Geometric Standard Deviation.
Compute the sample geometric standard deviation.
geoSD(x, na.rm = FALSE, sqrt.unbiased = TRUE)
x |
numeric vector of observations. |
na.rm |
logical scalar indicating whether to remove missing values from |
sqrt.unbiased |
logical scalar specifying what method to use to compute the sample standard
deviation of the log-transformed observations. If |
If x contains any non-positive values (values less than or equal to 0),
geoMean returns NA and issues a warning.
Let \underline{x} denote a vector of n observations from some distribution. The sample geometric standard deviation is a measure of variability. It is defined as:
s_G = exp(s_y) \;\;\;\;\;\; (1)
where
s_y = [\frac{1}{n-1} ∑_{i=1}^n (y_i - \bar{y})^2]^{1/2} \;\;\;\;\;\; (2)
y_i = log(x_i), \;\; i = 1, 2, …, n \;\;\;\;\;\; (3)
That is, the sample geometric standard deviation is the antilog of the sample standard deviation of the log-transformed observations.
The sample standard deviation of the log-transformed observations shown in Equation (2) is the square root of the unbiased estimator of variance. (Note that this estimator of standard deviation is not an unbiased estimator.) Sometimes, the square root of the method of moments estimator of variance is used instead:
s_y = [\frac{1}{n} ∑_{i=1}^n (y_i - \bar{y})^2]^{1/2} \;\;\;\;\;\; (4)
This is the estimator used in Equation (1) when sqrt.unbiased=FALSE.
A numeric scalar – the sample geometric standard deviation.
The geometric standard deviation is only defined for positive observations. It is usually computed only for observations that are assumed to have come from a lognormal distribution.
Steven P. Millard (EnvStats@ProbStatInfo.com)
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# Generate 2000 observations from a lognormal distribution with parameters # mean=10 and cv=1, which implies the standard deviation (on the original # scale) is 10. Compute the mean, geometric mean, standard deviation, # and geometric standard deviation. # (Note: the call to set.seed simply allows you to reproduce this example.) set.seed(250) dat <- rlnormAlt(2000, mean = 10, cv = 1) mean(dat) #[1] 10.23417 geoMean(dat) #[1] 7.160154 sd(dat) #[1] 9.786493 geoSD(dat) #[1] 2.334358 #---------- # Clean up rm(dat)
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