Definition:Class Interval/Integer Data

Definition
Let $D$ be a finite collection of $n$ data regarding some quantitative variable. 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 value of the largest datum in $D$. Let $P = \left\{{x_0, x_1, x_2, \ldots, x_{n-1}, x_n}\right\} \subseteq \Z$ 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 integer intervals of the forms $\left[{x_i \,.\,.\, x_{i+1}}\right]$ or $\left[{x_i \,.\,.\, x_i}\right]$ 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.