Some of these views have already been expressed. I'm just trying to
summarize them here, because I present these views to students when
they ask the same types of questions as have been asked here.
We traditionally attempt to linearize a graph for these reasons:
(1) If we don't have a computer to fit the best curve through the data
points, our eyes plus a see-through ruler can do a pretty good job of
drawing the best line through the data points.
(2) If we are trying to determine if the equation (the theory) is
upheld by the data, it is much easier to judge (by eye) if the data
points form a straight line, than to determine if they follow a
specific curve.
I usually have to further explain "specific curve" because some
students will quickly point out that they can easily see that the data
points are curved like a 1/r^2 curve. At which point I say, "How can
you tell it is 1/r^2 versus 1/r versus 1/r^3 versus 1/r^2.5 etc.?"
Today our students have several options (several very good commercial
programs) for doing non-linear curve fitting, therefore point (1) is
not so crucial any more. But point (2) is still valid. Yes, we can
obtain all sorts of statistical data from the non-linear fit, but that
statistical data often does not grab our attention us a much as seeing
the data points curve away from a straight line when they are supposed
to be straight.
I sometimes make an analogy between this and digital meters versus
analog meters. If I am in the control room of a plant, perhaps a
nuclear power plant, and I see a digital meter is reading 128.3, do I
need to perform any corrective measures? I'll have to think about that
a second or two. But if that same meter were an analog meter, and half
of the meter face were painted green and half painted red, and the
needle is currently way into the red region, I know in an instant that
I had better be doing something.
Linear graphs are that way. I can see in an instant if we're basically
on target or way off target.
That we traditionally leave the horizontal axis unmodified and apply
our "linearization algebra" on the vertical axis is, I believe, just
convention. But the convention has merits because it makes for an
easier-to-read graph; at least in my opinion. When I am looking for
trends in data, my mind just functions better if the horizontal axis
represents raw data rather than algebraic manipulations of the data.
However, sometimes offsets or similar artifacts in the data will
dictate which way things get plotted. Here is my favorite example.
Take data of capacitance versus plate separation for a variable-plate
capacitor. The expected relationship is C = eA/d where e is the
permittivity, A is the area, d is the separation. We should be able to
plot C versus 1/d and get a straight line, or we can plot 1/C versus d
and get a straight line. Most people would recommend plotting 1/C
versus d because that leaves the independent variable unmodified.
However, there is wire capacitance, and some capacitance meters have
zero offsets. Therefore we really have:
C(exp) = eA/d + C(offsets and wires)
This is still linear if we plot C versus 1/d, and we can determine our
offsets/wire-capacitance from the intercept. However, I can no longer
plot 1/C versus d to get a straight line. I have to plot C versus 1/d.
This violates the earlier rule of leaving the independent variable
alone. But in this case, it is clearly the best thing to do.
Michael D. Edmiston, Ph.D. Phone/voice-mail: 419-358-3270
Professor of Chemistry & Physics FAX: 419-358-3323
Chairman, Science Department E-Mail edmiston@bluffton.edu
Bluffton College
280 West College Avenue
Bluffton, OH 45817