Category:Definitions/Class Intervals

This category contains definitions related to Class Intervals.
Related results can be found in Category:Class Intervals.

Let $D$ be a finite set of $n$ observations of a quantitative variable.

Integer Data

Let the data in $D$ be described by integers.

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 $P = \set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n} \subseteq \Z$ be a subdivision of $\closedint a b$, where $a \le x_0 \le x_n \le b$.

The integer interval $\closedint a b$, where $a \le d_{\min} \le d_\max \le b$, is said to be divided into class intervals of integer intervals of the forms $\closedint {x_i} {x_{i + 1} }$ or $\closedint {x_i} {x_i}$ if and only if:

Every datum is assigned into exactly one class interval

Every class interval is disjoint from every other class interval

The union of all class intervals contains the entire integer interval $\closedint {x_0} {x_n}$

By convention, the first and last class intervals are not empty class intervals.

Real Data

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 class intervals of real intervals with endpoints $x_i$ and $x_{i + 1}$ if and only if:

Every datum is assigned into exactly one class interval

Every class interval is disjoint from every other class interval

The union of all class intervals contains the entire real interval $\closedint {x_0} {x_n}$

The class intervals may be any combination of open, closed, or half-open intervals that fulfill the above criteria, but usually:

Every class interval except the last is of the form $\closedint {x_i} {x_{i + 1} }$

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

By convention, the first and last class intervals are not empty class intervals.