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Several measures of central tendency can be characterized as solving a variational problem, in the sense of the calculus of variations, namely minimizing variation from the center. That is, given a measure of statistical dispersion, one asks for a measure of central tendency that minimizes variation: such that variation from the center is ...
The population MAD is defined analogously to the sample MAD, but is based on the complete population rather than on a sample. For a symmetric distribution with zero mean, the population MAD is the 75th percentile of the distribution. Unlike the variance, which may be infinite or undefined, the population MAD is always a finite number.
A measure of statistical dispersion is a nonnegative real number that is zero if all the data are the same and increases as the data become more diverse. Most measures of dispersion have the same units as the quantity being measured. In other words, if the measurements are in metres or seconds, so is the measure of dispersion.
However, in-sample measurements deliver values of the ratio of mean average deviation / standard deviation for a given Gaussian sample n with the following bounds: [,], with a bias for small n. [ 7 ] The mean absolute deviation from the mean is less than or equal to the standard deviation ; one way of proving this relies on Jensen's inequality .
An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation; for example, the variance of a sum of uncorrelated random variables is equal to the sum of their variances. A disadvantage of the variance for practical applications ...
In the theory of vector measures, Lyapunov's theorem states that the range of a finite-dimensional vector measure is closed and convex. [1] [2] [3] In fact, the range of a non-atomic vector measure is a zonoid (the closed and convex set that is the limit of a convergent sequence of zonotopes). [2]
While a simple measure, it is notable in that some texts and guides suggest or imply that the dispersion of nominal measurements cannot be ascertained. It is defined for instance by ( Freeman 1965 ). Just as with the range or standard deviation , the larger the variation ratio, the more differentiated or dispersed the data are; and the smaller ...
A simple example is a volume (how big an object occupies a space) as a measure. In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and ...