meanMean32002-06-04 09:47:302009-09-14 20:44:19Definition% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLoosely speaking, a mean is a way to describe a collection of numbers such
that the mean in some sense describe the ``average'' entry of these numbers.
The most familiar mean is the arithmetic mean, and unless otherwise noted, by mean,
we always mean the arithmetic mean.
\subsubsection*{Example}
The mean of the numbers $\{1,\,2,\,\ldots,\,n\}$ is $\frac{n+1}{2}$.
Mathematically, we define a mean as follows:
\subsubsection*{Definition}
A \emph{mean} is a function $f$ whose domain is the collection of
all finite multisets of $\mathbb{R}$ and whose codomain is $\mathbb{R}$,
such that
\begin{itemize}
\item $f$ is a homogeneous function of degree 1.\, That is, if $\{x_1, \ldots, x_n\}$ is a multiset, then
$$
f(\{ \lambda x_1, \ldots, \lambda x_n\}) = \lambda f(\{x_1, \ldots, x_n\}),\quad \lambda\ge 0.
$$
\item For any set $S = \{x_1,\ldots,x_n\}$ of real numbers,
$$ \min\{x_1,\ldots,x_n\} \leq f(S) \leq \max\{x_1,\ldots,x_n\}.$$
\end{itemize}
Pythagoras identified three types of means: the \PMlinkname{arithmetic mean}{ArithmeticMean}, the geometric
mean, and the harmonic mean. However, in the sense of the above definition,
there is a wealth of ther means too. For instance, the minimum function and maximum
functions can be seen as ``trivial'' means. Other well-known means include:
\begin{itemize}
\item median,
\item mode,
\item generalized mean
\item power mean
\item Lehmer mean
\item arithmetic-geometric mean,
\item arithmetic-harmonic mean,
\item harmonic-geometric mean,
\item root-mean-square (sometimes called the quadratic mean),
\item identric mean,
\item contraharmonic mean,
\item Heronian mean,
\item Cesaro mean,
\item \PMlinkname{maximum function, minimum function}{MinimalAndMaximalNumber}
\end{itemize}