Talk:Geometric mean

When to use the geometric mean plagiarism?

Not sure how to note this, but parts of the When to use Geometric means is cribbed without reference from http://www.math.utoronto.ca/mathnet/questionCorner/geomean.html --128.135.213.42 19:03, 27 March 2007 (UTC)

Apparently this section has been removed some time ago. I have just added a new section about applications with a simple example. Ole Laursen (talk) 10:40, 10 October 2009 (UTC)

I've got a couple of issues with the example given.

1. Why would you expect proportional growth in orange production from a single tree? I don't think the growth rate in the number of oranges produced by a given tree is proportional to the number of oranges produced the previous year. Barring a really good reason for this, I'll change the example to something about bacteria or rabbits or something else commonly found in texts and courses on exponential growth.
2. Because you truncate the annual growth rates, you end up getting the wrong number in the end. Starting at 100 and increasing to 300 in 3 years, the geometric mean is the cube root of 3, or 1.44225, not 1.443. Adam Lein (talk) 21:52, 19 January 2011 (UTC)

Representations for the G-Mean

I think it's easier to understand the geometric mean as an nth root of a product than as a product raised to the power of a reciprocal, so I added an alternative description in the beginning of the article. Should this become the only description present? Rbarreira 17:13, 15 January 2006 (UTC)

Yes, I think that it should so I took the liberty of removing it. The older less standard version read: ${\bigg (}\prod _{i=1}^{n}a_{i}{\bigg )}^{1/n}=(a_{1}\cdot a_{2}\dotsb a_{n})^{1/n}$

Comments to added picture

Nice picture, but what happens when $n>2$? Bob.v.R 14:21, 17 April 2006 (UTC)

Unclear sentence

I have no idea what this sentence means: This geometric interpretation of the mean is very likely what gave it its name. aevea (talk) 15:02, 11 January 2008 (UTC)

DELETED! In addition to not being referenced, I'm pretty sure the name actually came from the arithmetic/geometric distinction that is essentially synonymous with additive/multiplicative (Cf. the difference between an arithmetic sequence and a geometric one -- the nomenclature there is the same as here). Thus, I'm pretty sure what was there was wrong anyways. ILikeThings (talk) 00:23, 14 August 2008

Alternative definition

The definition that appears to be used in NY State Math B Regents exams textbooks is The geometric mean of two positive numbers $a$ and $b$ is the positive number $x$ for which ${\frac {a}{x}}={\frac {x}{b}}$. Perhaps this formulation of the definition can be added? 24.164.188.164 (talk) 21:55, 22 May 2009 (UTC)

This is a nice definition. It emphasizes that like the normal (arithmetic) mean of two numbers is half-way between them regarding addition (you add something to the small number to get the mean, and add it again to get the large number), the geometric mean of two numbers is half-way between them regarding multiplication (you multiply the small number by something (x in your formula) to get the mean, and multiply it again to get the large number. This makes it easier to understand what the "oranges" example in the text comes from, and perhaps also the explanation about the logarithmic average, and that the word "geometric" in this mean means the same thing as it does in geometric sequence. Nyh (talk) 08:38, 8 March 2010 (UTC)

The problem I see for including this "definition" is that it is a special case of describing the geometric mean of two numbers. If used as an example of this special case, it could be helpful; however, I don't see this as intrinsically necessary for the article. Drbb01 (talk) 17:16, 8 August 2013 (UTC)

Note #1 incomplete

Note 1 reads, in part, "The geometric mean only applies to positive numbers in order to avoid taking the root of a negative product..." Pedantically, it should read 'taking the positive root...'. I don't know how to change this, since it seems to be in the first section which has no Edit link. I apologize for my ignorance, and appreciate any help with this minor edit. Trelligan (talk) 19:09, 19 October 2009 (UTC)

I think the point the point is rather that one would not know which root to take, positive, negative or complex. For example, if one had two negative numbers, one would probably want a "mean" value to be negative, while if there wsere one positive and one negative, the product would be negative and hence the formal square root would be purely imaginary (which probably isn't a "good" value for a mean). While one could always work with the geometric mean of the absolute values, it is not really a measure of an average/typical value, in a meaningful sense. (On editing the lead section, either use the "edit this page" tab or there is a user option/preference setting which allows an edit link to appear for the lead section.) Melcombe (talk) 09:29, 20 October 2009 (UTC)

Applications - Benchmark results

This "section" (it's just one short sentence) is extremely ambiguous. It doesn't even explain what its subject is (I had to mouse over the link to realize it was talking about computing benchmarks). It offers no explanation as to why the geometric mean is the "correct" one. It doesn't specify for what types of benchmarks it is the correct one (all of them? or certain types?). For these reasons, I am removing this section. —Preceding unsigned comment added by Borromean-ring (talk • contribs) 00:08, 5 February 2010 (UTC)

Redirect from "mean proportional"

The geometric mean used to be also known by the term "mean proportional".89.242.136.4 (talk) 15:58, 22 March 2010 (UTC)

Reference Problem

I found that that the link http://hdr.undp.org/en/faq-page is no longer acessible and also I am not sure why the UNDP would be involved in explaining Geometric Mean

Illustration is baffling

I find the graphs at the top of the article to be useless. It's not clear what the color shading represents or what is the concept illustrated by these four charts? Mheberger (talk) 17:20, 4 September 2014 (UTC)

Seconded and I rm'ed it. Perhaps someone can think of a plot of illustration that adds to the article, but this one was not doing it. a13ean (talk) 18:20, 4 September 2014 (UTC)

Contested discussion in Properties section

The properties section contains a discussion of why the geometric mean is the appropriate mean for normalized numbers, referring to Flemming and Wallace, 1986. However, this reasoning has been contested by Smith, 1988. I added this reference together with a summary of the paper. Furthermore, these two papers refer to the specific case of averaging computer performance benchmark results, so it is strange that it is in the general Properties section of the Geometric mean article. I would advocate to put this discussion in the Applications section. Seyerman (talk) 10:46, 12 February 2015 (UTC)

However, I find the following sentence misleading:

"Metrics that are inversely proportional to time (speedup, IPC) should be averaged using the harmonic mean."

because the Smith, 1988, paper actually argues against simply taking the harmonic mean of speedup values; it suggests that "an aggregate performance measure such as [...] harmonic mean rate should be calculated before any normalizing is done." — Preceding unsigned comment added by 69.53.236.236 (talk) 08:19, 18 April 2015 (UTC)

Because of the notable relationship of the Pythagorean means I looked at how the harmonic means apply in each table and WP:BOLDed it onto the page. It agrees quite strongly with the position advanced by 69.53.236.236 and probably the one by Seyerman. Please compare that to the following table normalized to Computer C, where the arithmetic and geometric both champion Computer C whereas harmonic champions Computer A again. Warmest Regards, :)—_thecurran Speak your mind ^{my past} 09:20, 6 February 2016 (UTC)

Shall we add the table as normalized to Computer C?

	Computer A	Computer B	Computer C
Program 1	0.05	0.5	1
Program 2	50	5	1
Arithmetic mean	25.025	2.75	1
Geometric mean	1.581 . . .	1.581 . . .	1
Harmonic mean	0.099 . . .	0.909 . . .	1

I think academically this case should be present in the page but I'm wary that it might not flow well enough. Warmest Regards, :)—_thecurran Speak your mind ^{my past} 09:23, 6 February 2016 (UTC)

Significant figures versus decimal places in Properties computer table?

Scientifically it's far more appropriate to communicate fractional components within a related set of numbers out to a consistent number of significant figures rather than to a consistent number of 'decimal places' hence scientific notation but to maintain the article's original flow, all additional data was truncated after the third 'decimal place'. Please find below all four tables rounded out to a maximum of four significant figures.

	Computer A	Computer B	Computer C
Program 1	1	10	20
Program 2	1000	100	20
Arithmetic mean	500.5	55	20
Geometric mean	31.62	31.62	20
Harmonic mean	667.1	70.97	20

	Computer A	Computer B	Computer C
Program 1	1	10	20
Program 2	1	0.1	0.02
Arithmetic mean	1	5.05	10.01
Geometric mean	1	1	0.6325
Harmonic mean	1	0.1980	0.03996

	Computer A	Computer B	Computer C
Program 1	0.1	1	2
Program 2	10	1	0.2
Arithmetic mean	5.05	1	1.1
Geometric mean	1	1	0.6325
Harmonic mean	0.1980	1	0.3636

	Computer A	Computer B	Computer C
Program 1	0.05	0.5	1
Program 2	50	5	1
Arithmetic mean	25.025	2.75	1
Geometric mean	1.581	1.581	1
Harmonic mean	0.09990	0.9091	1

Opinions? Warmest Regards, :)—_thecurran Speak your mind ^{my past} 09:45, 6 February 2016 (UTC)

Error in Properties section

The first table in the Properties section contains two errors in computing the Harmonic mean. The table is currently

	Computer A	Computer B	Computer C
Program 1	1	10	20
Program 2	1000	100	20
Arithmetic mean	500.5	55	20
Geometric mean	31.622 . . .	31.622 . . .	20
Harmonic mean	667.110 . . .	70.967 . . .	20

But it ought to be

	Computer A	Computer B	Computer C
Program 1	1	10	20
Program 2	1000	100	20
Arithmetic mean	500.5	55	20
Geometric mean	31.622 . . .	31.622 . . .	20
Harmonic mean	1.998 . . .	18.182 . . .	20

This changes the validity of the next sentence, ″The arithmetic, geometric, and harmonic means "agree" that computer C is the fastest.″ This affects the entire narrative of the section.

MathatGrace (talk) 19:47, 9 February 2016 (UTC)

It appears I'm the second one to notice such a confusing mistake in the table. Just eyeballing it, one can see that it doesn't satisfy the identity HM <= GM <= AM. I'll fix the table now. 68.226.23.184 (talk) 04:13, 18 February 2016 (UTC)

Example image description ambiguous

The text for the picture currently states, "Geometric mean or mean proportional,[1] in an example with the length l2 is perpendicular to AB." 2 problems: "with the length 12 is ..." is incorrect grammar, someone should fix that to "in an example with the length l2 perpendicular to AB".

But the reason I don't do this is because the premise of the sentence is incorrect. l2 is a length, it can't be perpendicular to anything. l2's line, BC' is clearly parallel to line AB. BC is perpendicular to AB, and has the length l2, but I don't think this connection is clear just by the description. Someone more knowledgeable in geometry should be able to word this better to let the reader know the significance of the picture... --208.99.251.161 (talk) 13:13, 6 September 2017 (UTC)

Thanks for your entry, I hope it is now understandable. Greeting from Munich Petrus3743 (talk) 22:18, 7 September 2017 (UTC)

Should be used only for positive numbers

The entire article should be rewritten under the assumption that the input numbers are all greater than zero, because it makes more mathematical sense. Why ever risk taking the n-th root of a negative or zero number?

Daniel R. Grayson (talk) 22:37, 11 May 2018 (UTC)

@Daniel R. Grayson: If you verified it by a reference (either from Wolfram MathWorld etc.), you may mention this. HaydenWong (talk) 12:45, 17 December 2018 (UTC)

Geometric mean error propagation.

I was wondering, lets say I have a set of independent random variables, each having own mean and variance, and all of them having standard guassian distribution. Having mean and standard deviation estimations, how do I estimate an error (standard deviation) of the geometric mean? I am not talking about Geometric standard deviation, which is different thing. 2A02:168:F609:0:7855:CA84:B8D8:C659 (talk) 18:57, 11 August 2019 (UTC)

Bolzano–Weierstrass weasel

I've tagged a remark in the article as being weasel words, since it suggests using the Bolzano–Weierstrass theorem to prove that the geometric mean is the arithmetic–harmonic mean. While technically true—you could use Bolzano–Weierstrass at a point in such a proof—it is mostly confusing as it glosses over a lot of set-up that would be needed before Bolzano–Weierstrass could be applied. A good argument for showing that the arithmetic–harmonic mean exists is to modify that found in the Arithmetic–geometric mean article.

A minimal improvement would be to instead refer to the Squeeze theorem, since this is more likely to help a reader fill in the missing details, but that still leaves the reader having to demonstrate that $a_{n}-h_{n}$ in fact goes to $0$ (which a Bolzano–Weierstrass proof would also need). 130.243.94.123 (talk) 17:02, 29 March 2021 (UTC)

Kolmogorov Mean

Mean Average, Quatratic and Root mean arr a special cases of Kolmogorov Mean. Sergei Kazariants (talk) 21:44, 16 April 2023 (UTC)