# Talk:Harmonic mean

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## Image label incorrect?

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)

Yes, you are wrong, just at the end of your calculation: ${\displaystyle H={\frac {2}{(1/a)+(1/b)}}}$, that is, 1/H is the arithmetic mean (rather than the sum!) of 1/a and 1/b. Boris Tsirelson (talk) 15:53, 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)

## stats comment

added the definition when frequency is not unary. Nova67 16:16, 12 June 2005 (UTC)

I don't understand. Are you saying that you added such a definition? If so, there's really no need to mention it here; the article history will show that. Are you saying that such a definition needs to be added? Please explain what you mean. Ruakh 14:27, 15 Jun 2005 (UTC)

## ”Draining Pool” example

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 (talkcontribs) 00:31, 25 August 2009 (UTC)

## Electrical Resistance?

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)

The total resistance is given by the harmonic mean divided by the number of resistors, not just the harmonic mean. The resistance should then be around 24 ohms. 128.84.45.4 20:55, Jul 06, 2004 (UTC)
Exactly. The average resistance is 48 ohms, while the equivalent resistance is 24 ohms. (I don't think "average resistance" is a term any physicist/electrician/electrical engineer would use, since it would have different meanings for resistors in parallel as for resistors in series, but that's how this article is using the term, and I think that's fine.) Ruakh 02:32, 8 Mar 2005 (UTC)
Shouldn't that read arithmetic mean resistance rather than average?. Also resistance should strictly be impedance Smidoid 12:18, 23 Feb 2008 (UTC)
See my section on "taking the average of ratios", The ratios are voltages/currents which is the definition of resistance. In parallel, the voltages are all the same, but the currents are different. So the problem falls in the case of denominators unknown which implies use of the harmonic mean. We obtain the "ratio of averages" the voltage average is fixed at V. But what we need is the ratio of V to total current so we must multiply the average current denominator by N (the number of resistors) to get total current (same as dividing the "average of ratios" by N). Thus we get the value of the equivalent resistance V/I(total). Note that we can do this without assigning any numerical values to voltage and currents.David S. Lawyer 10:23, 23 November 2015 (UTC) — Preceding unsigned comment added by Dlawyer (talkcontribs)

## Negative Numbers?

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)

The article explicitly restricts the notion of harmonic mean to the positive reals. If it didn't, then we'd have the bigger issue of finding the harmonic mean of -1 and 1 (since 2/(1/-1 + 1/1) = 2/(-1 + 1) = 2/0, which is not defined). Ruakh 02:32, 8 Mar 2005 (UTC)

## Removed example.

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:

===An example===
* An experiment yields the following data: 34,27,45,55,22,34 To get the harmonic mean
# How many items? There are 6. Therefore n=6
# What is the sum on the bottom of the fraction? It is 0.181719152307
# Get the reciprocal of that sum. It is 5.50299727522
# To get the the harmonic mean multiply that by n to get <U>33.0179836513</U>
--

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)

## Average speed

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)

## Deriving arithmetic and geometric means when n>2

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) .

The article gives the relationship that for just two numbers, H = G2/A. This relationship doesn't hold for more than two numbers, and there's no corresponding relationship for more than two numbers: for any n > 2 and any A and G, there are infinitely many possibilities for H, unless A = G, in which case A = G = H. Ruakh 01:58, 8 March 2006 (UTC)
In the example area, it's shown that for pumps p1 p2 p3 ... working to empty the same pool, it takes ${\displaystyle {\frac {{p_{1}}\cdot {p_{2}}\cdot {p_{3}}}{p_{1}+p_{2}+p_{3}}}}$ hours to empty it, I believe this is the part you're mentioning? I think this only works for n==2. Assume you have n pumps which take 2 hours to empty it. A) for n==1, this gives ${\displaystyle {\frac {2}{2}}=1hour}$. B) for n==2 this gives ${\displaystyle {\frac {{2}\cdot {2}}{2+2}}=1hour}$ (correct) C) For n==1000 we get ${\displaystyle {\frac {2^{1000}}{2000}}=5.36*10^{297}hours}$. For this problem, it makes more sense to have it be 2/n for this problem. If the pool is 2m gallons, they drain it in 2 hours, giving a rate of m gallons per hour, then the rate at which n pumps can drain a pool of 2m gallons is 2m/mn=2/n. The general version seems to be exactly the harmonic mean over n (as intended), I.E.
${\displaystyle {\frac {H}{n}}={\frac {1}{{\frac {1}{a_{1}}}+{\frac {1}{a_{2}}}+\cdots +{\frac {1}{a_{n}}}}}={\frac {1}{\sum _{i=1}^{n}{\frac {1}{a_{i}}}}}}$

Cassiline (talk) 19:59, 13 April 2008 (UTC)

## Trigonometric example

I added this part and I want to thank the person who did the editing. It really flows well.
This example only work with the double angle tangent identity. I tried ${\displaystyle tanA=tan(3x)}$ and the formula failed to work out. It will fail to work out for any ${\displaystyle tan(nx)}$ where ${\displaystyle n/=2}$.

Opinionhead (talk) 20:09, 10 May 2011 (UTC)

## Omission - Relationship with other means.

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)

Thank you. It must have been a typo. I have read the reference and I cannot find this inequality in this paper. Im afraid I do not have an explanation for its inclusion. Well spotted. DrMicro (talk) 11:14, 22 April 2012 (UTC)

## "Theoretical values" section is confused

Currently the subsection "Theoretical values" of the section "Statistics" reads

Theoretical value
The variance of the harmonic mean is[2]
${\displaystyle \operatorname {Var} \left({\frac {1}{x}}\right)={\frac {m\left[\operatorname {E} (1/x-1)\right]}{nm^{2}}}}$
where m is the arithmetic mean of the reciprocals, x are the variates, n is the population size and E is the expectation operator. Asymptotically E(1 / x) is distributed normally.
The mean of the sample m is also distributed normally with variance s2.
${\displaystyle s^{2}={\frac {m[\operatorname {E} (1/x-1)]}{m^{2}n}}}$

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)

## Harmonics / history / music?

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)

## Taking an average of ratios(rates). New article? Add to "Rate" article?

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)

We already have the use of harmonic mean in electronics given in the article. I'm not sure what you are suggesting. There are lots of applications where a good understanding of ratio is required to calculate an appropriate average, but the method will vary according to the situation and type of weightings given. Dbfirs 10:59, 23 November 2015 (UTC)""
I'm suggesting that we need an article giving a few rules for finding the mean of rates. Many "situations" have a lot in common and an utilize common principles to determine what type of mean to use (arithmetic or harmonic) and what weights to use.
Would this new article be just a list of methods for different situations? Are there many practical situations where the geometric mean is used? You could create it in draft space so that we can see what it would look like. Dbfirs 08:55, 24 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)

That sounds appropriate. Doubling the size will be no problem because it is too short as it stands. Thank you for letting us know. Dbfirs 21:52, 25 November 2015 (UTC)

## Need proof of inequality

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)

Really, this is another special case of Generalized mean inequality. Boris Tsirelson (talk) 19:48, 11 June 2016 (UTC)

## Finance example is Not even wrong and subsection should be removed or heavily revised

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.

### Incorrect: unweighted harmonic mean

${\displaystyle P/E={\frac {1}{1/20+1/1000}}\approx 19.6}$

### Correct: ratio of aggregates, equivalent to market-cap-weighted harmonic mean

${\displaystyle P/E={\frac {\100\ billion+\1\ billion}{\5\ billion+\1\ million}}\approx 20.2}$

11:50, 20 April 2016‎ 198.180.162.5 (talk)‎

The finance section is correct, but under the assumption that it is an equally weighted portfolio; otherwise, you need to use the weighted harmonic mean. So you are right, I have made the change to clarify that. Also note that in your example of the calculation of the unweigthed harmonic mean, you forgot to divide the denominator by 2, which of course leads to a wrong P/E.7804j (talk) 09:28, 29 September 2016 (UTC)

## "In trigonometry" section

The section "In trigonometry" says

In the case of the double-angle tangent identity, if the tangent of an angle A is given as a/b, then the tangent of 2A is the product of (1) the harmonic mean of the numerator and denominator of tan A and (2) the reciprocal of (the denominator minus the numerator of tan A).
In general, the double-angle formula can be written as
${\displaystyle \tan(2A)={\frac {2ab}{a+b}}\cdot {\frac {1}{b-a}}}$

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:

• Reference : Anderson, James E & Neary, J Peter, 2014. "Revenue Tariff Reform," CEPR Discussion Papers 9838, C.E.P.R. Discussion Papers.

ExpertIdeasBot (talk) 17:56, 27 June 2016 (UTC)

The weighted harmonic mean seems to be the correct way to average P/E to me, but I agree that the section was misleading because it was referring to the unweighted harmonic mean. I have clarified the section and added an example. The weighted arithmetic average is definitely not the right way to calculate the P/E of the portfolio though, so I think you are wrong on that point. 7804j (talk) 14:23, 29 September 2016 (UTC)
• ^ Jean WH (1984) The harmonic mean and other necessary conditions for stochastic dominance. J Finance 39 (2) 527
• ^ Zelen M (1972) Length-biased sampling and biomedical problems. In Biometric Society Meeting, Dallas, Texas