Statistical dispersion

In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartile range. For example, if the variance of the data in a set is large, the data is widely spread. On the other hand, if the variance is small, the data in the set is clustered.

Dispersion is contrasted with location or central tendency, and together they are the most commonly used properties of distributions.

The variability or dispersion of a characteristic X, measured over n statistical units, is the propensity of that characteristic to manifest itself in different ways, that is, in different modes.

If the trait is quantitative, the variability can be measured using indices based on the distance of the modes from a position index (usually the arithmetic mean or median); the most commonly used indices of variability are the variance, the mean square deviation or standard deviation, and the coefficient of variation.

On the other hand, if the character is qualitative, the variability can be measured with indices of heterogeneity. The properties of variability are:

  • Non-negativity V(x) > 0; V(x) = 0 if and only if all modes of the distribution are equal.
  • Monotonicity: v(x) takes larger values the greater the diversity among the modes of the distribution.
  • Invariance by translation v(x+B) = V(x)
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