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Re: [Phys-l] Curve fitting versus averaging [was question onaveraging]



Brian Whatcott said:

I am visualizing taking an average of the differences
given by posn 1 - posn 0, posn2 - posn1
and so on supposing the errors are randomly distributed about
the mean value. This seems to provide a smaller error that dividing the
difference of (posn n - posn 0) / n.

Averaging the differences can't provide a smaller error because it provides exactly the same result.

Explicitly write down the algebra for the average:

[(P1-P0) + (P2-P1) + (P3-P2) + (P4-P3) + (P5-P4)] / 5

Notice that all the positions between P0 and P5 appear in the sum twice; once as positive and once as negative.

Remove the inner parentheses ( ) and do the sum, and you get [P5 - P0] / 5

You wasted your time obtaining data points P1 through P4, and you wasted your time doing the calculation using them.

The result rests on the first and last points, and the error is totally dependent on how well you established the first and last points.

Indeed, you can fabricate any outrageous data you want for the positions of node1 through node4 and you still get the correct result if the positions of node0 and node5 are correct. Conversely, if one of the positions of node0 and node5 is incorrect, your result will be lousy no matter how well you established the positions of nodes 1 through 4.


Michael D. Edmiston, Ph.D.
Professor of Chemistry and Physics
Bluffton University
1 University Drive
Bluffton, OH 45817
419.358.3270
edmiston@bluffton.edu