Statistical Dispersion Measure

A Statistical Dispersion Measure is variability measure of a probability distribution.

• AKA: Statistical Variability.
• Context:
  • …
• Example(s):
  • Range Measure, Interquartile Range.
  • Standard Deviation, Variance Measure.
  • Mean Absolute Deviation, Median Absolute Deviation.
• See: Overdispersion, Underdispersion, Variance, Standard deviation.

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References

2016

• (Shan et al., 2016) ⇒ Shan, M., Nastasa, V., & Popescu, G. (2016). “Statistical dispersion relation for spatially broadband fields". Optics letters, 41(11), 2490-2492. DOI:10.1364/OL.41.002490 [1]
  • Let us consider first the Helmholtz equation:

[math]\displaystyle{ \nabla^2 U(\mathbf{r},\omega)+n^2\beta_0^2U(\mathbf{r},\omega)=0\quad\quad(1) }[/math]

where [math]\displaystyle{ U }[/math] is the field in a medium, [math]\displaystyle{ n }[/math] is the refractive index of the medium, and [math]\displaystyle{ \beta_0 }[/math] is the wavenumber in vacuum, [math]\displaystyle{ \beta_0 =\omega/c }[/math]. Note that, if the medium is homogeneous, i.e., [math]\displaystyle{ n }[/math] is independent of [math]\displaystyle{ \mathbf{r} }[/math] (...)

Finally, we obtain the statistical dispersion relation for a field in weakly scattering medium, namely,

[math]\displaystyle{ \langle \kappa^2 \rangle =n^2_0\beta_0^2 \left(1+\frac{\sigma^2_n}{n_0^2}\right)\quad\quad (13) }[/math]

Equation (13) represents the main result of this Letter. It establishes the relationship between the second-order moment of the k-vector, [math]\displaystyle{ \langle \kappa^2\rangle=\langle \kappa_x^2\rangle\langle \kappa_y^2\rangle\langle \kappa_z^2\rangle }[/math] , and the statistics of the refractive index fluctuations. Clearly, when [math]\displaystyle{ \sigma_n \rightarrow 0 }[/math], we recover the homogeneous dispersion relation, [math]\displaystyle{ \langle \kappa^2\rangle=n_0^2\beta_0^2 }[/math].

2016

• (Eric W. Weisstein, 2016) ⇒ Weisstein, Eric W. “Statistical Dispersion." From MathWorld -- A Wolfram Web Resource. http://mathworld.wolfram.com/StatisticalDispersion.html Retrieved 2016-07-10
  • [math]\displaystyle{ (\Delta\;u)_i^2=(u_i-\bar{u})^2 }[/math]

where [math]\displaystyle{ \bar{u} }[/math] is the average of {[math]\displaystyle{ u_i }[/math]}.

2011

• (Manikanden, 2011) ⇒ Manikandan, S. (2011). Measures of dispersion. Journal of Pharmacology and Pharmacotherapeutics, 2(4), 315. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3198538/ DOI: 10.4103/0976-500X.85931 [2]
  • The measures of central tendency are not adequate to describe data. Two data sets can have the same mean but they can be entirely different. Thus to describe data, one needs to know the extent of variability. This is given by the measures of dispersion. Range, interquartile range, and standard deviation are the three commonly used measures of dispersion.

2009

• http://en.wikipedia.org/wiki/Statistical_dispersion
  • In statistics, statistical dispersion (also called statistical variability or variation) is variability or spread in a variable or a probability distribution. Common examples of measures of statistical dispersion are the variance, standard deviation and interquartile range.
  • Dispersion is contrasted with location or central tendency, and together they are the most used properties of distributions.
  • A measure of statistical dispersion is a real number that is zero if all the data are identical, and increases as the data becomes more diverse. It cannot be less than zero.
  • Most measures of dispersion have the same scale as the quantity being measured. In other words, if the measurements have units, such as metres or seconds, the measure of dispersion has the same units. Such measures of dispersion include:
    • Standard deviation.
    • Interquartile range.
    • Range.
    • Mean difference.
    • Median absolute deviation.
    • Average absolute deviation (or simply called average deviation)
  • These are frequently used (together with scale factors) as estimators of scale parameters, in which capacity they are called estimates of scale.

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