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Gaussian Curve Summations
Name: Jay
Status: educator
Grade: 9-12
Location: MA
Country: USA
Date: April 2011
Question:
I have noticed that when similar Gaussian curves are placed
next to each other with 50% overlap between the curves, the summed
curves give a constant value equal to the curve peak. Is there a name
for this phenomenon, and is it useful in science?
Replies:
You have to be careful that what appears to be a real
mathematical result is analytically true and not a consequence of the
scale of the graphing. For example, let us look at two Gaussian curves,
G1 = (1/2pi)^1/2 x
exp(-1/2(x-1)^2) and G2 = (1/2pi)^1/2 x exp(-1/2(x+1)^2) and their
weighted sum Gt(x)= 2 x G1 + 2 x G2. These are two Gaussian curves
with mean values
(+/-) 1 and standard deviation = 1. That is, they are equally "fat". If
you graph Gt(x) between (+/-)0.5, G(t) appears to be a constant very
close to unity. However, if you increase the values of 'x' to (+/-)2
you see that the values of Gt = (+/-)0.5. Certainly not a constant. It
is a flaw in our scaling of Gt. Or put another way, we are pulling G1
and G2 apart, with no change in the standard deviation. You could make
this trickier if you do not hold the standard deviation constant, but
make the standard deviation proportional to the mean, so that the
curves get "fatter" as you move G1 and
G2 apart.
By no means does this minimize your observation. On the
contrary, your observation is thought provoking, and it illustrates an
important concept. No matter how "close" you get to a number, that does
not mean that finite precision results in equality. That is, "2" does
not equal "1.99999999" nor does it equal "2.00000001".
Sometimes theoretical physicists choose exact values for
experimental constants. So the speed of light in a vacuum, for example,
may be assigned a value of "1" (exactly). This avoids the pesky problem
of how well that speed is known. It also takes some of the clutter out
of complicated derivations.
Vince Calder
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Update: June 2012
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