# 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

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 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

This page or section has statements made on it that ought to be extracted and proved in a Theorem page.In particular: This section needs to be superseded by a page stating the case preciselyYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by creating any appropriate Theorem pages that may be needed.To discuss this page in more detail, feel free to use the talk page. |

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.

Although this article appears correct, it's inelegant. There has to be a better way of doing it.In particular: Do not gloss over the fact that integer data and real data are QUALITATIVELY different. "Integer data" represents a count, while "real data" represents a measurement. Even though a measurement may be rounded to an integer, it is not the same as data which can genuinely be understood as a "how many" question. In cases where "how many" numbers are extremely large (e.g. human population counts) the distinction is blurred as the classes tend to have boundaries in the 1000s or 1 000 000s, and so the practicalities of the distinction are less important. But the distinction is there.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Improve}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Sources

- 2011: Charles Henry Brase and Corrinne Pellillo Brase:
*Understandable Statistics*(10th ed.): $\S 2.1$