Graphing From Data

TI 86

From Dr. Henry Donato’s Web Page: For more information on the use of the TI 86, visit Dr. Henry Donato’s Web page.

In this exercise, we will examine how scientists find relationships between experimentally observed

quantities. For illustrative purposes, let’s assume that I have measured the length in cm of my

goldfish at yearly intervals and have collected the following data

Age (in years) Length (in cm)

1 11

2 19

3 30

4 41

5 49

Note that this data is completely fictitious but it is intuitively reasonable. In the days before

scientists had access to any computing devices, this data would be analyzed in the following way.

First one would graph the data points on graph paper. This would make clear what one has

already guessed, namely that the increase in length is approximately linear with time. The

expression "linear with time" means that the increase in length occurring during a fixed interval of

time, say a year, is always the same. Of course, the intervals in the data shown above are not all

the same, but they are close to 10 cm a year. Scientists are always looking for the simplest

possible description and explanation for observations so they would be likely to assume that the

data is saying that the increase in length of my goldfish is linear and the deviation from linearity

observed in the data is due to experimental error. Here, experimental error would include

problems involved with actually measuring the length of the fish, as well as problems in the

experimental protocol, such as having more food available to the fish in some time periods (say

between years 2 and 3) compared to others (say between years 4 and 5). So to represent the

presumed linear increase in length of my fish, I will draw the "best" straight line through the data

points. The simplest way to do this is to place a ruler on the graph paper and move it around so

that the line it describes passes as close as possible to each data point. Note that this line need not

actually pass through each data point or for that matter any of the data points. So the data points

were used to find the line and the line represents the functional relationship between age of the fish

and the length of the fish. The equation for a straight line is

y = a + bx

where a is the y intercept and b is the slope both of which can be determined from the graph. So

our analysis of the data yields:

Length = a + b(Age)

With modern computing devices, these operations can be performed more quickly and with

greater objectivity than the procedure discussed above. In this exercise I will describe the use of

the TI-86 calculator to perform this analysis.

1.Enter the Data: Data is entered into the list screen of the TI-86. After turning on the

calculator press 2nd, -, F4(edit).

xStat is the default name for the x values, yStat is the default name for the y values, and

fStat is the default name for the frequency values or the number of times that a particular x

and y value were measured (usually these values are 1). I will refer to the fStat values from

now on as weights. To enter the data, enter each number followed by the ENTER key.

2.Perform Linear Least Squares. The linear least squares calculation is under the STAT

menu on the TI-86. Leave the LIST menu by pressing EXIT, and then 2nd + to enter the

STAT menu. Press F1 (CALC), F3 (LinR), 2nd - (to again bring up the LIST menu), F3

(NAMES), F2 (xStat), , (one must enter a comma), F3 (yStat), ,, F1, ENTER.

So the best straight line through the data points is

Length = 0.6 + 9.8(Age)

3.Plot the Data and the Linear Least Squares Line. 2nd + (Go back to the STAT

screen). F3 (PLOT), F1 (PLOT1), Set plot ON, Type = SCATTER, Xlist Name =

xStat, Ylist Name= yStat, and Mark = (any mark you would like).

Then GRAPH, F2 (Window) to set the upper and lower values on the x and y

coordinates of the graph.

Pressing F5 (GRAPH) will show you what the graph looks like.

To obtain the linear least squares line plotted through the data points, then press F1 (y(x)=) and

enter the regression equation into the y1 variable. The easiest way to do this is to enter RegEq into

the y1 = screen. RegEq is the variable name into which the last regression equation calculated is

stored. The variable can be easily reached at the CATLG-VARS screen. Press 2nd CUSTOM

(CATLG-VARS), F2 (ALL), move the cursor down until it highlights RegEq and press ENTER.

Note that the equation in the y1 variable is selected because the equals sign is highlighted. Pressing

2nd F5 (GRAPH) adds a plot of the regression equation to your graph.