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

→ Main article : Location parameters (descriptive statistics)

• 

  The median (here ) divides the mass of the distribution ${\displaystyle {\tfrac {\ln 2}{\lambda }}}$

• 

  The expected value (here ) is the focus of the distribution ${\displaystyle {\tfrac {1}{\lambda }}}$

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

→ Main article : Measure of dispersion (statistics)

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 :

• empirical variance , also called (ambiguous) sample variance, the mean squared deviation from the arithmetic mean
• Empirical standard deviation , the square root of the empirical variance
• Span , (the difference between the largest and smallest observation English range )
• Mean absolute deviation from the arithmetic mean
• Interquartile range , which contains the middle 50% of the observations (English interquartile range )

Concentration parameters

• Absolute concentration
• Relative concentration
• Atkinson measure
• Gini coefficient from the Lorenz curve
• Herfindahl index
• Hoover inequality
• Rosenbluth index
• Theil index

Shape dimensions or parameters

• Skewness (statistics)
• Camber (statistics)

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 .