median, mean, mode as minimizers

Median, mean, and mode are the most common measures of central tendency.

Norms

In mathematics, a norm measures the “distance” from a point to another. A norm is denoted by double vertical lines: $||x||$.

Absolute-value norm is a norm on a single value, see $\text{(1)}$.

$$\begin{array}{}\text{(1)}& ||x||=|x|\end{array}$$

Euclidean norm is a norm on a list, see $\text{(2)}$.

$$\begin{array}{}\text{(2)}& ||\mathbf{x}||=\sqrt{x_1^2+\cdots +x_n^2}\end{array}$$

p-norm is a generalization of Euclidean norm. It is a norm on a list.

Let $p\ge 1$ be a real number. The p-norm of list $\mathbf{x}=(x_1,\dots ,x_n)$ is

$$||\mathbf{x}|{|}_p:={\left(\sum _{i=1}^n{\left|x_i\right|}^p\right)}^{1/p}.$$

When p=1, p-norm reduces to $||\mathbf{x}|{|}_1:=\sum _{i=1}^n\left|x_i\right|$. It takes absolute values, but it is not the absolute-value norm due to the $\sum$ symbol.

When p=2, we get Euclidean norm.

As p approaches $\mathrm{\infty }$, the p-norm approaches the infinity norm or maximum norm: $||\mathbf{x}|{|}_{\mathrm{\infty }}:=\underset{i}{max}\left|x_i\right|$ .

When p=0, 0-norm is not defined.

Statistical dispersions

In statistics, dispersion is the extent to which a distribution is stretched or squeezed.

For a given (finite) data set $\mathbf{x}$, the dispersion about a point c is the “distance” from $x_i$ to the c in the p-norm:

$$f_p(c)={\Vert \mathbf{x}–\mathbf{c}\Vert }_p={\left(\frac{1}{n}\sum _{i=1}^n{\left|x_i–c\right|}^p\right)}^{1/p}$$

Average Absolute Deviation has the identical form of Mean Absolute Error (MAE) which is from the viewpoint of prediction.

The 2-norm is Mean Squared Error (MSE) in addition with the power of $\frac{1}{2}$. MSE is $\frac{1}{n}\sum _{i=1}^n{(x_i-c)}^2$.

The 0-norm counts the number of unequal points. Mode minimizes this 0-norm.

For $p=\mathrm{\infty }$ the largest number dominates, and thus mid-range $M=\frac{maxx+minx}{2}$ minimizes $\mathrm{\infty }$-norm.

See also

Minimum Cost to Make Array Equalindromic