Re: Q: Arithmetic, Harmonic, Geometric, etc., Means

[Robert J. MacG. Dawson]

Stanley110 wrote:
> 
> Ladies and Gentlemen,
> 
> What is the physical significance or meaning regarding a manufacturing process
> whose output over an extended period of time has the same value for the
> arithmetic, geometric and harmonic mean of a property, its purity,  for
> example? ... > Or if any two are same and the third is different?

If any two are the same, the significance is that the output is
CONSTANT. For instance, for the arithmetic and geometric means:

(A) if n=2, AM = GM => x_1 = x_2.

Proof: sqrt(x_1 x_2) = (x_1 + x_2)/2
  <=>   x_1 x_2 = x_1^2/4 + x_1 x_2/2 + x_2^2/4
  <=>0= x_1^2/4 - x_1 x_2/2 + x_2^2/4
  = (x_1 - x_2)^2
  <=>   x_1 = x_2

(B) if n is a power of 2, by induction all the x_i are equal.

(C) even if n isn't a power of 2 a bit of algebra shows that
the same result holds (forgive me for omitting the details 
but it's late in the day and I have to get home to my family.)

Similar results hold for the other pairs of means, left as an
exercise (quite a pleasant one) for the curious.

If any of the data differ, there are inequalities (proved as above)
that show that the AM > GM > HM in all cases.

Analogous results hold for distributions.

-Robert Dawson


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Q: Arithmetic, Harmonic, Geometric, etc., Means

[Stanley110]

Ladies and Gentlemen,

What is the physical significance or meaning regarding a manufacturing process
whose output over an extended period of time has the same value for the 
arithmetic, geometric and harmonic mean of a property, its purity,  for
example?

What is the physical significance or meaning if the different means are not the
same?

Or if any two are same and the third is different?

Or if the different means agree for one property associated with the
manufacturing process and not another?

And can you direct me to literature or texts that address this subject?

Please reply to this newsgroup and to me at <[EMAIL PROTECTED]>.

Thank you.

Stan Alekman


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