Means

Given:

• A sequence $x=(x_1,x_2,\dots ,x_n)$ of $n$ real numbers
• A sequence $w=(w_1,w_2,\dots ,w_n)$ of $n$ non-negative weights

Mean	Formula
Arithmetic Mean	$\overline{x}=\frac{1}{n}\sum _{i=1}^n{x}_i$
Geometric Mean	${\left(\prod _{i=1}^n{x}_i\right)}^{\frac{1}{n}}$
Harmonic Mean	$\frac{n}{\sum _{i=1}^n\frac{1}{x_i}}$
Weighted Arithmetic Mean	$\frac{\sum _{i=1}^n{w}_i{x}_i}{\sum _{i=1}^n{w}_i}$
Root-Mean Square (RMS)	$\sqrt{\frac{1}{n}\sum _{i=1}^n{x}_i^2}$
Mid-range	$M=\frac{\max (x)+\min (x)}{2}$
Median	$\tilde{x}=\text{median}(x)=\{\begin{array}{ll}{x}_{\lfloor n/2\rfloor +1}& \text{if }n\text{ is odd}\\ \frac{1}{2}\left(x_{n/2}+x_{n/2+1}\right)& \text{if }n\text{ is even}\end{array}$	given $x$ is non-decreasing
Mode	$\mathrm{mode}(x)=\left\{x_i\in x\mid w_i=\max (w)\right\}$	the set of element(s) in $x$ with the maximum frequency, where $w\in {\mathbb{N}}^n$ are frequencies, and elements in x are distinct

Arithmetic Mean, Geometric Mean, and Harmonic Mean are known collectively as the Pythagorean Means.

Inequalities

• Mean Inequalities: ($HM\le GM\le AM$). $x_1,\dots ,x_n$ are positive reals $\frac{n}{{\sum }_{i=1}^n\frac{1}{x_i}}\le \sqrt[n]{{\prod }_{i=1}^n{x}_i}\le \frac{1}{n}{\sum }_{i=1}^n{x}_i$

Continuous Functions

Let $f(t)$ be a continuous function on the interval $[T_1,T_2]$.

Mean	Formula
Root-Mean Square (RMS)	$\sqrt{\frac{1}{T_2-T_1}{\int }_{T_1}^{T_2}f(t{)}^{2}dt}$