Frequency data

In the statistics are referred to as frequency data , the summary of the characteristics of the values of a feature with the respective frequencies respectively . ${\displaystyle a_{1},a_{2},\ldots ,a_{k}}$ ${\displaystyle x_{1},x_{2},\ldots ,x_{n}}$ ${\displaystyle X}$ ${\displaystyle h_{1},h_{2},\ldots ,h_{k}}$ ${\displaystyle f_{1},f_{2},\ldots ,f_{k}}$

Explanation

Let be the absolute frequency of occurrences and thus the number of values ​​for which applies and let be the relative frequency of , i.e. H. the proportion of values ​​for which applies. The absolute frequency distribution and the relative frequency distribution are often summarized in a frequency table. The characteristics together with the frequencies or are also referred to as frequency data. ${\displaystyle h_{j}=h(a_{j})}$ ${\displaystyle a_{j}}$ ${\displaystyle x_{i}=a_{j}}$ ${\displaystyle f_{j}=f(a_{j})=h_{j}/n}$ ${\displaystyle a_{j}}$ ${\displaystyle x_{i}=a_{j}}$ ${\displaystyle h_{1},h_{2},\ldots ,h_{k}}$ ${\displaystyle f_{1},f_{2},\ldots ,f_{k}}$ ${\displaystyle a_{1},a_{2},\ldots ,a_{k}}$ ${\displaystyle h_{1},h_{2},\ldots ,h_{k}}$ ${\displaystyle f_{1},f_{2},\ldots ,f_{k}}$

Application examples

Arithmetic mean for frequency data

For frequency data with the characteristics and the associated relative frequencies , the arithmetic mean results as ${\displaystyle a_{1},a_{2},\ldots ,a_{k}}$ ${\displaystyle h_{1},h_{2},\ldots ,h_{k}}$

${\displaystyle {\overline {x}}:=a_{1}h_{1}+a_{2}h_{2}+\ldots +a_{k}h_{k}=\sum _{j=1}^{k}{a_{j}h_{j}}}$.

Empirical variance for frequency data

For frequency data with the characteristics and relative frequencies , the empirical variance is calculated as follows ${\displaystyle a_{1},a_{2},\ldots ,a_{k}}$ ${\displaystyle f_{1},f_{2},\ldots ,f_{k}}$

${\displaystyle {\tilde {s}}^{2}=\sum \limits _{j=1}^{k}\left(a_{j}-{\overline {x}}\right)^{2}f_{j}}$,

with , ${\displaystyle {\overline {x}}:=\sum _{j=1}^{k}{f_{j}a_{j}}={\frac {1}{n}}\sum _{j=1}^{k}{h_{j}a_{j}}}$

Individual evidence

1. ^ Ludwig Fahrmeir, Rita artist, Iris Pigeot, Gerhard Tutz: Statistics. The way to data analysis. 8., revised. and additional edition. Springer Spectrum, Berlin / Heidelberg 2016, ISBN 978-3-662-50371-3 , p. 30
2. ^ Ludwig Fahrmeir, Rita artist, Iris Pigeot, and Gerhard Tutz: Statistics. The way to data analysis. 8., revised. and additional edition. Springer Spectrum, Berlin / Heidelberg 2016, ISBN 978-3-662-50371-3 , p. 50.
3. ^ Ludwig Fahrmeir , Rita artist, Iris Pigeot , Gerhard Tutz : Statistics. The way to data analysis. 8., revised. and additional edition. Springer Spectrum, Berlin / Heidelberg 2016, ISBN 978-3-662-50371-3 , p. 65.