# Statistics and plotting on the TI-83 family

## by Sam Jordan

### edited by Ray Kremer

For those who find the TI-83 manual hard to follow when it comes to using the built in statistics functions, I have revamped Sam Jordan's TI-86 statistics tutorial so that it applies to the TI-83 and TI-83 Flash family.

Another guide for TI-83 family statistics has been written by William Larson.

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#### Statistical Analysis

The first step in performing a statistical analysis on the TI-83 is determining the type of data you will be working with. The two types are single variable data sets, and paired variable data sets. On the TI-83 these are called 1-Var Stats and 2-Var Stats (short for One variable and Two variable) data sets.

An example of a single variable data set would be the height in inches of all male students in the Statistics 101 class. Let us assume that there are 9 male students in the statistics class with the following heights measured to the nearest inch.

64, 66, 68, 69, 70, 70, 71, 72, 74

The TI-83 will perform a single variable analysis of this data if we place it in the proper "LIST" format, and store it in a list variable. By default, the TI-83 uses the list variable "L1" for its analysis. We may save it in "L1" via:

{64,66,68,69,70,70,71,72,74}L1

where "" is the TI-83 "store" operator. You can also use the on-screen list editor by entering [STAT] then [1].

We can now find the "1-Var Stats" or "one variable" statistics for this set of data. All we need to do is enter the command:

1-Var Stats

This may be done by selecting it from the "STAT" menus by entering:
[STAT][Right Arrow]
[1]
while on the home screen.

This will display the following:

1-Var Stats
x-bar=69.33333333
Ex=624
Ex2=43338
Sx=3.041381265
ox=2.867441756
n=9
minX=64
Q1=67
Med=70
Q3=71.5
maxX=74

Actually, I couldn't type the exact variable names that appeared in the above output, but the actual ones are described in the manual and are the:

Mean
Sum of X
Sum of squared X
Sample Standard Deviation of X
Population Standard Deviation of X
Number of data values
Minimum X value
1st Quartile
Median X
3rd Quartile
Maximum X

Since at least one of the values in the "L1" list is repeated, we could have entered "L1" as:

{64,66,68,69,70,71,72,74}L1

While also defining an "L2" list which gives the "frequency" of each of the values in the "L1" list

{1,1,1,1,2,1,1,1}L2

which in this case means that the value "70" occurs 2 times in the original list.

Assuming that we have entered these two lists, and that we have exactly the same number of entries (8) in both lists, we enter the command this way:

1-Var Stats L1,L2

This may be done by entering:
[STAT][Right Arrow]
[1]
[2nd][1][,]
[2nd][2]
while on the home screen.

That pretty much covers single variable statistics.

Two variable statistics are handled in much the same way as One variable stats.

Assume we have the following (x,y) pairs:

(1,3) (2,4.1) (3,5) (4,5.9)

These would be entered into the stat lists:

{1,2,3,4}L1
{3,4.1,5,5.9}L2

Notice that both lists contain the same number of items. THIS IS IMPORTANT! If they don't contain the same number of items, you will get weird errors when trying to perform analysis of the data later! Once again, an optional third list may be used for frequency.

You can obtain the two variable stats with the command "2-Var Stats" by selecting it from the [STAT] menus. 2-Var Stats will give the following:

2-Var Stats
x-bar=2.5
Ex=10
Ex2=30
Sx=1.290994449
ox=1.118033989
n=4
y-bar=4.5
Ey=18
Ey2=85.62
Sy=1.240967365
oy=1.074709263
Exy=49.8
minX=1
maxX=4
minY=3
maxY=5.9

Again, I couldn't type the exact output, but the actual stuff is described in the manual. The values are actually stored to variables, you can use them afterward by taking the variable names from the VARS 5:Statistics menu.

To do regressions on the same data as entered in L1 and L2, you can use the regressions built into the [STAT] menu. For example, you can perform a linear regression on that data by "LinReg(ax+b)".

This will produce:
LinReg
y=ax+b
a=.96
b=2.1

which indicates the line:
y=.96*x+2.1
If you have run the DiagnosticOn command from the Catalog ([2nd][0]) the linear correlation will also appear:

r2=.9974025974
r=.9987004543

#### Statistics Command Defaults

I can't say that I've tried this, but in Appendix A of the TI-83 manual, it lists some expanded ways of invoking the statistical analysis commands.

In general, if you specify nothing, the default parameters will be L1, L2, frequency of 1 and the equation goes into RegEq.

If the only thing you want to change is the location of the regression equation then just type the name of the equation variable. Say you want to use L1, L2, frequency of 1, and Y3 as the regression equation storage for a linear regression. Do:
LinReg(ax+b) Y3
(Y3 is in the VARS, Y-VARS, 1:Function menu)
What I think the calculator is doing is looking at the parameters after LinReg(ax+b) for first the list names, and then the Regression Equation name. If it doesn't see any predefined lists, then it uses all the default lists and assumes that the next parameter must be the Regression Equation name. The optional parameters vary depending on the type of regression.

#### Stat Plotting

Assuming that you have data in a list:

{1,1,2,2,3,4}L1

First select the Plot mode.
1. [2nd][STAT PLOT] for the plot menu to turn on a plot and select a mode. Turn on PLOT 1 by selecting PLOT1 with [1] and use the cursor to move to blinking ON and press [ENTER]. Then cursor down to the line "Type:" and use the arrow keys to select the plot type that you want. For single variable data, the only valid types are the Histogram or ModBoxplot or Boxplot (the 3rd, 4th, and 5th pictograms). You can then cursor down again to the "Xlist:" line and enter the name of the list in which you have stored your data and then press enter to have it remembered. Finally you can cursor down one more time to the "Freq:" line and use either "1" to indicate that each value in your list is to be counted 1 time, or enter the name of a frequency list which MUST have the same number of entries as your single variable list.

2. You can plot your data by hitting [GRAPH] which will draw the selected plot.

To plot two variable data, then instructions are similar to the above, except that the valid plot types then become either Scatter or xyLine (the 1st and 2nd pictograms).

#### Regression Equation Graphs

After you've read everything above on how to enter data and calculate basic stats on it and on how to Plot the data you entered you might also want to be able to graph any Regression equation that you got during the analysis just to see how it looks against the data that you entered and plotted.

The good news is that this is fairly simple. After you've entered all your data and run your Regression function (usually "LinReg(ax+b)" for a Linear Regression) the calculator will store the actual Regression expression into one of the Statistics variables called "RegEQ". To graph it, all you need to do is go into the Y= screen, arrow to an empty line, and press [VARS][5][Right Arrow][Right Arrow][1]. This will paste the contents of RegEQ to the Y= line. Now simply make sure that this selected and graph it like you would any other equation.

It is often instructive to use the previously posted "Plot" instructions to generate a "Scatter Plot" of your x and y lists (for two variable statistics) and then generate and graph the Regression Equation so that they are both displayed at the same time to get a visual view of how well the equation fits the data.

#### Computing and Plotting Residuals

Residuals are defined as the difference between the "observed" and "predicted" values of a regression.

You begin with a set of x,y data pairs:
(1,3) (2,4.1) (3,5) (4,5.9)

You enter these into the x and y stat lists as:
{1,2,3,4}L1
{3,4.1,5,5.9}L2

(Note that all lists have the same number of elements. If not, it won't work!)

After you've entered these lists, you can use the STAT PLOT menu to plot them as a scatter plot.

You can also run "LinReg(ax+b)" to generate their regression equation which provides the line of "best fit" to the points. After doing this, you can assign the resulting "RegEQ" to one of the graph function variables so that you can graph it against your original scatter plot. The easiest way is simply to paste RegEQ to an equation on the Y= screen. You can also set it equal to the regression equation by pasting RegEQ to the home screen, enclosing it in quotation marks, following it with a store arrow and Y1 and then hitting enter to store it.

If you need to compute the "residual" difference between the regression equation and the original L2 values and RegEQ has been copied to Y1, that is done by:
L2-Y1(L1)
The calculator has already done this for you in fact, and has stored it to a list named RESID, which can be found in the LIST menu under NAMES.

Which in this case should be set to:
{-.06,.08,.02,-.04}

If you then want to plot the residuals, you may do so by defining a scatter plot with:
Xlist name=L1
Ylist name=RESID

and make sure that you set your Ymin and Ymax values small enough based on the largest and smallest values in "RESID" so that you can see the plot.

This same technique can be used to plot the residuals of ANY regression.

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