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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2efca2bb1f437e5c2c4f8990e079047baedff34)
![{\ displaystyle x_ {1}, x_ {2}, \ ldots, x_ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8694289524164f895d6665f163e14c4dc5ec648d)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![{\ displaystyle h_ {1}, h_ {2}, \ ldots, h_ {k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9bd277ee91a34dd36ca7c22f94885243ca17f42)
![{\ displaystyle f_ {1}, f_ {2}, \ ldots, f_ {k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d638f2e6d46ec29cf80847e723577c3b38fa33e6)
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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1158a2f977c6903c2b19b861681b91b90138bad6)
![{\ displaystyle a_ {j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0096fb78d6843c9fb67a840dc796b61ad93eec2)
![{\ displaystyle x_ {i} = a_ {j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cbd6ae15e35554ef14e264d39164ea23cdca8cb)
![{\ displaystyle f_ {j} = f (a_ {j}) = h_ {j} / n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9215dad6618b7043aab496064622e9d5c95bf92d)
![a_ {j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0096fb78d6843c9fb67a840dc796b61ad93eec2)
![{\ displaystyle x_ {i} = a_ {j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cbd6ae15e35554ef14e264d39164ea23cdca8cb)
![{\ displaystyle h_ {1}, h_ {2}, \ ldots, h_ {k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9bd277ee91a34dd36ca7c22f94885243ca17f42)
![{\ displaystyle f_ {1}, f_ {2}, \ ldots, f_ {k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d638f2e6d46ec29cf80847e723577c3b38fa33e6)
![{\ displaystyle a_ {1}, a_ {2}, \ ldots, a_ {k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2efca2bb1f437e5c2c4f8990e079047baedff34)
![{\ displaystyle h_ {1}, h_ {2}, \ ldots, h_ {k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9bd277ee91a34dd36ca7c22f94885243ca17f42)
![{\ displaystyle f_ {1}, f_ {2}, \ ldots, f_ {k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d638f2e6d46ec29cf80847e723577c3b38fa33e6)
Application examples
Arithmetic mean for frequency data
For frequency data with the characteristics and the associated relative frequencies , the arithmetic mean results as
![{\ displaystyle h_ {1}, h_ {2}, \ ldots, h_ {k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9bd277ee91a34dd36ca7c22f94885243ca17f42)
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.
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2efca2bb1f437e5c2c4f8990e079047baedff34)
![{\ displaystyle f_ {1}, f_ {2}, \ ldots, f_ {k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d638f2e6d46ec29cf80847e723577c3b38fa33e6)
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,
with ,
![{\ displaystyle {\ overline {x}}: = \ sum _ {j = 1} ^ {k} {f_ {j} a_ {j}} = {\ frac {1} {n}} \ sum _ {j = 1} ^ {k} {h_ {j} a_ {j}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/059801f82987900547bb2aceaabff4cd833654ba)
Individual evidence
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^ 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
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^ 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.
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^ 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.