# Parameters (statistics)

In statistics , aggregating parameters or measures summarize the essential properties of a frequency distribution , e.g. B. a longer series of measurement data, or a probability distribution .

Some parameters of descriptive statistics correspond to the moments of random variables .

The term parameter is also used in distribution models, in which case one speaks of distribution parameters . It is then usually one of several quantities that, together with the distribution class, determine the exact form of a distribution.

## Location parameters

The definitions of location parameters aim to describe the location of the sample elements or the elements of the population in relation to the measurement scale . They combine a number of values ​​or a variable that depends on chance (e.g. time until the next decay of an atom) into a single number - the central tendency - which represents the values ​​as well as possible.

### definition

Be a sample. A function is called a measure of position if it is translation-equivalent: ${\ displaystyle x_ {1}, \ dots, x_ {n} \ in \ mathbb {R}}$ ${\ displaystyle l: x_ {1}, \ dots, x_ {n} \ mapsto \ mathbb {R}}$ ${\ displaystyle l (x_ {1} + a, \ dots, x_ {n} + a) = l (x_ {1}, \ dots, x_ {n}) + a}$ With ${\ displaystyle a \ in \ mathbb {R}}$ ### Examples

In descriptive statistics, the location parameters of a distribution are:

• Arithmetic mean :${\ displaystyle l _ {\ text {arithm.}} (x_ {1}, \ dots, x_ {n}) = {\ frac {1} {n}} (x_ {1} + \ cdots + x_ {n} )}$ • empirical quantiles : median , quartile , decile , percentile
• mode

For the three first mentioned position parameters as well as mode and median see also mean .

In the case of random variables, one speaks of the expected value .

According to the definition above, the following parameters are not position measurements:

• Geometric mean :${\ displaystyle l _ {\ text {geom.}} (x_ {1}, \ dots, x_ {n}) = {\ sqrt [{n}] {x_ {1} \ cdot x_ {2} \ dotsm x_ { n}}}}$ • Harmonic mean :${\ displaystyle l _ {\ text {harm.}} (x_ {1}, \ dots, x_ {n}) = {\ frac {n} {{\ frac {1} {x_ {1}}} + \ cdots + {\ frac {1} {x_ {n}}}}}}$ ## Dispersion parameters

A measure of dispersion or measure of dispersion (also known as dispersion parameter ) is understood to be statistical key figures , the determination of which enables statements to be made about the distribution of measured values, for example from weighing and counting, around the center point. In descriptive statistics , one describes the scatter (or dispersion; also variation ) with the following measures :

## Individual evidence

1. ^ Norbert Henze: Measure and probability theory (Stochastics II) . Karlsruhe 2010, p. 127 .
2. ^ Andreas Büchter, H.-W. Henn: Elementary Stochastics - An Introduction . 2nd Edition. Springer, 2007, ISBN 978-3-540-45382-6 , pp. 71 .