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Questions tagged [harmonic-mean]

The harmonic mean is a kind of mean (or average) for positive scalar data. The harmonic mean can be expressed as the reciprocal of the arithmetic mean of the reciprocals. See https://en.wikipedia.org/wiki/Harmonic_mean

The harmonic mean of n numbers $x_1 \dots x_n, x_i > 0$ is:

$H = \frac{n}{\sum{\frac{1}{x_i}}}$

Perhaps the most common use of H is when you are averaging rates or averages.

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Harmonic mean minimizes sum of squared relative errors

I am looking for a reference where it is proven that the harmonic mean $$\bar{x}^h = \frac{n}{\sum_{i=1}^n \frac{1}{x_i}}$$ minimizes ( in $z$) the sum of squared relative errors $$\sum_{i=1}^n \left( \frac{(x_i - z)^2}{x_i}\right).$$
Martin Van der Linden
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Can the standard deviation be calculated for harmonic mean?

Can the standard deviation be calculated for the harmonic mean? I understand that the standard deviation can be calculated for arithmetic mean, but if you have harmonic mean, how do you calculate the standard deviation or CV?
user3353
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Why we don't use weighted arithmetic mean instead of harmonic mean?

I wonder what is an intrinsic value of using harmonic mean (for instance to calculate F-measures), as opposed to weighted arithmetic mean in combining precision and recall? I am thinking that weighted arithmetic average could play the role of…
olga
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Harmonic mean with zero value

How does harmonic mean handle zero values? what would the harmonic mean of {3, 4, 5, 0} be since $1/0=\infty$?
Dez Udezue
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In calculating the F-measure with precision and recall, why is the harmonic mean used?

The article for F-measure in Wikipedia says: The traditional F-measure or balanced F-score (F1 score) is the harmonic mean of precision and recall: $F_1=2\times\frac{precision \times recall}{precision+recall}$ Why is the harmonic mean used in…
e9t
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What is the difference between 'Laplace approximation' and 'Modified harmonic mean'?

this question is about Bayesian and computational statistics. I am learning them right now, I have two very common output from my software, one is Laplace approximation and the other is Modified harmonic mean. Both of them are used for approximation…
Flying pig
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Is the harmonic mean the maximum likelihood estimator for some common continous distribution's parameter?

If $y$ is a vector of continuous data the arithmetic mean is the maximum likelihood estimator for $\mu$ when assuming $y \sim \text{Normal}(\mu,\sigma)$ (not uniquely, of course). The geometric mean is the maximum likelihood estimator of $\mu$ when…
Rasmus Bååth
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Harmonic mean of random variables

Is there an analytic solution/approximation to the PDF/CDF and mean of an harmonic mean of random variables? I'm wondering about beta distributions ($\beta$) or truncated exponential distributions ($E$)? Generally, what is the PDF/CDF and mean of $X…
Diogo Santos
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convergence of geometric mean/harmonic mean

Does any one know papers regarding the convergence of geometric mean or harmonic mean in probability, parallel to central limit theorem?
Nidhin
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The Harmonic Mean of the Likelihood, n sample normal-normal case

This is very related to Radford Neal's awesome blogpost, but his example uses a single data point x, and I want to know how to replicate the results using $n$ data points. This is essentially what my question will "boil down" to. Below I'll try to…
shitoushan
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Why is F-score called "F-score"?

All other algorithm scores have names that make sense, but the F-score is just "F". Was the letter chosen at random?
T. Spikes
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What is the proper method for calculating the Coefficient of Variation of a rate?

I have been asked to evaluate the variation between movement rates of fish within different systems (rates are in km/day). I have between 25-100 samples in each system. I understand that when reporting the average rate for a given system, it is…
Von
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Harmonic mean for averaging predictions?

I have 14 different predictions about each observation in my dataset and I want to compute an overall prediction for each observation based on these. Would it be better to use the harmonic mean instead of the arithmetic mean?
pir
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Median versus Harmonic Mean As Log Normal Data Summary

I have a set of data that follows a lognormal distribution (it is fixed-distance, variable-speed situation https://stats.stackexchange.com/a/23130/55305). I am trying to summarize the data in a single value and I would normally expect to use the…
cjbarth
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convergence in probability of harmonic mean

I have $X_1,...,X_n$ iid as gamma($\alpha,\beta$). Defined also is the harmonic mean $Y_n=\frac{n}{\sum{x_i^{-1}}}$. I'm trying to figure out if $Y_n$ converges in probability to some constant $c$. My intuition says c=0, but I do not really know any…
Mark
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