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August 12th, 2011, 04:39 AM   #11
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Re: Arithmetic mean: Why use its computation as its definiti

Quote:
Originally Posted by Evroe
Why do teachers/profs and textbooks define the arithmetic mean using its method of computation?
The average of two numbers is usually taken to mean the number precisely halfway between them on the number line. This typical definition is called the "arithmetic mean". So if you start at x, then the distance to y is y-x, and then half that distance is ˝(y-x). So the average of x and y is half way from x to y, or x+˝(y-x), which simplifies to ˝(x+y).

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.
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August 12th, 2011, 05:52 AM   #12
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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

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August 12th, 2011, 10:15 AM   #13
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Re: Arithmetic mean: Why use its computation as its definiti

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Originally Posted by Pell's fish
Do you have such definition?
Yes, I found the definition 25 years ago, and I'm sorry, but I'm involved in a family emergency that began on Wednesday, but I will reply again here as soon as possible.
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August 12th, 2011, 11:52 AM   #14
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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 non-trivial (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.
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August 13th, 2011, 04:05 AM   #15
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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 so-called "least squares" method are related to one another.
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August 13th, 2011, 11:24 AM   #16
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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).
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August 13th, 2011, 11:53 AM   #17
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Re: Arithmetic mean: Why use its computation as its definiti

Quote:
Originally Posted by Evroe
?(â – ai) = 0

?â – ?ai = 0

?â = ?ai

nâ = ?ai

â = (1/n)?ai
nâ = ?ai if and only if all â are equal.
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August 13th, 2011, 06:57 PM   #18
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Re: Arithmetic mean: Why use its computation as its definiti

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Originally Posted by greg1313
Quote:
Originally Posted by Evroe
?(â – ai) = 0

?â – ?ai = 0

?â = ?ai

nâ = ?ai

â = (1/n)?ai
nâ = ?ai if and only if all â are equal.
Sorry, yes, I should have specified that, for n > 1, â is a single value meant to characterize the entire set of a values collectively.
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August 13th, 2011, 07:15 PM   #19
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Re: Arithmetic mean: Why use its computation as its definiti

Quote:
Originally Posted by CherryPi
Evroe, thank you for the enlightening analysis. That's actually very brilliant, assuming it's correct (I see no reason it should not be).
Thanks, CherryPi. The arithmetic mean derivation was easy enough to remember, but not the "least squares" derivation, although it's in my old notebook somewhere. I was discouraged over the years by many college/university professors who staunchly averred that the LSM could be derived only by squaring the residuals and then minimizing their sum. I finally found a professor who confirmed that the method that I had worked out was already known and that, as I also found, the LSM and the arithmetic mean were related to one another.
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August 14th, 2011, 04:49 AM   #20
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Re: Arithmetic mean: Why use its computation as its definiti

Quote:
Originally Posted by Evroe
Quote:
Originally Posted by CherryPi
Evroe, thank you for the enlightening analysis. That's actually very brilliant, assuming it's correct (I see no reason it should not be).
Thanks, CherryPi. The arithmetic mean derivation was easy enough to remember, but not the "least squares" derivation, although it's in my old notebook somewhere. I was discouraged over the years by many college/university professors who staunchly averred that the LSM could be derived only by squaring the residuals and then minimizing their sum. I finally found a professor who confirmed that the method that I had worked out was already known and that, as I also found, the LSM and the arithmetic mean were related to one another.
Since we're talking about things that have been forgotten by the populous:

The method of squaring, cubing, and 'n'ing--finding roots of equations manually--has 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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