To get [H]T , we put the numbers in matrix [H]. Then press 2nd, QUIT to quit the matrix editor.
b) Press 2nd, MATRIX, move the cursor to highlight [H] and press ENTER.
c) Press 2nd, MATRIX, move the cursor to MATH, and press 2 for T . You then should have [H]T .
d) If you have the µ matrix in [I], then press x, the multiply sign; then press 2nd, MATRIX, and press 2
for matrix [I].
e) Press ENTER to get the answer of 6.94.

Portfolio Standard Deviation:
The portfolio standard deviation requires considerably more work. Remember that the formula for the standard deviation
is as follows:
σp =(wT V w)^(1/2)
The sticky point is getting the variance-covariance matrix, V. Remember that the formula for that matrix is as follows:
σxy =1/n Ʃi=1,n (Xi-X̄)(Yi-
ȳ
Where xbar and ybar are the arithmetic means.
So put the returns from the table above in matrix [A]. Put the S&P values in the first column and those for Fund A in
the second column. Then in matrix [B], enter a 10 x 10 matrix with all 1s. To save some time in entering all of the 1s,
you can go to the matrices and dimension [B] as a 10 x 10. Then go to the home screen. Press 2nd, MATRIX, move
the cursor to MATH, select Fill( and press ENTER. Then enter information so that you have Fill(1, [B]. Press ENTER
to fill the matrix. Note that [B] must be entered from the list of matrices.
Let's do the variance-covariance matrix in two steps as follows:
Diff = [A]-1/10*[B]*[A]--> [C]    (Eq 1)
The symbol --> is obtained by pressing the STO button.
Covr = V =1/10*[C]T [C] -->[D]

Now, we want to implement the following equation for the standard deviation:
σp =(wT V w)^(1/2)
Since we can't take the square root of a matrix, we'll have to solve for the square; then take the square root of the number
inside the 1x1 matrix.
p)2=([H]T *[D]* [H])
σp =
p2 ) (NOTE: Don't take the square root  of Ans because it's a matrix. Take the square root of the number inside the
matrix.
There's a little program that I have written to do these calculations below in the APPENDIX.
I may be back later to add more detail as to what the matrices are doing for the variance-covariance matrix,
but for now, that's it.

APPENDIX:

Here's a little program to do the above calculations. You'll need to put the returns in matrix [A] and make a k x k matrix of 1s,
and the averages of the returns in ]I].

:MVOPT
:"FKIZER 06/09/14"
:Input "1ST. WEIGHT" , F
:Input "2ND WEIGHT , S
:Input "NO. RETURNS ",N
:[[F] [S]]--> [H]
:[H]T *[I]-->[E]
:[A]-1/N*[B]*[A]-->[D]
:1/N*[D]T*[D]-->[C]
:[H]T *[C]*[H]-->[J]
:Matr►list([J]L6)
:ClrHome
:Disp "RETN ",[E]
:Disp "STD DEV ", √(L6)

PROGRAM for calculating InverseT
: ”FKIZER 91906”
: INPUT “DF=”, D
: Menu(“SELECT”, Lft TL”, 1, “RT TL”, 2, “2-TL”, 3)
: Lbl 1
: solve(tcdf(-1E9, X, D) – A, X, -1.7) →T
: Goto 4
: Lbl 2
: solve(tcdf(-1E9, X, D) –(1- A), X, 1.7) →T
: GoTo 4
: Lbl 3
: solve(tcdf(-1E9, X, D) – A/2, X, 1.7) →T
: Disp abs(T
:Lbl 4
:Disp T
Using the Program:

a) After you’ve entered the program, highlight the program name and press ENTER.
b) The program will ask for the confidence level,  α, and then the degrees of freedom,  df.  For this program,
α
is not divided by 2 when doing a two-tailed test. Remember that for a
c)  You will then be presented with a menu to select either right-tail, left-tail, or 2-tail.  Select the one appropriate by
either pressing the appropriate number or highlighting the number and pressing ENTER.  The answer will be
displayed in approximately 20  seconds.

Program for sorting data into classes:
NOSCAL
:FKIZER090210
:SortA(L1)
:min(L1)-->S
:dim(L1)-->Q
:max(L1)-->M
:int(M/W)+1-->dim(L2)
:Input "CLS WDTH  ",W
:0-->T:1-->X:W-->F:0-->C
:ClrHome
:Lbl 1
:While L1(X)
≥S and L1≤F
:T+1-->T
:X+1-->X
:If X>Q
:Then
:T-->L2(C+1)
:Goto 2
:End
:End
:C+1-->C
:T-->L2(C)
:0-->T
:S+W-->S
:F+W-->F
:Goto 1
:Lbl 2
:L2
After you’ve entered the program, use it in this manner.
a)  First enter the data in list L1. The data need not be in any order.
b)  To execute the program, highlight the program name and press ENTER.
c) The program will ask for the class width, CLS WDTH.  Enter the class width and press ENTER.
d)  The numbers for the classes will be stored in list L2 and that list will be displayed after execution.  Note that you can
move the numbers after the ellipses (the three dots) with the cursor arrows.  When finished press CLEAR to stop the
program. . Making it Better:  I would be grateful if you would report any errors or suggestions for improvements to me.  Just click "E-mail Webmaster," site the item number, and tell me your suggested change.

Printing Hint:  Most browsers will send both the navigation bar and the text to the printer, and, as a result, some printers will cut off the right edge of this document if  the file is printed directly.  To prevent this, you can use landscape, of course.   But if you'd like to get rid of the navigation panel,  highlight the instructions portion only (not the navigation panel) and check "Selection" on the Print dialog box; then click "Apply."  This will eliminate the navigation panel and get all of the instructions on the printed pages.  Some newer printers have a special Web Page function for printing that will print the page without cutting part of it off.

Copy Restrictions:  You may make single copies of this document for your own personal use and for the use of other students, but inclusion in another document, publication,  or any use for profit requires my permission.  Teachers may make multiple copies of this document for their students if they first get my permission.  Merely send me an email (Just click on Webmaster in the navigation bar.) with a one-sentence explanation of what you’re using the document for.  I’ll give you permission in a timely manner. 