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Re: [Phys-L] LSF Slope Versus Average of Single Point Slopes



I don't see why the result would be all that surprising. On the other hand, I'm also quite sure that it isn't true. For instance, a standard linear least squares fit to the points

X Y
2 1
3 3
10 7.756757

passes within 10^-7 of the origin but its slope is 0.783 compared to the average value of Y/X which is 0.759.

John Mallinckrodt
Cal Poly Pomona

On May 4, 2012, at 1:49 PM, Donald Polvani wrote:

The following result puzzles me. I have found that for a straight line,
with a zero y-intercept, simply averaging the slope values obtained from
single points produces equivalent results to obtaining the slope from a
least squares fit (LSF). In our beginning physics lab course, we have been
studying how appropriate data transformations can produce a straight line
whose slope can be used to compute the desired quantity. The example I
worked out involves determining Planck's constant from measurements of the
turn-on voltage of LEDs emitting known wavelengths. The students plot
turn-on voltage on the y axis and 1/wavelength on the x-axis to obtain the
slope m = hc/e. Given the values of c (speed of light) and e (electron
charge) they compute a value of h (Planck's constant) from m (the slope of
the line). Some of the students want to get m by simply using a single
plotted point (i.e. m = y/x). This, of course, usually produces a poor
value (unless the point happened to fall near the ideal line). I decided to
make a Monte Carlo simulation, with zero mean Gaussian errors in both the
wavelengths and voltages, to demonstrate to the students the value of doing
a LSF. From the simulation, the LSF slope and value of h were usually much
superior to using any SINGLE point slope. However, when I AVERAGED the
point determined slopes (over 6 different turn-on voltages for 6 different
wavelengths), I got an average slope equivalent in accuracy to the LSF
result. This was true in my Monte Carlo simulation after 100, 500, and 5000
trials.



I thought that the LSF technique would produce the optimum result. But the
average point slope result seems to be just as good and much simpler to
calculate. This result must be particular to the case where the y-intercept
is zero (as otherwise the point slope approach, which assumes the line goes
through the origin, isn't even a correct model of the situation);
nonetheless, I'm surprised and puzzled.



Can anyone enlighten me as to why simply averaging single point derived
slopes appear to work as well as the LSF?



Don



Dr. Donald G. Polvani

Adjunct Faculty, Physics

Anne Arundel Community College



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