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.


 * Maybe it would be better to define the stuff for rational/real data, and then mention that the intervals can be taken to be integer intervals when the data comprises integers or natural numbers. That would save about half the page without loss of information. --Lord_Farin (talk) 12:03, 1 October 2012 (UTC)

Def'ns
(Empty class)

Class width

Class mark

Class limit

Class boundary

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