Amaton Posted May 4, 2013 Share Posted May 4, 2013 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? Link to comment Share on other sites More sharing options...
John Cuthber Posted May 5, 2013 Share Posted May 5, 2013 It's a right angle with the rather odd symbol given in the 4th picture here http://en.wikipedia.org/wiki/Right_angle Link to comment Share on other sites More sharing options...
Amaton Posted May 8, 2013 Author Share Posted May 8, 2013 (edited) 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 Link to comment Share on other sites More sharing options...
John Cuthber Posted May 8, 2013 Share Posted May 8, 2013 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. Link to comment Share on other sites More sharing options...
Amaton Posted May 8, 2013 Author Share Posted May 8, 2013 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. Link to comment Share on other sites More sharing options...
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