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In probability theory and statistics, the index of dispersion, [1] dispersion index, coefficient of dispersion, relative variance, or variance-to-mean ratio (VMR), like the coefficient of variation, is a normalized measure of the dispersion of a probability distribution: it is a measure used to quantify whether a set of observed occurrences are clustered or dispersed compared to a standard ...
To find a second-order approximation for the covariance of functions of two random variables (with the same function applied to both), one can proceed as follows.
A key step in the derivation of the binary power law by Hughes and Madden was the observation made by Patil and Stiteler [61] that the variance-to-mean ratio used for assessing over-dispersion of unbounded counts in a single sample is actually the ratio of two variances: the observed variance and the theoretical variance for a random ...
In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed. [1] Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartile range. For instance, when the variance of data in a set is large, the data is widely scattered.
This follows from the fact that the variance and mean are independent of the ordering of x. Scale invariance: c v (x) = c v (αx) where α is a real number. [22] Population independence – If {x,x} is the list x appended to itself, then c v ({x,x}) = c v (x). This follows from the fact that the variance and mean both obey this principle.
For any index, the closer to uniform the distribution, the larger the variance, and the larger the differences in frequencies across categories, the smaller the variance. Indices of qualitative variation are then analogous to information entropy, which is minimized when all cases belong to a single category and maximized in a uniform ...
Firstly, if the true population mean is unknown, then the sample variance (which uses the sample mean in place of the true mean) is a biased estimator: it underestimates the variance by a factor of (n − 1) / n; correcting this factor, resulting in the sum of squared deviations about the sample mean divided by n-1 instead of n, is called ...
Algorithms for calculating variance play a major role in computational statistics.A key difficulty in the design of good algorithms for this problem is that formulas for the variance may involve sums of squares, which can lead to numerical instability as well as to arithmetic overflow when dealing with large values.