Construction of means

[Amaton]

From the Wikipedia article on Pythagorean means, a geometric construction...

 

 

where [math]a[/math] and [math]b[/math] are two numbers represented as line segments. Their arithmetic, geometric, harmonic, and quadratic means are shown expectedly.

 

Now, the harmonic mean is a sector of the line segment from endpoints of A and G. What is the significance of that angle, i.e. is it a right angle? Because otherwise I can see H taking on any other value.

 

Also, the Wikipedia article also gives the following properties for the Pythagorean means:

 

[math]M(n,n,...,n)=n[/math]

[math]M(bx_1,...,bx_n)=bM(x_1,...,x_n)[/math]

[math]M(...,x_i,...,x_j,...)=M(...,x_j,...,x_i,...)[/math]

[math]\mbox{min}(x_1,...,x_n)\le M(x_1,...,x_n)\le \mbox{max}(x_1,...,x_n)[/math]

 

It then introduces the quadratic mean for the ordering [math]\mbox{min}<H<G<A<Q<\mbox{max}[/math]. However, the four above properties also seem to hold for the quadratic mean. Is this correct?

[John Cuthber]

It's a right angle with the rather odd symbol given in the 4th picture here

http://en.wikipedia.org/wiki/Right_angle

[Amaton]

It's a right angle with the rather odd symbol given in the 4th picture here

http://en.wikipedia.org/wiki/Right_angle

 

Thanks. That is a rather odd notation.

 

Also, for a data set containing only non-negative real numbers, doesn't [math]Q(x_1,...,x_n)[/math] hold the same four properties described above? Also considering the geometric construction, one wonders if the quadratic mean should be considered a Pythagorean mean, as it seems so closely related.

Edited May 8, 2013 by Amaton

[John Cuthber]

I think it would only be Pythagorean if Pythagoras knew about it.

However, as far as I can tell, yes it does hold those 4 properties.

[Amaton]

I think it would only be Pythagorean if Pythagoras knew about it.

 

I see the Pythagorean means as a more mathematical-based grouping, rather than only historical, but that is sensible.