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Harmonic Mean

Date: 12/15/96 at 11:10:40
From: Gary Hasul
Subject: When and Why to use the Harmonic Mean

I have searched to net to find an explanation of why anyone would use 
the harmonic mean. Most sites just define it or state the formula.  
Some state that it is useful for speed and velocity problems.  Your 
site talks about the mean as a "representative" value for a set of 
data. Can you give me an example stating why the harmonic mean is 
"more representative" than the arithmetic mean?

Date: 12/17/96 at 12:42:35
From: Doctor Bombelli
Subject: Re: When and Why to use the Harmonic Mean

The Greeks were into means, and the harmonic mean in particular.  Here 
is the Greek definition from Porphyry in the "Commentary on Ptolemy's 
Harmonics": The subcontrary mean, which we call harmonic, is such that 
by whatever part of itself the first term exceeds the second, the 
middle term exceeds the third by the same part of the third.  

That is, b is the harmonic mean between a and c if (a-b)/a = (b-c)/c.  
Can you get b = 2ac/(a+c) out of this?

The Greeks (Pythagoreans specifically) used these means in music: 
holding strings in certain ratios and plucking them, for example.  
Iamblichus says that the harmonic mean "was then called subcontrary, 
but which was renamed harmonic by the circle of Archytas and Hippasus, 
because it seemed to furnish harmonius and tuneful ratios."

There are lots of other neat properties of means.  Here is a sampling:

1. If b is the harmonic mean between a and c, then 1/c -1/b = 1/b -1/a 
   so that 1/c, 1/b, 1/a form an arithmetical progression.

2. Let s be the side of a square inscribed within a triangle and 
   having one side lying along the base of the triangle.  s is half 
   the harmonic mean of the base of the triangle and the altitude of 
   the triangle on the base.

3. Let s be the side of a square inscribed within a right triangle and 
   having one angle coinciding with the right angle of the triangle.  
   s is half the harmonic mean of the legs of the triangle.

4. If s, a, b are chords of 1/7, 2/7, and 3/7 of the circumference of 
   a circle, then s is half the harmonic mean of a and b.

5. If a car travels at the rate of r miles per hour from A to B and 
   then returns at the rate of s miles per hour, the average rate for 
   the trip is the harmonic mean of r and s.

I hope that helps a little.

-Doctor Bombelli,  The Math Forum
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Date: 12/16/2002 at 13:16:26
From: Katie
Subject: What is the definition of harmonic mean?

Can you give me a layman's definition for harmonic mean? It would also 
be helpful if you could give me an equation.

Thank you.

Date: 12/18/2002 at 20:32:33
From: Doctor Schwa
Subject: Re: What is the definition of harmonic mean?

Hi Katie,

To find the mean, or the "arithmetic" mean, you add up all the numbers 
and divide by how many numbers there are.

The other means, like "quadratic" or "harmonic," are similar.

For the quadratic mean, you square all the numbers, then take the 
usual average (add them up and divide by how many numbers there are), 
and then take the square root (un-squaring).

For the harmonic mean, which was your question, you first take the 
reciprocal of each number, then take the usual average, then take the 
reciprocal again (because reciprocal is the same as "un-reciprocal" 
... the operation is its own inverse).

For example, if you want the harmonic mean of 10 and 20, you first 
take 1/10 and 1/20, find their average, which is 3/40, and then take 
the reciprocal of that, 40/3.

In algebra, the harmonic mean h of two numbers a and b is 

   1 / ( (1/a + 1/b) / 2),

or in other words 

   1/m = 1/2 (1/a + 1/b).

I hope that helps clear things up!

- Doctor Schwa, The Math Forum
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