Thanks to all who responded. The discussion has been helpful, and I
think has also confirmed more thought is often warranted before forging
ahead with your first guess of how a data set ought to be analyzed.
I am inclined to agree with Alvin Bachman that averaging first makes the
most sense for this kind of data. I also agree with John Denker that
proper data analysis can be difficult, and it certainly depends on what
you are trying to accomplish.
Performing repeated trials of a particular experiment then performing
some type of averaging was not done in any experiments I participated in
as a professional scientist. This means that in my professional
research career I don't think I ever took or analyzed data the way many
student labs are organized. Perhaps this is one reason many of the
experiments I have students perform and analyze are not this type of
experiment... but a few are, including this air track experiment.
If an experiment is repeated multiple times to get "an average value,"
it seems to me the raw data should be averaged. The raw value is what
has been measured; it is what we assume has the normal distribution; and
as I pointed out by example, others stated, and Alvin showed by
analysis... the average of a function of the data does not necessarily
approach the function of the average of the data.
If we calculate first then average I think it can lead to a false
impression of what actually happened. In the example I gave we might
say the goal was to determine the average velocity, but that is not what
was measured. What was measured was the time to travel a particular
delta-x.
Here is a specific description of what I mean. In my particular
example, the average delta-t for a 20-cm delta-x was 0.563 s yielding an
average velocity of 35.55 cm/s. For this data set of 15 trials, if each
individual velocity is calculated then these are averaged, the result is
37.23 cm/s. If we report the 37.23 cm/s as the measured velocity, then
the reverse calculation implies we measured an average delta-t of 0.537
s which is clearly incorrect. That is, our actual data do not yield an
average time of 0.537 s, rather our data yield an average time of 0.563
s. Thus, averaging later would give a false impression of the average
value of the actual data for the experiment.
I would also like to repeat that I think this is somewhat of a new
problem for student labs that involve this type of repetitive
measurement. In "the old days" before spreadsheets we would have never
calculated a final result for each data point then averaged the results.
We would always average the raw data then calculate a single result. I
always thought we did it that way because it was by far the easiest way
to do it; i.e. far fewer calculations (which I was doing by hand and/or
slide rule). Now I am inclined to think it goes deeper than that. I am
inclined to think it was the proper way to do it.
At the same time, I repeat that I know a lot of science involves
acquiring data for which averaging is not appropriate at any point. I
think this discussion pertains specifically to the type of experiment in
which repetitive measurements of the same thing are measured in hope of
determining a good mean value and good estimate of the standard
deviation.
I am also curious about what was done for some of the historic
experiments. For example, I know that Millikan's assistants made many,
many measurements with the oil-drop experiment over many months. What
kind of averaging was done? Without trying to check it out (which I
simply don't have time to do right now) I suspect they did some of both.
I suspect that for a particular data set for a particular drop with a
particular charge they averaged the raw timing data. But then they had
to average calculated results from that same drop having different
charge and calculated results from other drops. But... when the same
drop with particular charge was watched for a series of up and down
motions, I would guess they averaged the up times and averaged the down
times. I'll bet they did not average the velocities.
Michael D. Edmiston, Ph.D.
Professor of Physics and Chemistry
Bluffton University
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu
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