I got hit with a question today that has never come up before in many
years of teaching. I suspect some of you have already been through
this, but I never thought much about it until today.
I'll describe the general question, then I'll describe the specific case
that brought it about.
General question... You make several measurements of a number that will
be used in a calculation to obtain a final result. You can average the
measurements into one average measurement then use that average in the
calculation, or you can do the calculation with each measurement then
average the results of the calculation. Depending on the type of
calculation performed, and depending on the standard deviation of the
measurements, the two methods may produce almost identical results, or
may produce noticeably different results. If the results are different,
which method would you say is the correct or preferred method... average
first, or average last?
When calculations were done by hand or by hand and slide rule, I would
say we pretty much always averaged the measurements then plugged that
average through the calculation. However, with the advent of
spreadsheets it is trivial to crank through the full calculation with
each trial, then average all the calculated results.
For my initial general physics lab I have students perform an air-track
experiment in which they examine the reproducibility of launching the
air-track glider with a rubber-band system using a magnetic release.
They launch the glider 15 times, measure the delta-t to travel between
two photogates, and calculate the velocity by dividing the delta-x
between the photogates by the delta-t. Almost all students use MS-Excel
to calculate the average and standard deviation. Some average the
delta-t values first; some calculate individual velocities and average
those.
In the limit that the standard deviation approaches zero, the two
results are identical. With the rubber-band launch system, a typical
set of 15 trials yields a standard deviation of about 2% in the measured
times. A particular calculated velocity with this type of spread in the
delta-t values is 32.92 cm/s if you average the velocities, or 32.91
cm/s if you average the times then calculate the velocity. This 0.03%
difference is hardly worth worrying about.
This year for the first time, I thought it would be fun for the students
to launch the glider by simply giving it a push with their hand. This
should impress them with how difficult it is to get reproducible
velocities by just pushing the glider. One student's result, which
appears typical, was a 22% standard deviation in the measured times.
When the velocity is calculated from the hand-push data the result is
37.23 cm/s if the velocities are averaged, or 35.55 cm/s if the times
are averaged first. This difference of almost 5% was easily noticed by
a student who calculated the result both ways with his spreadsheet. He
then proceeded to ask me which result was the correct result.
Upon reflection I realize this has not come to my attention earlier for
three reasons. (1) We generally don't do experiments in a manner
yielding a standard deviation as bad as 22%. (2) Prior to six or seven
years ago students did not use spreadsheets. (3) Even when students
have used spreadsheets an individual student generally does not
calculate the result both ways and compare them. Well... it all came
together today.
Since the standard deviation of the 37.23 cm/s result (and the 35.55
cm/s result) is about 22% or about 8 cm/s, it doesn't matter much... but
the student wanted to report the result as either 37.2 cm/s +- 8 cm/s or
as 35.6 cm/s +- 8 cm/s and he wondered which one would be more proper.
Comments?
Michael D. Edmiston, Ph.D.
Professor of Physics and Chemistry
Bluffton University
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu
_______________________________________________
Phys-L mailing list
Phys-L@electron.physics.buffalo.edu https://www.physics.buffalo.edu/mailman/listinfo/phys-l