Geometric standard deviation

Geometric standard deviation

In probability theory and statistics, the geometric standard deviation describes how spread out are a set of numbers whose preferred average is the geometric mean. If the geometric mean of a set of numbers {"A"1, "A"2, ..., "A""n"} is denoted as μ"g", then the geometric standard deviation is

: sigma_g = exp left( sqrt{ sum_{i=1}^n ( ln A_i - ln mu_g )^2 over n } ight). qquad qquad (1)

Derivation

If the geometric mean is

: mu_g = sqrt [n] { A_1 A_2 cdots A_n }.,

then taking the natural logarithm of both sides results in

: 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 A_i is positive for all i), so

: ln mu_g = {1 over n} [ ln A_1 + ln A_2 + cdots + ln A_n ] .,

It can now be seen that ln , mu_g is the arithmetic mean of the set { ln A_1, ln A_2, dots , ln A_n } , therefore the arithmetic standard deviation of this same set should be

: ln sigma_g = sqrt{ sum_{i=1}^n ( ln A_i - ln mu_g )^2 over n }.

Thus

::ln(geometric SD of "A"1, ..., "A""n") = arithmetic (i.e. usual) SD of ln("A"1), ..., ln("A""n").

Relationship to log-normal distribution

The geometric standard deviation is related to the log-normal distribution.The log-normal distribution is a distribution which is normal for the logarithmtransformed values. By a simple set of logarithm transformations we see that thegeometric standard deviation is the exponentiated value of the standard deviation of the log transformed values (e.g. exp(stdev(ln("A"))));

As such, the geometric mean and the geometric standard deviation of a sample ofdata from a log-normally distributed population may be used to find the bounds of confidence intervals analogously to the way the arithmetic mean and standard deviation are used to bound confidence intervals for a normal distribution. See discussion in log-normal distribution for details.

External links

* [http://www.thinkingapplied.com/means_folder/deceptive_means.htm Deceptive Means]

ee also

*Geometric mean
*Log-normal distribution
*Natural logarithm


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Standard deviation — In probability and statistics, the standard deviation is a measure of the dispersion of a collection of values. It can apply to a probability distribution, a random variable, a population or a data set. The standard deviation is usually denoted… …   Wikipedia

  • Geometric mean — The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, which is what most people think of with the word average, except… …   Wikipedia

  • Standard error (statistics) — For a value that is sampled with an unbiased normally distributed error, the above depicts the proportion of samples that would fall between 0, 1, 2, and 3 standard deviations above and below the actual value. The standard error is the standard… …   Wikipedia

  • Absolute deviation — In statistics, the absolute deviation of an element of a data set is the absolute difference between that element and a given point. Typically the point from which the deviation is measured is a measure of central tendency, most often the median… …   Wikipedia

  • Median absolute deviation — In statistics, the median absolute deviation (MAD) is a robust measure of the variability of a univariate sample of quantitative data. It can also refer to the population parameter that is estimated by the MAD calculated from a sample. For a… …   Wikipedia

  • Gold standard (test) — For other uses, see Gold standard (disambiguation). In medicine and statistics, gold standard test refers to a diagnostic test or benchmark that is the best available under reasonable conditions. It does not have to be necessarily the best… …   Wikipedia

  • Log-normal distribution — Probability distribution name =Log normal type =density pdf μ=0 cdf μ=0 parameters =sigma > 0 infty < mu < infty support = [0,+infty)! pdf =frac{1}{xsigmasqrt{2piexpleft [ frac{left(ln(x) mu ight)^2}{2sigma^2} ight] cdf =frac{1}{2}+frac{1}{2}… …   Wikipedia

  • List of statistics topics — Please add any Wikipedia articles related to statistics that are not already on this list.The Related changes link in the margin of this page (below search) leads to a list of the most recent changes to the articles listed below. To see the most… …   Wikipedia

  • List of mathematics articles (G) — NOTOC G G₂ G delta space G networks Gδ set G structure G test G127 G2 manifold G2 structure Gabor atom Gabor filter Gabor transform Gabor Wigner transform Gabow s algorithm Gabriel graph Gabriel s Horn Gain graph Gain group Galerkin method… …   Wikipedia

  • List of probability topics — This is a list of probability topics, by Wikipedia page. It overlaps with the (alphabetical) list of statistical topics. There are also the list of probabilists and list of statisticians.General aspects*Probability *Randomness, Pseudorandomness,… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”