Talk:Generalized mean

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It appears that this article should be renamed to Power mean and have Generalized mean redirect to Power mean instead of having Power mean be a redirect for Generalized mean. Google show 34 million results for power mean and only 3.3 million results for generalized mean. Furthermore, many of the latter results are "mirrors" of the wikipedia article.--b4hand 15:17, 30 Apr 2005 (UTC)

Your results are off. If you Google for "power mean" and "generalized mean" in quotes, then you only get 11,800 and 8,370 results, respectively; however, many sites use the phrase "power mean" in unrelated ways ("Does political power mean economic power?"), driving up the former. (I'm sure a few sites also use "generalized mean" in unrelated ways, but it seems much more commonly the case for "power mean.") My best guess is that "power mean" is only somewhat more popular than "generalized mean," and the current presence of generalized f-means in this article makes me loath to move it wholesale to power mean without a stronger reason. Ruakh 21:31, 30 Apr 2005 (UTC)

In that case, shouldn't this page be named "generalised mean" with an 's', to parallel the spelling of generalised f-mean (which is apparently the victor after multiple changes)? I don't know the WP rules for US vs. UK spelling, but I assume you want consistency between closely-related articles. 63.150.32.42 01:29, 19 April 2006 (UTC)

1 Generalized Mean with Exponent 0
2 Incorrectly using the name "Generalized Mean"
3 Generalized Weighted Mean?
4 Simpler

[edit] Generalized Mean with Exponent 0

I have tried to reproduce the claim that $\lim_{s \rarr t} M \left ( s \right )$ , but it doesn't seem to work out. Perhaps it's due to rounding errors, but I find $M \left ( 0 \right )$ seems to most closely approach the geometric mean at about $1 \times 10^7$ . Can somebody explain this, please, with calculations? Keep it as simple as possible. --72.140.146.246 02:53, 4 June 2006 (UTC)

You mean, the claim that $\lim_{t \rarr 0} M\left( t \right) \rarr G$ ?

If we assume that $a_1, a_2, \ldots, a_n$ are all positive real numbers (and hence that $M \left( t \right)$ is positive for all nonzero real

t

) — a reasonable assumption, since otherwise the geometric mean is undefined — then the claim is equivalent to the statement that $\lim_{t \rarr 0} \left( \exp \left( \ln \left( M \left( t \right) \right) \right) \right) \rarr G$ . Replacing $M \left( t \right)$ with its definition, applying the law of logarithms that turns an exponent inside to a factor outside, applying l'Hôpital's rule once, and doing some algebra, you can confirm this limit. (I'd write the whole thing out, but the wiki markup is not very quick-and-easy for those of us who don't much use T_EX. If you can't figure it out, comment here and I'll try to type it up for you.)

Ruakh 00:49, 5 June 2006 (UTC)

Thanks. I tried this, and got as far as $\lim_{t \rarr 0} \left ( \exp \left ( {1 \over t} \ln \left ( {1 \over n} \sum_{i=0}^{n} a_i^t \right ) \right ) \right )$ , but I couldn't see where to apply l'Hopital's rule. Could you enlighten me here? --72.140.146.246 17:39, 6 June 2006 (UTC)

Yup, so far so good. Now,

exp

is continuous over $\mathbb{R}$ , so $\lim_{t \rarr 0} \left ( \exp \left ( {1 \over t} \ln \left ( {1 \over n} \sum_{i=0}^{n} a_i^t \right ) \right ) \right )$ is the same as $\exp \left ( \lim_{t \rarr 0} \left ( {1 \over t} \ln \left ( {1 \over n} \sum_{i=0}^{n} a_i^t \right ) \right ) \right )$ . Inside the limit, both the numerator — $\ln \left ( {1 \over n} \sum_{i=0}^{n} a_i^t \right )$ — and the denominator —

t

— go to 0 as $t \rarr 0$ (do you see why?), so you can apply l'Hôpital's rule. Ruakh 19:16, 6 June 2006 (UTC)

Okay, I did this and came up with $\exp \left ( {1 \over n} \sum_{i=1}^{n} \ln a_i \right )$ , which is the same result as using ln for the generalised f-mean (I used i=0 above by mistake; it should be i=1). I was about to ask how to prove this is equivalent to the geometric mean, but after visiting that article, I figured it out for myself.

Thanks for your help. --72.140.146.246 23:36, 6 June 2006 (UTC)

P.S. I'm wondering whether this should be included in the article?

You're welcome. And I don't know if it should be in this article; it takes up a lot of space. Maybe it should go in its own article, with both this article and Geometric mean linking to it? Ruakh 02:31, 7 June 2006 (UTC)

[edit] Incorrectly using the name "Generalized Mean"

The use of the name "Generalized Mean" in this case is absolutely incorrect, the true "Generalized Mean" that certainly embraces ALL the MEANS (Arithmetic, Harmonic, golden, geometric, arithmonic, Power Mean, etc.) is the Rational Mean.

201.249.205.33 00:25, 27 December 2006 (UTC)Ing. Domingo Gomez Morin

[edit] Generalized Weighted Mean?

What about a generalized WEIGHTED mean? It's more general, and all the weighted means are derived from it.
Adar Weidman, January 2007.

[edit] Simpler

I think this article should be a little less technical. The generalized mean is not such an obscure topic in mathematics that a non-mathematician would never need to look it up. Maybe an introduction with basics would make things easier to understand, and the rest of the article may be kept intact. Goldencako 03:10, 19 October 2007 (UTC)

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Talk:Generalized mean

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[edit] Generalized Mean with Exponent 0

[edit] Incorrectly using the name "Generalized Mean"

[edit] Generalized Weighted Mean?

[edit] Simpler

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