Geometric standard deviation

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In probability theory and statistics, the geometric standard deviation (GSD) describes how spread out are a set of numbers whose preferred average is the geometric mean. For such data, it may be preferred to the more usual standard deviation. Note that unlike the usual arithmetic standard deviation, the geometric standard deviation is a multiplicative factor, and thus is dimensionless, rather than having the same dimension as the input values. Thus, the geometric standard deviation may be more appropriately called geometric SD factor.^[1][2] When using geometric SD factor in conjunction with geometric mean, it should be described as "the range from (the geometric mean divided by the geometric SD factor) to (the geometric mean multiplied by the geometric SD factor), and one cannot add/subtract "geometric SD factor" to/from geometric mean.^[3]

Definition

If the geometric mean of a set of numbers {A₁, A₂, ..., A_n} is denoted as μ_g, then the geometric standard deviation is

${\displaystyle \sigma _{g}=\exp \left({\sqrt {\sum _{i=1}^{n}\left(\ln {A_{i} \over \mu _{g}}\right)^{2} \over n}}\right).\qquad \qquad (1)}$

Derivation

If the geometric mean is

${\displaystyle \mu _{g}={\sqrt[{n}]{A_{1}A_{2}\cdots A_{n}}}.\,}$

then taking the natural logarithm of both sides results in

${\displaystyle \ln \mu _{g}={1 \over n}\ln(A_{1}A_{2}\cdots A_{n}).}$

The logarithm of a product is a sum of logarithms (assuming ${\displaystyle A_{i}}$ is positive for all ${\displaystyle i}$), so

${\displaystyle \ln \mu _{g}={1 \over n}[\ln A_{1}+\ln A_{2}+\cdots +\ln A_{n}].\,}$

It can now be seen that ${\displaystyle \ln \,\mu _{g}}$ is the arithmetic mean of the set ${\displaystyle \{\ln A_{1},\ln A_{2},\dots ,\ln A_{n}\}}$, therefore the arithmetic standard deviation of this same set should be

${\displaystyle \ln \sigma _{g}={\sqrt {\sum _{i=1}^{n}(\ln A_{i}-\ln \mu _{g})^{2} \over n}}.}$

This simplifies to

${\displaystyle \sigma _{g}=\exp {\sqrt {\sum _{i=1}^{n}\left(\ln {A_{i} \over \mu _{g}}\right)^{2} \over n}}.}$

Geometric standard score

The geometric version of the standard score is

${\displaystyle z={{\ln(x)-\ln(\mu _{g})} \over \ln \sigma _{g}}={\log _{\sigma _{g}}(x/\mu _{g})}.\,}$

If the geometric mean, standard deviation, and z-score of a datum are known, then the raw score can be reconstructed by

${\displaystyle x=\mu _{g}{\sigma _{g}}^{z}.}$

Relationship to log-normal distribution

The geometric standard deviation is used as a measure of log-normal dispersion analogously to the geometric mean.^[3] As the log-transform of a log-normal distribution results in a normal distribution, we see that the geometric standard deviation is the exponentiated value of the standard deviation of the log-transformed values, i.e. ${\displaystyle \sigma _{g}=\exp(\operatorname {stdev} (\ln(A)))}$.

이와 같이 로그 정규 분포 모집단에서 추출한 데이터 표본의 기하학적 평균과 기하학적 표준 편차를 사용하여 산술 평균과 표준 편차가 정규 분포의 경계 신뢰 구간에 사용되는 방법과 유사한 신뢰 구간의 경계를 찾을 수 있다. 자세한 내용은 로그 정규 분포의 토론을 참조하십시오.

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