Analyzing Graphed Data
Everyone enrolled in this class should already be familiar with how to number and label the axes of a graph,
and plot data points on it. If you are uncertain about how to do this, please request extra help from Mr.
Samson.
Rather than focusing on how to prepare a graph, we are going to concern ourselves with how to maximize
our ability to statistically analyze graphed data. Our main tool for analyzing data in this class is the
Logger Pro computer program which allows us to input data manually or by computer probe, plot it on a
graph, and statistically analyze it.
The way we graph data in Science is not always the same as we would do it in Math class. Here are a few
graphing/data table rules to keep in mind in Science class:
The manipulated variable is always listed first on the data table, followed by the responding variable.
Most variables that we measure in class can not have negative values (what are negative grams?). Unless
there is good reason to do otherwise, we will use the positive quadrant for graphing, with (0,0) in the lower
left corner.
The manipulated variable (independent variable) is always placed along the x-axis, and the responding
variable (dependent variable) is placed along the y-axis. Never label the axis “manipulated” or
“responding”, it is understood which is which.
Both axes of a graph must be labeled with the variable with the unit in parentheses.
Examples: Time (sec)
Velocity (m/s)
Mass (g)
Wrong: Axes are not labeled
Wrong: Axes labeled, but units not included
Correct: axes labeled, units shown in parenthesis
The axes of a graph should always be numbered in such a way that the data makes it more than half way
across, otherwise it is hard to see the relationship between the variables.
Wrong: data is crammed into one portion of the graph, curve in data is not evident
Correct: same data now occupies the majority of the graph space, curve in data is now evident
Our goal is find the “best-fit-line”, which represents our estimate of what the true, mathematical
relationship between the variables. The best-fit line fits is a straight line or smooth curve that fits the data in
such a way that some data points fall above it and some fall below it, but are as close to the line as possible.
There is never, ever, any reason to play connect-the-dots on a graph. Remember, natural relationships
almost always graph out as straight lines or smooth curves.
Wrong: do not play connect the dots!!
Connecting the dots implies that every data point is perfect and that the relationship between the variables
is very weird. Relationships in nature show up on graphs as straight lines and smooth curves, which our
data only approximates (and therefore lands on either side of the best-fit line)
Correct: smooth curve that fits the data with some points landing on either side
Wrong: a smoothly curved best-fit line is shown, but the data tends to land on one side more than the other
When we graph our data, we first look at the data points and make estimate which of five types of
relationships the data fits, then fit a line or curve for that relationship through the data points
The five relationships we will look for are: directly proportional, exponential, inverse, inverse square, and
unrelated.
Directly Proportional
Best-fit line is straight and starts at zero.
The general form of the equation is: y = Ax,
where “A” is the slope of the line.
It is very unusual to have and x or y intercept with naturally occurring relationships that are
directly proportional.
To test for a directly proportional best-fit with Logger Pro, select “(proportional)” from the curve-fit
options.
Example: the data below yields the general equation
y = 10.16 x
since the y axis is velocity, and the x axis is time, we would write the specific equation velocity = 10.16
(time)
We can now use the equation to solve for either velocity or time
Example: how fast will the object be moving after 12 seconds?
Velocity = 10.16 (12) = 121.9
How long will it take the object to reach as speed of 180 m/s?
180 = 10.16 (time)
time = 180  10.16 = 17.72 sec
Exponential
Best-fit line starts at (0,0) and curves upward. The general form of the equation is: y = Ax2 where “A” is a
constant that is specific to the particular data set.
To test for an exponential best-fit with Logger Pro, select “(power)” from the curve-fit options. This will
test the data for the equation y = AxB, where “B” can have any value.
Although this the curve shown above has a good fit, it is highly unusual for any natural relationship to have
any number other than “2” as the exponent of “x”. In other words, x2 relationships happen all the time, but
x1.81 almost never do. For this reason you need to manually change the value of “B” to “2” as shown below.
Notice that when you change the value of “B” from “1.81” to “2”, your best-fit-line no longer fits very well.
In order to improve the fit, change the value of the constant “A” until you once again have a good fit as
shown below.
Clicking the “OK” button gives a graph with an equation for the best-fit line:
Now you can take the equation, substitute in the x and y variables, and use it as a predictor:
Distance = 4.9( Time2 )
Example: how far will the object go in 8 seconds?
Distance = 4.9 (82 ) = 4.9  64 = 313.6 meters
RMSE stands for “root mean square error” and is a measure of the reliability of the equation.
The lower the RMSE value, the more reliably the equation fits the data.
Notice the difference in the RMSE values between the second and third graphs shown above. The lower
number of 6.7847 indicates a better fit than 59.532
Inverse
As the manipulated variable gets bigger, the responding variable gets smaller, so the best-fit line curves
from the upper left to the lower right. If the manipulated variable gets three times bigger, then the
responding variable gets three times smaller.
In order to test for an inverse best-fit line in Logger Pro, choose “inverse” from the curve fit menu.
This will test the data for the general equation is y = A / x
The graph on the below shows a best fit line with the specific equation: pressure = 664 / volume
We could use this equation to find out the volume at which the pressure would be 17 atm
17 = 664 / vol
vol = 664 / 17 = 39 ml
The low RMSE value indicates a high degree of reliability
Inverse Square
As the manipulated variable gets bigger, the responding variable gets exponentially smaller, so the best-fit
line appears to drop in a steep curve from the upper left to the lower right. If the manipulated variable gets
three times bigger, then the responding variable gets nine times smaller (3 2 times smaller).
In order to test for an inverse square best-fit line in Logger Pro, choose “inverse square” from the curve fit
menu. This will test the data for the general equation is y = A / x2
The graph on the below shows a best fit line with the specific equation: brightness = 1090 / distance2
We could use this equation to solve for the brightness at any distance, even one that is well beyond the
rangeof our data, for instance 200 meters
brightness = 1090  2002 = 1090  40,000 = .0273 lumens
It is a little trickier to solve for distance, because when you solve for distance2, you need to take the square
root in order to get the final answer.
Example: at what distance is the brightness equal to 150 lumens?
150 lumens = 1090 / dist2
dist2 = 1090  150
dist2 = 7.27
(take the square root of both sides)
dist = 2.70
No Relationship
Sometimes we find that there is no relationship between the variables, so that as we increase the value of
the manipulated variable, there are only small, error-related fluctuations in the value of the responding
variable. In this case the best-fit line is a horizontal line with a y-intercept.
Unfortunately, there isn’t a “no relationship” option in Logger Pro, so instead you will choose the “linear”
option from the curve fit menu. This will produce a best-fit line with a y-intercept.
If there is no relationship between the variables, then the best-fit line should be horizontal, so you have to
manually change the slope of the line to zero.