# Definition:Class (Descriptive Statistics)

This page is about Class in the context of Descriptive Statistics. For other uses, see Class.

## Definition

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

## Integer Data

Let the data in $D$ be described by natural numbers or 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 classes 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
Every class is disjoint from every other
The union of all classes contains the entire integer interval $\closedint {x_0} {x_n}$

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

## 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 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.

## Class Mark

The midpoint of a class is called the class mark.

## Empty Class

A class is empty if and only if it is of frequency zero.

## Comment

It is often the case that rational data are presented in decimal notation with a small and uniform number of digits for each datum.

In such cases the data may be artificially treated as integer data by "ignoring" the decimal point when creating the classes.