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 usually:


 * 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.

Comment
It is often the case that rational data are presented in decimal notation with a small and uniform number of digits for each of the datum.

In such cases the data may be artificially treated as integer data by "ignoring" the decimal point when creating the classes.

How do I present this? Transclude them?

Also, I am making two separate definition because Brase & Brase specifically define classes such as the following:


 * $D = (1, 4, 6, 9, 9)$


 * $\text{classes} = [1..5],[6..9]$

or


 * $D' = (1,2,3,3,3,4)$


 * $\text{classes}' = [1..1][2..2][3..3][4..4]$

...which under the "real data" definition does not contain the whole of $[1..9]$ etc. Does anyone have a better suggestion then defining it twice? --GFauxPas (talk) 01:28, 4 October 2012 (UTC)

Def'ns
(Empty class)

Class width

Class mark

Class limit

Class boundary

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