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Is this true?: "in an electrical circuit you have two resistors connected in parallel, one with 40 ohms and the other with 60 ohms, then the average resistance is 48 ohms" 210.224.218.13 15:51, Nov 20, 2003 (UTC)
What if the two numbers are negative? The harmonic mean of -1 and -2 is 2/(-1 + -(1/2)), which equals 2/(-1.5) or 2 * -(2/3) or -4/3, which is larger than the arithmetic mean, which is -1.5 (as compared to the geometric mean being -1.3......). However, the geometric mean would actually be the square root of 2... messed up for negative numbers, I suppose. ugen64 03:26, Dec 16, 2003 (UTC)
added the definition when frequency is not unary. Nova67 16:16, 12 June 2005 (UTC)
I removed the following text that a non-registered user inserted into the article, as its placement and internal organization were confusing and ill-suited to the article:
If anyone wants to rescue it, feel free. It needs to be rewritten, given lead text, and placed appropriately (perhaps in a sidebar; the "ExampleSidebar" template could be used). Ruakh 19:48, 8 September 2005 (UTC)
This statement warrants a little qualification:
if for half the distance of a trip you travel at 40 miles per hour and for the other half of the distance you travel at 60 miles per hour, then your average speed for the trip is given by the harmonic mean of 40 and 60, which is 48; that is, the total amount of time for the trip is the same as if you traveled the entire trip at 48 miles per hour.
it may be true, of course, but not independent of the premises of the problem which are rather subtely expressed only. It is equally true that:
if for half the time of a trip you travel at 40 miles per hour and for the other half of the time you travel at 60 miles per hour, then your average speed for the trip is given by the arithmetic mean of 40 and 60, which is 50; that is, the total distance you covered is the same as if you traveled the entire trip at 50 miles per hour.
While this isn't central to a discussion of harmonic means, as I discovered recently people are easily lulled, by reading such concise statements, into the mistaken belief that the harmonic mean is THE correct manner of averaging independent speed measures. And this, it is not. It is A correct measure given certain premises concerning the problem. Bwechner 05:00, 25 November 2005 (UTC)
The statement as it stands isn't even strictly correct. The harmonic mean of 40 and 60 is hardly an integer -- in fact, it is closer to 49 than it is to 48! 142.162.10.169 ( talk) 01:33, 20 March 2008 (UTC)
The arithmetic mean of the speed example is stated to be 50 in the "Relationship with other means" section, but 40 in the "Examples" section. — Preceding unsigned comment added by 67.249.89.107 ( talk) 15:15, 2 October 2019 (UTC)
OK, so in the article, there is a derived method of finding arithmetic mean and geometric mean when n=2. What about when n=3 or 4? I;m sorry I don't know this- I'm just a high school student at the Illinois Mathematics and Science Academy —The preceding unsigned comment was added by 143.195.150.93 ( talk • contribs) .
Cassiline ( talk) 19:59, 13 April 2008 (UTC)
I don’t believe the pool-draining example highlights a special case, and I suspect the example might warrant clarification or removal. As this is not my area of expertise, I submit the following for your review: The rates provided (4 and 6) are “hours per pool”. The harmonic mean, then, is the average number of hours it would take one pump to empty the pool (4.8 hours). With twice as many pumps working at that average rate, it would of course take only half the time, or 2.4 hours. —Preceding unsigned comment added by Jenburton ( talk • contribs) 00:31, 25 August 2009 (UTC)
Opinionhead ( talk) 20:09, 10 May 2011 (UTC)
I just deleted a statement that cannot be correct, or is at least not properlyy explained/defined. DrMicro wrote that
"If all the real variables in the set are > 0 then H ≥ M2 / m where m is the minimum in the set." [1]
If this were the case (and M is the maximum as defined in the main article above) then the H would be greater than the maximum, meaning that it no longer qualified as a mean. If someone knows the correct statement of this inequality as stated in the paper (it's too late for me to be reading a maths paper right now) feel free to re-add and correct.
I'll message DrMicro about it too.
Cheers,
JPBrod ( talk) 23:28, 21 April 2012 (UTC)
References
Currently the subsection "Theoretical values" of the section "Statistics" reads
Unfortunately the citation for this is a paper at a 1972 conference, so there's no way to check the source to confirm that it's been copied correctly (or even that it was right in the first place since it appears not to have been refereed or published). Moreover, it's been changed in the present article from what it said earlier.
(1) This looks suspicious to me because the variance of the harmonic mean is given as the same expression as the variance of the arithmetic mean of the reciprocals.
(2) It looks suspicious because surely the original source would not have left something with m not cancelled from the numerator and denominator.
(3) It says the variance of the harmonic mean is denoted as var(1/x) where "x are the variates". That doesn't make sense to me -- if 1/x is the harmonic mean, then x is the mean of the reciprocals of the variates, not "the variates". So this expression, even if correct, may refer to the variance of the reciprocals of the variates rather than the variance of the sample harmonic mean.
Does anyone have a source for the true mean and variance of the sample harmonic mean? Duoduoduo ( talk) 15:08, 5 June 2013 (UTC)
Anyone mind if I add a musical (harmonic ratio) example, since that's where the name of the mean comes from? Thanks! -- Michael Scott Cuthbert (talk) 21:26, 19 June 2014 (UTC)
The harmonic mean is sometimes used to take "true' averages of ratios. In many cases, weighting is required. In some cases, only the arithmetic mean is needed and gives the same results as the harmonic mean (since different weightings were used). Shouldn't this (or another article) explain how to do this in general. A formula showing weights is given in this article, but no rules for determining weights. I've formulated this and it's really pretty simple. What one often wants to determine when averaging ratios Ai/Bi (i terms from 1 to N) is (SumAi)/(SumBi) = "Ratio of averages" (both numerator and denominator are divided by N to get an "average" but the two N's cancel out). Let Ri = Ai/Bi, the ratios to average. If one knows only the Ri's and Bi's, then to get the "ratio of averages" one takes the arithmetic mean of the Ri's using the Bi's as weights (the denominator of the ratios). If one knows only the Ri's and Ai's then one uses Ai's as weights and takes the harmonic mean. The sum of the weights must add to 1 so each Ai weight is divided by sumAi. Likewise for Bi's. These rules show how to determine the weights. Proving these rules takes only several lines and is very simple algebra.
For electrical components in series and parallel, there exist implied ratios. For example resistance is voltage/current (a ratio). Another case is where only the Ri's are known. Then one can use the above rules by assuming that all the Ai's (or all the Bi's) are the same number (say 1). This results in an unweighted harmonic mean. David S. Lawyer 09:31, 23 November 2015 (UTC)
I've found a small article Rate (mathematics) and plan to add to it: taking the mean of rates. It might more than double the size of that article. It would be referenced in this article and the Arithmetic mean article. It would show when to use the harmonic mean and when to use the arithmetic mean and what weights to use for finding certain results. If and when I do this, I'll post in this Talk since someone here might be willing to review it, edit it, etc.David S. Lawyer 00:16, 25 November 2015 (UTC)
The article needs a new section entitled "Proof that the harmonic mean is less than the geometric mean", preferably with a source. Or, if there is enough to say, there could be a new article Inequality of harmonic and geometric means, analogous to the article Inequality of arithmetic and geometric means. Loraof ( talk) 16:27, 13 December 2015 (UTC)
I put in a link to an article with a proof. Maybe that will help someone write a new section or article. Loraof ( talk) 17:16, 13 December 2015 (UTC)
The [ reference to "Fairness Opinions", on page 217] presents an example with two equal-sized positions. This is rarely the case in real life and the weighted average should be recommended instead.
Suppose you have a $100-billion firm with a P/E of 20 ($5 billion in earnings). Average this with a $1-billion firm with a P/E of 1000 ($1 million in earnings). The aggregate price/earnings of this combination should be dominated by the large firm.
11:50, 20 April 2016 198.180.162.5 (talk)
The weights used in the article depend only on the investment fraction. There is nothing precisely wrong, but the reference to market capitalization may be misleading (it is used only to calculate the P/E, not the susbsequent weights). — Preceding unsigned comment added by 163.1.165.0 ( talk) 12:35, 24 October 2022 (UTC)
This image is from the article "Harmonic Mean:"
It appears to me, that the length of the purple line H is actually TWICE the harmonic mean for the following reason:
H = (G*G) / A = (a*b) / ( (a+b) / 2 ) = 2 * ( 1 / ( (1/a) + (1/b) ) )
Ergo:
H is TWICE as big as the harmonic mean.
Is this true? Please correct me, if I am wrong.
73.4.14.51 ( talk) 14:19, 6 May 2016 (UTC)
You are right. Thank you for the explanation. — Preceding unsigned comment added by 73.4.14.51 ( talk) 17:21, 10 May 2016 (UTC)
Without doing any calculations, this figure is incorrect by inspection: The rank order of lengths, from the drawing, is Q > A = H > M, not what's asserted (correctly) in the legend (Q>A>G>H). Note that H = A here because both are the radius of the same circle. — Preceding unsigned comment added by Memeri ( talk • contribs) 18:22, 23 November 2022 (UTC)
The section "In trigonometry" says
and then gives a numerical example. I find this pointless—all it says is that the double angle formula does not use the harmonic mean, but can be expressed by starting with the harmonic mean and then adjusting it for the fact that the harmonic mean doesn't give the right answer. So I'm deleting it. Loraof ( talk) 16:31, 12 June 2016 (UTC)
Dr. Anderson has reviewed this Wikipedia page, and provided us with the following comments to improve its quality:
The harmonic mean is the preferable method[dubious – discuss] for averaging multiples, such as the price–earnings ratio, in which price is in the numerator. If these ratios are averaged using an arithmetic mean (a common error), high data points are given greater weights than low data points. The harmonic mean, on the other hand, gives equal weight to each data point.[6] The simple arithmetic mean when applied to non-price normalized ratios such as the P/E is biased upwards and cannot be numerically justified, since it is based on equalized earnings; just as vehicles speeds cannot be averaged for a roundtrip journey.[7]
This discussion is misleading. Neither the arithmetic nor the harmonic mean of P/E ratios is useful for typical financial purposes. A typical concern is with a portfolio of assets with differing P/E ratios. The weighted arithmetic average P/E would typically be used to evaluate the P/E of the portfolio, the weight on each P/E ratio being the proportion of total assets allocated to that individual asset.
The harmonic mean is related to the other Pythagorean means, as seen in the third formula in the above equation. This is noticed if we interpret the denominator to be the arithmetic mean of the product of numbers n times but each time we omit the j-th term. That is, for the first term, we multiply all n numbers except the first; for the second, we multiply all n numbers except the second; and so on. The numerator, excluding the n, which goes with the arithmetic mean, is the geometric mean to the power n. Thus the nth harmonic mean is related to the nth geometric and arithmetic means.
This paragraph should go, with some editing, to immediately follow the equation defining H.
We hope Wikipedians on this talk page can take advantage of these comments and improve the quality of the article accordingly.
Dr. Anderson has published scholarly research which seems to be relevant to this Wikipedia article:
ExpertIdeasBot ( talk) 17:56, 27 June 2016 (UTC)
In Section 3.1 "Two numbers", could it be that A, G, H refer to different concepts in the discussion than in the diagram? If so, someone more mathematically more adept than me will need to step in to make things right.
-- Beatrice57 ( talk) 19:28, 24 August 2020 (UTC)
The statistics section mentions, without introduction, the harmonic means of several continuous probability distributions. There should be a definition of the continuous case somewhere. I suppose it would be something like integral(f(x))/integral(f(x)/x), where f(x) is a PDF that supplies the weights and the integrals are both taken over the whole domain. Elias ( talk) 12:05, 27 November 2020 (UTC)