User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

Definition
Let $D$ be a finite collection of $n$ data regarding some quantitative variable.

Integer Data
Let the data in $D$ be described by natural numbers or by integers.

Let $d_{\text{min}}$ be the value of the smallest datum in $D$.

Let $d_{\text{max}}$ be the largest datum in $D$. Let $P = \left\{{x_i \in \Z: x_0, x_1, x_2, \ldots, x_{n-1}, x_n}\right\}$ be a subdivision of $\left[{a \,.\,.\, b}\right]$, where $a \le x_0 \le x_n \le b$.

The integer interval $\left[{a \,.\,.\, b}\right]$, where $a \le d_{\text{min}} \le d_{\text{max}} \le b$, is said to be divided into classes of the form $\left[{x_j \,.\,.\, x_{j+1}}\right]$ iff:


 * Every datum is assigned into exactly one class


 * Every class is disjoint from every other


 * The union of all classes contains the entire integer interval $\left[{x_0 \,.\,.\, x_n}\right]$

By convention, the first and last classes are not empty classes.

Real Data
Let the data in $D$ be described by rational numbers or by real numbers.

Let $d_{\text{min}}$ be the value of the smallest datum in $D$.

Let $d_{\text{max}}$ be the largest datum in $D$. Let $P = \left\{{x_i \in \R: x_0, x_1, x_2, \ldots, x_{n-1}, x_n}\right\}$ be a subdivision of $\left[{a \,.\,.\, b}\right]$, where $a \le x_0 \le x_n \le b$.

The closed real interval $\left[{a \,.\,.\, b}\right]$, where $a \le d_{\text{min}} \le d_{\text{max}} \le b$, is said to be divided into classes of real intervals with endpoints $x_i$ and $x_{i+1}$ iff:


 * Every datum is assigned into exactly one class


 * Every class is disjoint from every other


 * The union of all classes contains the entire real interval $\left[{x_0 \,.\,.\, x_n}\right]$

The classes may be any combination of open, closed, or half-open intervals that fulfill the above criteria, but the usual convention is:


 * Every class except the last is of the form $\left[{x_i \,.\,.\, x_{i+1}}\right)$


 * The last class is of the form $\left[{x_{n-1} \,.\,.\, x_n}\right]$

By convention, the first and last classes are not empty classes.

Is this definition clear? I feel like it could be better.

Def'ns
(Empty class)

Class width

Class mark

Class limit

Class boundary

--GFauxPas (talk) 17:25, 27 September 2012 (UTC)