How to Analyze an "Is This Proportional to That?" Experiment

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The question "Is Y proportional to X?" is a very common experimental question, as in "Is the acceleration of an object proportional to the applied net force?" or "In free fall, is distance fallen proportional to the square of the elapsed time?"

Mathematical Background:

The question "Is Y proportional to X?" has traditionally been answered graphically - if Y is proportional to X, then the equation relating Y and X is Y = mX + b for some constants m and b, and the graph of Y versus X will be a straight line with slope m and y-intercept b.

From algebra, we know that the graph of y = mx is a straight line with slope m. Y is proportional to X.

graph of y = mx

So, if the graph of two quantities is a straight line, then the two quantities are proportional and the equation relating them is analogous to y = mx. In the diagram at right, if the graph of d vs. t2 is a straight line, then d is proportional to t2, and the equation relating them is d = mt2. The slope of the line may (or may not) be physically significant - in this case, theory says that m = g/2, where g is the free-fall acceleration.

d = mt^2 graph

The technique of finding two quantities whose graph is a straight line is an extremely common experimental technique. First, theory often suggests which quantities to graph, so guesswork is generally not involved. Secondly, it is relatively easy (once you get used to it) to tell whether data points fall along a straight line or not. On the other hand, if a graph curves, do the points fall along a parabola, a cubic, a trigonometric function, or some even a more-exotic transcendental function? It is enormously difficult to tell. Also, the mathematical method to find the equation of the straight line that best fits a data set (called linear regression) is arduous, but "doable" by hand. This is not a consideration in the computer age, when many software packages will fit data to lots of different functions at the touch of a button, but physicists have traditionally used straight lines.

A couple of examples are shown below.

A Simple Example:

Suppose we are attempting to answer the first question above: Is the acceleration of an object proportional to the applied net force?" Theory (Newton's Second Law) says that the answer should be "yes" - but is it?

Some (made-up) experimental data is shown below.

Here are some (made-up) values for the acceleration of an object and the net force that produced that acceleration.

Here is a graph of acceleration vs. force for the data shown at left. It is customary to put the independent variable (the quantity you change in the experiment - here, force) on the x-axis, and the dependent variable (the quantity that changes on its own in the experiment - here, acceleration) on the y-axis. The graph was made using Exceltm from the data at left. A best-fit (least squares linear regression) line (what Exceltm calls a "trendline") has been drawn through the data points, with the equation of the line shown, and an R2 value has been calculated. R2 is the square of a statistic called the "correlation coefficient" which measures how closely data values fit a particular equation. A value close to 1 indicates a good fit, while a value of R2 close to 0 indicates a random correlation.

f vs. a data
f vs. a graph

If you look at the data points by themselves, they seem to fall along a straight line. (A trick that old-timers use to see if the graph is more-or-less straight is to hold the edge of the graph paper up to your eye and "sight" down the points. You can then easily "get a feel" for the shape of the graph.) Notice that the data points all fall on or near the best-fit line, and R2 is very close to 1. This indicates that this data is consistent with theory.

Aa a bonus, if you compare the equation y = 0.151x and Newton's Second Law in the form a = (1/m)F (Note that we have plotted a on the y-axis and F on the x-axis) you get 1/m = 0.151, or m = 6.62 kg which we can compare to the measured mass of the experimental object.

Another Example:

"In free fall, is distance fallen (from rest) proportional to the elapsed time squared?" Theory says "yes", but does it really work? Suppose we set up an experiment and measure the distances that an object falls in several time intervals. Given this data, how do we answer the question?

Below is some (made up) data for the experiment described above. The left column is time (in seconds) and the right column is distance (in meters). This data is in an Exceltm spreadsheet.

Here is a graph of the data at left produced by Exceltm. Note that the graph doesn't tell us a lot about the relationship between t and d - the graph might be a parabola, but it could be a cubic, a tangent function, or something else. In other words, this graph just isn't very useful.

distance vs. time data
distance vs time graph

A third column has been added to the data table below, which calculates time2 by squaring each value in the left-hand column.

Below is a graph of distance vs. time2, and a least-squares line has been calculated. First, note that the points all fall very close (suspiciously close, actually) to the line (R2 = 0.990) - therefore, we can say with confidence that our experiment strongly supports the hypothesis that d is proportional to t2.

d vs t^2 data
d vs t^2 graph


In order to answer the question "Is Quantity Y proportional to Quantity X?":

  1. Gather a reasonable amount of (X, Y) data (or data from which X and Y can be calculated).
  2. Plot a graph of Y vs. X.
  3. If the points of the graph seem to fall along a straight line (and you get an R2 value close to 1) then you can conclude that your data supports the hypothesis - otherwise, not.

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last update September 3, 2006 by JL Stanbrough