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August 12th, 2011, 04:39 AM  #11  
Senior Member Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3  Re: Arithmetic mean: Why use its computation as its definiti Quote:
The reason that textbooks don't spell all this out is that it's usually assumed that you can figure it out yourself, but I agree it would probably help most students to walk through it. If the computation is the definition, then there is some sort of assumption being made, rightly or wrongly, that the reader doesn't need assistance to deduce the intuitive concept motivating it.  
August 12th, 2011, 05:52 AM  #12 
Senior Member Joined: Feb 2010 Posts: 711 Thanks: 147  Re: Arithmetic mean: Why use its computation as its definiti
Perhaps the OP has in mind this: For a set of data , the mean is that value of which minimizes 
August 12th, 2011, 10:15 AM  #13  
Newbie Joined: Aug 2011 Posts: 9 Thanks: 0  Re: Arithmetic mean: Why use its computation as its definiti Quote:
 
August 12th, 2011, 11:52 AM  #14 
Newbie Joined: Aug 2011 Posts: 9 Thanks: 0  Re: Arithmetic mean: Why use its computation as its definiti
I have time. Let's begin with the idea that the mean is the estimate, â, under “optimal conditions”, so we’ll call it the “optimal estimate”. Let’s consider the trivial situation where n = 1. What is the optimal estimate of the value a? The answer is obviously â = a. Why? Because it’s the value of the estimate when the error or residual between the estimate and the value (ie, â – a) is driven to zero. And that’s an essential fact of the mean’s foundation: driving the residual(s) to zero. Another essential fact is that residuals are additive. With the nontrivial (and most common) situation where n > 1, the optimal estimate of the values {a1, a2, a3, … an} occurs when the sum of the residuals is driven to zero, that is, ?(â – ai) = 0. Working through the derivation: ?(â – ai) = 0 ?â – ?ai = 0 ?â = ?ai nâ = ?ai â = (1/n)?ai So, that’s the mathematical foundation for the arithmetic mean. Residuals are driven to zero. Residuals are additive. Driving the sum of the residuals to zero results in the optimal estimate, aka the mean. 
August 13th, 2011, 04:05 AM  #15 
Newbie Joined: Aug 2011 Posts: 9 Thanks: 0  Re: Arithmetic mean: Why use its computation as its definiti
Yes mrtwhs, the arithmetic mean can be characterized as "least squares"; however, the least squares methodology unnecessarily introduces its own problems. By squaring residuals, their signs are obliterated, which in turn, requires the minimizing (ie, "least") of the sum of the squares. So by necessity, "least" must follow "squares" (in the computational scheme of things). I'm also saying here (although off topic) that, as with the arithmetic mean, one can derive the computing algorithms of the Least Squares Method without squaring or minimizing anything. Driving the three inherently fundamental characterizing values of a y = m?x + b relationship to zero allows one to derive its computing algorithms. So the arithmetic mean and the socalled "least squares" method are related to one another. 
August 13th, 2011, 11:24 AM  #16 
Senior Member Joined: Jul 2011 Posts: 245 Thanks: 0  Re: Arithmetic mean: Why use its computation as its definiti
Evroe, thank you for the enlightening analysis. That's actually very brilliant, assuming it's correct (I see no reason it should not be).

August 13th, 2011, 11:53 AM  #17  
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,963 Thanks: 1148 Math Focus: Elementary mathematics and beyond  Re: Arithmetic mean: Why use its computation as its definiti Quote:
 
August 13th, 2011, 06:57 PM  #18  
Newbie Joined: Aug 2011 Posts: 9 Thanks: 0  Re: Arithmetic mean: Why use its computation as its definiti Quote:
 
August 13th, 2011, 07:15 PM  #19  
Newbie Joined: Aug 2011 Posts: 9 Thanks: 0  Re: Arithmetic mean: Why use its computation as its definiti Quote:
 
August 14th, 2011, 04:49 AM  #20  
Senior Member Joined: Jul 2011 Posts: 245 Thanks: 0  Re: Arithmetic mean: Why use its computation as its definiti Quote:
The method of squaring, cubing, and 'n'ingfinding roots of equations manuallyhas been forgotten. I recall there being a method similar to division, but I also recall a common sensical method. So here is my challenge to you: What are those two methods? (Bonus points if applied uniquely and specially to fractional powers!)  

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