Definition:Class (Descriptive Statistics)/Real Data

Definition

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

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

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

Let $d_{\max}$ be the value of the largest datum in $D$.

Let $P = \left\{{x_0, x_1, x_2, \ldots, x_{n-1}, x_n}\right\} \subseteq \R$ 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 $x_{i+1}$ if and only if:

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.