Definition:Class (Descriptive Statistics)/Real Data

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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 $x_{i + 1}$ 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 $\closedint {x_0} {x_n}$


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.


Sources