Editing Quasi-arithmetic mean (section)

== Properties ==
The following properties hold for <math>M_f</math> for any single function <math>f</math>:

'''Symmetry:''' The value of  <math>M_f</math> is unchanged if its arguments are permuted.

'''Idempotency:''' for all ''x'',  <math>M_f(x,\dots,x) = x</math>.

'''Monotonicity''': <math>M_f</math> is monotonic in each of its arguments (since  <math>f</math> is [[Monotonic function|monotonic]]).

'''Continuity''':  <math>M_f</math> is continuous in each of its arguments  (since  <math>f</math> is continuous).

'''Replacement''': Subsets of elements can be averaged a priori, without altering the mean, given that the multiplicity of elements is maintained. With <math>m=M_f(x_1,\dots,x_k)</math> it holds:

:<math>M_f(x_1,\dots,x_k,x_{k+1},\dots,x_n) = M_f(\underbrace{m,\dots,m}_{k \text{ times}},x_{k+1},\dots,x_n)</math>

[[Partition of a set|'''Partitioning''']]: The computation of the mean can be split into computations of equal sized sub-blocks:<math>
M_f(x_1,\dots,x_{n\cdot k}) =
  M_f(M_f(x_1,\dots,x_{k}),
      M_f(x_{k+1},\dots,x_{2\cdot k}),
      \dots,
      M_f(x_{(n-1)\cdot k + 1},\dots,x_{n\cdot k}))
</math>

'''Self-distributivity''': For any quasi-arithmetic mean <math>M</math> of two variables: <math>M(x,M(y,z))=M(M(x,y),M(x,z))</math>.

'''Mediality''': For any quasi-arithmetic mean <math>M</math> of two variables:<math>M(M(x,y),M(z,w))=M(M(x,z),M(y,w))</math>.

'''Balancing''': For any quasi-arithmetic mean <math>M</math> of two variables:<math>M\big(M(x, M(x, y)), M(y, M(x, y))\big)=M(x, y)</math>.

'''[[Central limit theorem]]''' : Under regularity conditions, for a sufficiently large sample, <math>\sqrt{n}\{M_f(X_1, \dots, X_n) - f^{-1}(E_f(X_1, \dots, X_n))\}</math> is approximately normal.<ref>{{cite journal|last=de Carvalho|first=Miguel|title=Mean, what do you Mean?|journal=[[The American Statistician]]|year=2016|volume=70|issue=3|pages=764‒776|doi=10.1080/00031305.2016.1148632|url=https://zenodo.org/record/895400|hdl=20.500.11820/fd7a8991-69a4-4fe5-876f-abcd2957a88c|s2cid=219595024 |hdl-access=free}}</ref>
A similar result is available for Bajraktarević means and deviation means, which are generalizations of quasi-arithmetic means.<ref>{{Cite journal |last1=Barczy |first1=Mátyás |last2=Burai |first2=Pál |date=2022-04-01 |title=Limit theorems for Bajraktarević and Cauchy quotient means of independent identically distributed random variables |url=https://link.springer.com/article/10.1007/s00010-021-00813-x |journal=Aequationes Mathematicae |language=en |volume=96 |issue=2 |pages=279–305 |doi=10.1007/s00010-021-00813-x |issn=1420-8903}}</ref><ref>{{Cite journal |last1=Barczy |first1=Mátyás |last2=Páles |first2=Zsolt |date=2023-09-01 |title=Limit Theorems for Deviation Means of Independent and Identically Distributed Random Variables |url=https://link.springer.com/article/10.1007/s10959-022-01225-6 |journal=Journal of Theoretical Probability |language=en |volume=36 |issue=3 |pages=1626–1666 |doi=10.1007/s10959-022-01225-6 |issn=1572-9230|arxiv=2112.05183 }}</ref>

'''Scale-invariance''': The quasi-arithmetic mean is invariant with respect to offsets and scaling of <math>f</math>:  <math>\forall a\ \forall b\ne0 ((\forall t\ g(t)=a+b\cdot f(t)) \Rightarrow \forall x\ M_f (x) = M_g (x)</math>.