Definition:Class Interval/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 $P = \set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n} \subseteq \R$ be a subdivision of $\closedint a b$, where $a \le x_0 \le x_n \le b$.

The closed real interval $\closedint a b$, 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 $x_{i + 1}$ :


 * 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 $\closedint {x_0} {x_n}$

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 $\closedint {x_i} {x_{i + 1} }$


 * The last class is of the form $\closedint {x_{n - 1} } {x_n}$

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