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I was doing a thought experiment on the harmonic mean. Let's say we have a sequence of 4 numbers:

100, 110, 90, 100

The arithmetic mean (AM) of this sequence is 100 but the harmonic mean (HM) is 99.4975. Let's imagine that the initial and final values remain constant but the middle ones get more and more extreme.

100, 120, 80, 100

The AM is still 100 but the HM is now 97.9592

100, 130, 70, 100

AM = 100, HM = 95.2880

and so on

I think I understand why this happens. The AM is symmetric, if you will. So an equal movement above the mean will cancel out an equal movement below it. However, the HM is more skewed toward the lowest value in the sequence. As such, it only takes one low value to bring the HM down. And with each step, we have a lower minimum and thus a lower HM. This is basically a crude verbal proof. Is there a more rigorous, mathematical one?

I'm also interested to know if there's a relationship between the HM and the fluctuation (I guess the arithmetic standard deviation is a good way to represent this). I mean, there has to be, as higher fluctuations mean higher likelihood of lower values and thus a lower HM, right?

Basically, HM = f(σ)?

Feynstein 100
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  • https://en.wikipedia.org/wiki/QM-AM-GM-HM_Inequalities. More generally: https://en.wikipedia.org/wiki/Generalized_mean#Generalized_mean_inequality – Dmitry Jan 17 '24 at 19:18
  • There is as much a correlation between the harmonic mean and the standard deviation as there is between the harmonic mean and any of the other means. Which means if you change one, you will see a change in the other, but the size of one puts very little resriction on the size of the other. – Paul Sinclair Jan 18 '24 at 19:58
  • @PaulSinclair But in my example, the AM remains the same while the HM decreases. Doesn't that contradict what you're saying? – Feynstein 100 Jan 19 '24 at 16:43
  • @Dmitry Thank you but I was already aware of that and it's not what I'm looking for. Like I mentioned in the post, I'm looking for a relationship between the HM and the SD – Feynstein 100 Jan 19 '24 at 16:44
  • I was a little careless in my phrasing. I did not mean to imply changing one mean would require other means to also change. It does not contradict my point, which is that there is almost no correlation between them. – Paul Sinclair Jan 19 '24 at 16:48
  • @PaulSinclair Ah okay. But what about AM x HM = GM^2? That's only for a dataset of 2 but the wikipedia article says this can be generalized to any dataset. Although, the formula for that is much more complicated. – Feynstein 100 Jan 22 '24 at 16:58
  • As a general rule of thumb, if you have $n$ independent equations, you can solve them for up to $n$ unknowns. Each of the means supplies an equation for the data points. So if you have three or fewer data points, you can find the original data (or nearly so) just from the means. With only two data points, the three equations actually have excess information, requiring a condition to be satisfied by the means in order for a solution to exist. You've given the condition. I do not know the "generalized" formula, but my bet is it involves more values than just these 3 means. – Paul Sinclair Jan 22 '24 at 17:28
  • @PaulSinclair That's quite interesting. Here's the link to the Wikipedia article: https://en.wikipedia.org/wiki/Harmonic_mean#Relationship_with_other_means I'd love to hear your thoughts on it – Feynstein 100 Jan 27 '24 at 11:51
  • Okay - the harmonic mean is expressible in terms of the geometric and arithmatic means - but of different sequences. You cannot in general find the harmonic mean of ${x_n}$ in terms of just the arithmetic and geometric means of ${a_n}$. Instead the arithmetic mean is of a different sequence that can be derived from ${a_n}$. – Paul Sinclair Jan 27 '24 at 13:11

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