Cumulative frequency

The cumulative (also cumulative ) frequency or cumulative frequency is a measure of descriptive statistics . It indicates the number of feature carriers in an empirical study for which the feature value is smaller than a certain limit. The cumulative frequency is calculated as the sum of the frequencies of the characteristic values ​​from the smallest value up to the limit under consideration.

Example of a graphical representation of the absolute cumulative frequencies of the frequency distribution below

Graphic representation of the corresponding absolute frequency distribution

Explanation

At least ordinally scaled features are assumed, the characteristics can then be sorted according to size. The frequency of occurrence of the characteristics up to a certain upper limit is considered. Depending on whether absolute or relative frequencies are added up, one speaks of absolute cumulative frequency or relative cumulative frequency .

A question that could be solved with the help of the cumulative frequency is the question of the number of grades no worse than 4 in an exam. Here one would count and add up all ones, twos, threes and fours (or their frequencies) in order to calculate the cumulative frequency of the school grade up to the upper limit of four.

The equivalent of the cumulative frequency in probability theory is the distribution function .

Definition in formula notation

The measured values ​​are divided into classes selected according to a suitable criterion , and the classes are sorted and numbered from to . The absolute frequency of the measured values ​​belonging to these classes are denoted by. The associated relative frequencies are denoted by. The limit up to which the frequencies are to be summed is denoted by. ${\displaystyle 1}$ ${\displaystyle n}$ ${\displaystyle H_{1},\dots ,H_{n}}$ ${\displaystyle R_{1},\dots ,R_{n}}$ ${\displaystyle k}$

So the absolute cumulative frequency is defined by

${\displaystyle F_{\text{abs}}(k)=\sum _{n=1}^{k}H_{n}}$

and the relative cumulative frequency by

${\displaystyle F_{\text{rel}}(k)=\sum _{n=1}^{k}R_{n}}$.

See also

• Cumulative frequency function

Individual evidence

1. ↑ Hans Benninghaus: Introduction to social science data analysis . 7th edition. Oldenbourg Wissenschaftsverlag, Munich 2005, ISBN 3-486-57734-4 , p. 96 ( limited preview in Google Book search).
2. ↑ Christel Weiss: Cumulative Frequencies. In: Statistics Lexicon. Christel Weiß, Medical Statistics - Biometry, University of Heidelberg, 2003, accessed on July 26, 2008 . 

Web links

• Eric Weisstein: Cumulative Frequency on MathWorld (engl.)
• Nikos Drakos, Ross Moore; Matthias Stukenberg (Übers): Cumulative frequency (cumulative frequency). In: Statistics. July 7, 2004, accessed July 26, 2008 .