The harmonic mean $$ of two positive numbers $x_1$ and $x_2$ can
be written as a weighted average $=w_1 x_1+w_2 x_2$ where the
weights are $w_1=x_2/(x_1+x_2)$ and $w_2=x_1/(x_1+x_2)$. As such is
very responsible and respectful of different opinions. Can this
property be extended to more than two numbers?
Let me add that another interesting way of looking at the harmonic mean h of two positive numbers is to consider that
h=a-tau(a,a)/a
where a is the arithmetic mean and tau(a,a) the arithmetic variance of the two numbers.
Dear Prof. Germano,
This is an interesting question.
Let's denote h(x_1,x_2,...,x_n) the harmonic mean of n positive numbers x_1, x_2,...,x_n. And g(x_1,x_2,...,x_n;x_j) is the harmonic mean of (n-1) numbers x_1, x_2,...,x_(j-1),x_(j+1),...,x_n, i.e. those numbers without x_j.
Then h=w_1*x_1+w_2*x_2+...+w_n*x_n with
w_i = g(x_1,x_2,...,x_n;x_i)/(g(x_1,x_2,...,x_n;x_i)+(n-1)*x_i).
If n=2, then we have w_1=x_2/(x_2+x_1), w_2=x_1/(x_1+x_2).
Best Wishes,
Zhenhua Xia
Dear Prof. Xia
Thanks for your very interesting contribution! To look at the harmonic mean
as a weighted average is plenty of suggestions. I really like to collect them.
Another one is consider that the weight in the arithmetic mean is in any case
1/n. In the harmonic mean h your contribution as xi is \/n times h/xi. In other words the harmonic mean compensates your weakness.
The harmonic mean of x_1 and x_2 is x= 2/(1/x_1+1/x_2). Another formulation is x = w_1 x_1 + w_2 x_2; w_1 = 1/(x_1(1/x_1+1/x_2)); w_2 = 1/(x_2(1/x_1+1/x_2)). Therefore, the harmonic mean itself is a weighted arithmetic mean. The harmonic mean of a set of (usually positive) numbers is the reciprocal of the arithmetic mean of the reciprocals of the numbers.
Correct. What is discussed here is the "physical" meaning of these weights
in order to have more insight on the harmonic mean and more generally on the weighted averages.
In fact, the arithmetic mean is always the largest, and the harmonic mean is always the smallest. However, equality holds if a set of numbers are equal. Physically, the harmonic mean signifies the minimum magnitude of a set of numbers. When evaluating the local turbulence intensity, the harmonic mean of turbulent kinetic energy (k) has a noticeable impact on transition modeling. The use of harmonic mean emphasizes the small upwind values over the large local and downstream values. This is essential in cases where a turbulent front needs to be pushed from an upstream to a downstream location. The harmonic mean of k in the formulation of local turbulence intensity avoids multiple solutions dependent on initial conditions, which can occur with transition models.
Dear All
Could I have your comments on the following mixed
harmonic/arithmetic mean $m$ given by $2/m=1/a+1/h$, where $a$ is
the arithmetic mean and $h$ the harmonic mean of n positive numbers
$x_i$ ? Which is its meaning and the possible applications?
Dear Prof. Germano
You ask for the Ref. , which uses harmonic average in transition modeling. An Algebraic LCTM Model for Laminar–Turbulent Transition
Prediction by Menter et al.: Journal: Flow, Turbulence and Combustion
https://doi.org/10.1007/s10494-022-00336-8
Dear Professor Rahman
many thanks for your information. I will read it carefully.
Dear All
Always in the philosophy of constructing appropriate weighted averages
could you give me some information on the following weighted
average m. Let x_i be a set of positive n numbers, and let w_i be the weights given by
w_i = α/n + βx_i/(na) ; sum_i w_i = 1 ; sum_i w_i x_i=m
where α and β are constants and a is the arithmetic average. We should have a mitigated or augmented average given by
m= a +β v/a
where v is the variance. Are they considered in some application?