Talk:Generalized mean

Name and structure

This article is about 95% about power means, with an assertion that some people call them "generalized means" (certainly true, but it is clear from below that there are many other things called generalized means). About 5% of the article is about a link to generalized f-means, but they have their own article.

I propose:

• Generalized mean should be a disambiguation page, with pointers to power mean, Lehmer mean, and f means. Also rational mean, if that's a real thing (the link doesn't seem to work).
• This page should be called power mean; it can be rewritten slightly to fit

JoDu987 (talk) 15:28, 20 April 2012 (UTC)

Earlier top comments

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)

Generalized Mean with Exponent 0

I have tried to reproduce the claim that $\lim _{s\rightarrow t}M\left(s\right)$, 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 $\lim _{t\rightarrow 0}M\left(t\right)\rightarrow G$?

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 $\lim _{t\rightarrow 0}\left(\exp \left(\ln \left(M\left(t\right)\right)\right)\right)\rightarrow G$. 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 $\lim _{t\rightarrow 0}\left(\exp \left({1 \over t}\ln \left({1 \over n}\sum _{i=0}^{n}a_{i}^{t}\right)\right)\right)$, 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 $\lim _{t\rightarrow 0}\left(\exp \left({1 \over t}\ln \left({1 \over n}\sum _{i=0}^{n}a_{i}^{t}\right)\right)\right)$ is the same as $\exp \left(\lim _{t\rightarrow 0}\left({1 \over t}\ln \left({1 \over n}\sum _{i=0}^{n}a_{i}^{t}\right)\right)\right)$. 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 $t\rightarrow 0$ (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)

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

it's all a matter of convention. you can always generalize further (while it doesn't neccesarily make sense). —Preceding unsigned comment added by 78.8.67.67 (talk) 19:39, 14 February 2010 (UTC)

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.

• I would expect that the weighted version would divide by the sum of the weights, but it does not. Is the definition, as presented in the article at the moment, correct? -- IbexNu (talk) 22:58, 31 March 2018 (UTC)

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)

Mean and norm

The concepts of norm and generalized mean I find are very similar with the difference that mean is normalised by sample size, but I am confused as to why one does not mention the other. I known one is stats and the other is vector calculus, but can they be safely interlinked or would that open a can of worms in terminology? --Squidonius (talk) 04:25, 12 November 2010 (UTC)

Typo in "Inequality between any two power means" section

The exponent is inverted on the LHS of the second to last equation (after applying $f(x)$). The $(p/q)$^th root should be taken, not the $(q/p)$^th root. Blandant (talk) 17:30, 16 June 2011 (UTC)

History

Does anyone know the history of the development of the generalized mean? — Preceding unsigned comment added by 45.44.78.201 (talk) 17:15, 22 January 2019 (UTC)

Proof of lim p -> 0

Hello, I don´t understand why in this expression

${\frac {\sum _{i=1}^{n}w_{i}x_{i}^{p}\ln {x_{i}}}{\sum _{i=1}^{n}w_{i}x_{i}^{p}}}$

$\sum _{i=1}^{n}w_{i}x_{i}^{p}$ does not get cancelled out and in the next step the denominator just disappears:

$\sum _{i=1}^{n}w_{i}\ln {x_{i}}$

--TranslationTalent (talk) 22:37, 10 August 2019 (UTC)

Ok, I figured that the proof in this form only holds for $\sum _{i=1}^{n}w_{i}=1$. I think we can use L´Hopital here because of $\infty ^{0}\cdot \infty$. --TranslationTalent (talk) 13:47, 11 August 2019 (UTC)

I'm not sure what you're asking really. Everything in the proof looks fine. –Deacon Vorbis (carbon • videos) 14:45, 11 August 2019 (UTC)

@Deacon Vorbis: I have written on your talk page but you can answer here too if you want. L´Hopital only applies for 0/0 or infinity/infity. In "ln(sum(w*x^p))/p" p goes to zero, but the numerator goes to ln(sum(w)) which is not zero. IMO l´hopital can not be applied because of this. --TranslationTalent (talk) 15:04, 11 August 2019 (UTC)

$\textstyle \sum w_{i}$ is 1, so the denominator is 0. Again, the proof is fine. –Deacon Vorbis (carbon • videos) 16:09, 11 August 2019 (UTC)

@Deacon Vorbis: Ok, thanks. It´s kind of embarassing that I missed that.

PS: But this means that it´s basically unprovable if the sum of the weights is greater than 1? --TranslationTalent (talk) 17:56, 11 August 2019 (UTC)

ok,ok. I have understood it. The proof is totaly correct. Thanks for being so patient. I will add a little extension to the proof. which also holds for

$\textstyle \sum w_{i}\neq 1$. --TranslationTalent (talk) 18:06, 11 August 2019 (UTC).

I've reverted again because it didn't make any sense in the context of the rest of the article; please be more careful. –Deacon Vorbis (carbon • videos) 19:04, 11 August 2019 (UTC)

@Deacon Vorbis: No problem, I thought that it directly follows from the proof, but as it´s not sensical at this place I will add the second proof to the main geometric mean article where it belongs. --TranslationTalent (talk) 20:56, 11 August 2019 (UTC)

Kolmogorov Mean 94.137.182.204 (talk) 21:38, 16 April 2023 (UTC)

What is the meaning of 'generalised'?

It is not clear what 'generalised' means in this context? The first statement is: "In mathematics, generalized means (or power mean or Hölder mean from Otto Hölder) are a family of functions for aggregating sets of numbers." Will it make any difference if 'generalised' is removed? — Preceding unsigned comment added by 192.38.134.15 (talk • contribs) 08:18, 16 April 2023 (UTC)

It is generalised in the sense that it is a more general meaning than the usual, more specific, meaning of "mean". This is a standard meaning of "generalised" in mathematics: a general concept covers a wider range of cases that a related specific concept. JBW (talk) 16:02, 11 June 2023 (UTC)