Editing Average (section)

==Summary of types==
{{see also|Mean#Other means|Central tendency#Solutions to variational problems}}

{| class="wikitable"
|-
! Name !! Equation or description !! As solution to optimization problem
|-
| [[Arithmetic mean]] || <math>\bar{x} = \frac{1}{n}\sum_{i=1}^n x_i  =  \frac{1}{n} (x_1 + \cdots + x_n)</math> || <math>\underset{x \in \mathbb{R}}{\operatorname{argmin}}\, \sum_{i=1}^n (x - x_i)^2</math>
|-
| [[Median]] || A middle value that separates the higher half from the lower half of the data set; may not be unique if the data set contains an even number of points || <math>\underset{x \in \mathbb{R}}{\operatorname{argmin}}\, \sum_{i=1}^n |x - x_i|</math>
|-
| [[Geometric median]] || A [[rotation (mathematics)|rotation]] [[invariant (mathematics)|invariant]] extension of the [[median]] for points in <math>\mathbb{R}^d</math> || <math>\underset{\vec{x} \in \mathbb{R}^d}{\operatorname{argmin}}\, \sum_{i=1}^n ||\vec{x} - \vec{x}_i||_2</math>
|-
| [[Tukey median]] || Another rotation invariant extension of the median for points in <math>\mathbb{R}^d</math>—a point that maximizes the [[Tukey depth]] || <math>\underset{\vec{x} \in \mathbb{R}^d}{\operatorname{argmax}}\, \underset{\vec{u} \in \mathbb{R}^d}{\operatorname{min}} \, \sum_{i=1}^n \left(\begin{cases}1, \text{ if }(\vec{x}_i-\vec{x})\cdot\vec{u} \geq 0 \\ 0, \text{ otherwise}\end{cases}\right)</math>
|-
| [[Mode (statistics)|Mode]] || The most frequent value in the data set || <math>\underset{x \in \mathbb{R}}{\operatorname{argmax}}\, \sum_{i=1}^n \left(\begin{cases}1, \text{ if }x = x_i \\ 0, \text{ if }x \neq x_i\end{cases}\right)</math>
|-
| [[Geometric mean]] || <math>\sqrt[n]{\prod_{i=1}^n x_i} = \sqrt[n]{x_1 \cdot x_2 \dotsb x_n}</math> || <math>\underset{x \in \mathbb{R}_{> 0}}{\operatorname{argmin}}\, \sum_{i=1}^n (\ln(x) - \ln(x_i))^2,\qquad \text{if }x_i > 0\,\forall\, i \in \{1,\dots,n\}</math>
|-
| [[Harmonic mean]] || <math>\frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}}</math> || <math>\underset{x \in \mathbb{R}_{\neq 0}}{\operatorname{argmin}}\, \sum_{i=1}^n \left(\frac{1}{x} - \frac{1}{x_i}\right)^2</math>
|-
|[[Contraharmonic mean]]
|<math>\frac{x_1^2 + x_2^2 + \cdots + x_n^2}{{x_1} +{x_2} + \cdots + {x_n}}</math>
|<math>\underset{x \in \mathbb{R}}{\operatorname{argmin}}\, \sum_{i=1}^n x_i(x - x_i)^2</math>
|-
| [[Lehmer mean]]|| <math>\frac{\sum_{i=1}^n x_i^p}{\sum_{i=1}^n x_i^{p-1}}</math>|| <math>\underset{x \in \mathbb{R}}{\operatorname{argmin}}\, \sum_{i=1}^n x_i^{p-1}(x - x_i)^2</math>
|-
| [[Quadratic mean]]<br />(or RMS) || <math>\sqrt{\frac{1}{n} \sum_{i=1}^{n} x_i^2} = \sqrt{\frac{1}{n}\left(x_1^2 + x_2^2 + \cdots + x_n^2\right)}</math>|| <math>\underset{x \in \mathbb{R}_{\geq 0}}{\operatorname{argmin}}\, \sum_{i=1}^n (x^2 - x_i^2)^2</math>
|-
| [[Cubic mean]]|| <math>\sqrt[3]{\frac{1}{n} \sum_{i=1}^{n} x_i^3} = \sqrt[3]{\frac{1}{n}\left(x_1^3 + x_2^3 + \cdots + x_n^3\right)}</math>|| <math>\underset{x \in \mathbb{R}_{\geq 0}}{\operatorname{argmin}}\, \sum_{i=1}^n (x^3 - x_i^3)^2,\qquad \text{if }x_i \geq 0\,\forall\, i \in \{1,\dots,n\}</math>
|-
| [[Generalized mean]]|| <math>\sqrt[p]{\frac{1}{n} \cdot \sum_{i=1}^n x_{i}^p}</math>|| <math>\underset{x \in \mathbb{R}_{\geq 0}}{\operatorname{argmin}}\, \sum_{i=1}^n (x^p - x_i^p)^2,\qquad \text{if }x_i \geq 0\,\forall\, i \in \{1,\dots,n\}</math>
|-
| [[Quasi-arithmetic mean]]|| <math> f^{-1}\left(\frac{1}{n} \sum_{k=1}^{n}f(x_k) \right)</math>|| <math>\underset{x \in \operatorname{dom}(f)}{\operatorname{argmin}}\, \sum_{i=1}^n (f(x) - f(x_i))^2,\qquad \text{if } f</math> is [[Monotonic function|monotonic]]
|-
| [[Weighted arithmetic mean|Weighted mean]] || <math>\frac{ \sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i} = \frac{w_1 x_1 + w_2 x_2 + \cdots + w_n x_n}{w_1 + w_2 + \cdots + w_n}</math> || <math>\underset{x \in \mathbb{R}}{\operatorname{argmin}}\, \sum_{i=1}^n w_i(x - x_i)^2</math>
|-
| [[Truncated mean]] || The arithmetic mean of data values after a certain number or proportion of the highest and lowest data values have been discarded
|-
| [[Interquartile mean]] || A special case of the truncated mean, using the [[interquartile range]].  A special case of the inter-quantile truncated mean, which operates on quantiles (often [[decile]]s or [[percentile]]s) that are equidistant but on opposite sides of the median.
|-
| [[Midrange]] || <math>\frac{1}{2}\left(\max x + \min x\right)</math> || <math>\underset{x \in \mathbb{R}}{\operatorname{argmin}}\, \underset{i \in \{1,\dots,n\}}{\operatorname{max}}\, |x - x_i|</math>
|-
| [[Winsorized mean]] ||  Similar to the truncated mean, but, rather than deleting the extreme values, they are set equal to the largest and smallest values that remain
|-
| [[Medoid]] || A representative object of a set <math>\mathcal X</math> of objects with minimal sum of dissimilarities to all the objects in the set, according to some dissimilarity function <math>d</math>. || <math>\underset{y \in \mathcal X}{\operatorname{argmin}} \sum_{i=1}^n d(y, x_i)</math>
|}

Even though perhaps not an average, the <math>\tau</math>th [[quantile]] (another [[summary statistic]] that generalizes the median) can similarly be expressed as a solution to the optimization problem

:<math>\underset{x \in \mathbb{R}}{\operatorname{argmin}}\, \sum_{i=1}^n \max\big((1-\tau)(x_i - x),\, \tau(x - x_i)\big) = \underset{x \in \mathbb{R}}{\operatorname{argmin}}\, \sum_{i=1}^n \big(|x - x_i| + (1 - 2\tau)\,x\big)</math>,

which aims to minimize the total [[quantile regression#Sample quantile|tilted absolute value loss]] (or ''quantile'' loss or ''pinball'' loss).

The [[table of mathematical symbols]] explains the symbols used below.{{explain|reason=Below? There is nothing below this sentence in this section.|date=November 2024}}