Graphing From Data

TI 85

From Dr. Henry Donato’s Web Page: For more information about using the TI 85, 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-85 calculator to perform this analysis.

1.Enter the Data: Press STAT to enter the statistics menu, F2 (EDIT), accept xStat as the

xlist Name by pressing ENTER, accept yStat as the ylist Name, and finally enter the x

and y coordinates of each experimental point.

2.Perform Linear Least Squares. Press 2nd F1 (CALC), ENTER (to accept xstat as the

list of x values, ENTER (to accept ystat as the list of y values), and finally F2 (LINR).

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. First one must define the graph

range, i.e. the portion of the graph that one wants to display. Press GRAPH, F2

(RANGE), and enter values for xMin, xMax, yMin, and yMax.

Now display the graph by pressing STAT, F3 (DRAW), F2 (SCATTER). One

may clear the commands off the screen by pressing to CLEAR to better see the

graph.

Press STAT again and then 2nd F4 to display the regression line on the graph.