Coefficient of variation

The coefficient of variation (also: coefficient of deviation ) is a statistical parameter in descriptive statistics and mathematical statistics . In contrast to the variance , it is a relative measure of dispersion , that is, it does not depend on the unit of measurement of the statistical variable or random variable . It is only useful for measurement series with exclusively positive (or exclusively negative) values ​​or measurement series comparisons.

The motivation for this characteristic value is that a statistical variable with a large mean or a random variable with a large expected value generally has a greater variance than one with a small mean or expected value. Since the variance and the standard deviation derived from it are not normalized, it is not possible to assess whether a variance is large or small without knowing the mean value. For example, the prices for a pound of salt, which costs around 50 cents on average, fluctuate in the cents range, while prices for a car, which costs an average of 20,000 euros, for example, vary in the 1,000 euro range.

The coefficient of variation is a normalization of the variance : If the standard deviation is greater than the mean or the expected value, then the coefficient of variation is greater than 1.

The quartile dispersion coefficient is a robust version of the coefficient of variation.

Coefficient of variation for a random variable

definition

The coefficient of variation for a random variable with an expected value is defined as the relative standard deviation , that is, the standard deviation divided by the expected value of the random variable, in formulas ${\displaystyle \operatorname {VarK} }$ ${\displaystyle X}$ ${\displaystyle \operatorname {E} (X)\neq 0}$

${\displaystyle \operatorname {VarK} (X)={\frac {\mathrm {Standardabweichung} (X)}{\mathrm {Erwartungswert} (X)}}={\frac {\sqrt {\operatorname {Var} (X)}}{\operatorname {E} (X)}}}$.

The coefficient of variation is often given in percent .

example

The real random variable is standard normal , that is, the expected value and standard deviation of have the value 0 and 1. The coefficient of variation cannot be defined for this random variable (division by zero). The shifted random variable also has the standard deviation 1, but the expected value 1000. A coefficient of variation of is calculated here . ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X+1000}$ ${\displaystyle 1/1000=0,\!001}$

Squared coefficient of variation for a random variable

The variance of the random variable is called the squared coefficient of variation or . Like the coefficient of variation, it does not depend on the dimension in which the size is measured. ${\displaystyle X/\operatorname {E} (X)}$ ${\displaystyle \operatorname {SCV} }$ ${\displaystyle c_{X}^{2}}$ ${\displaystyle X}$

${\displaystyle \operatorname {SCV} =c_{X}^{2}=\operatorname {Var} \left({\frac {X}{\operatorname {E} (X)}}\right)={\frac {\operatorname {E} (X^{2})-\left[\operatorname {E} (X)\right]^{2}}{\left[\operatorname {E} (X)\right]^{2}}}={\frac {\operatorname {E} (X^{2})}{\left[\operatorname {E} (X)\right]^{2}}}-1}$

Empirical coefficients of variation

If, instead of the distribution of the random variables, there is a specific measurement series of values , the empirical coefficient of variation is formed as the quotient of the empirical standard deviation and the arithmetic mean : ${\displaystyle x_{1},\dots ,x_{n}}$ ${\displaystyle s}$ ${\displaystyle {\bar {x}}}$

${\displaystyle v={\frac {s}{\bar {x}}},\;{\bar {x}}>0}$.

If so, a normalized coefficient of variation can be defined as ${\displaystyle x_{i}\geq 0}$

${\displaystyle v^{*}={\frac {v}{\sqrt {n}}}}$,

applies to . ${\displaystyle 0\leq v^{*}\leq 1}$

If the empirical standard deviation is instead not calculated from the corrected sample variance (i.e. used instead ), then the value is to be used instead of in the denominator of . ${\displaystyle {\tilde {s}}}$ ${\displaystyle s}$ ${\displaystyle {\sqrt {n}}}$ ${\displaystyle v^{*}}$ ${\displaystyle {\sqrt {n-1}}}$

Empirical quartile dispersion coefficient

The quartile dispersion coefficient is a robust version of the coefficient of variation

${\displaystyle v_{r}={\frac {x_{0{,}75}-x_{0{,}25}}{x_{0{,}5}}}}$,

i.e. the interquartile range divided by the median .

Web links

Wiktionary: coefficient of variation  - explanations of meanings, word origins, synonyms, translations

Individual evidence

1. ↑ Joachim Hartung: Statistics: teaching and manual of applied statistics . 7th through Edition. Oldenbourg, 1989, ISBN 3-486-21448-9 , pp. 47 .
2. ↑ Wolfgang Kohn: Statistics: data analysis and probability calculation . Springer, 2004, ISBN 978-3-540-21677-3 , pp. 81 .