page_title: Beta distribution
section_title: Mean absolute deviation around the mean
hierarchical_section_title: Beta distribution / Properties / Measures of statistical dispersion / Mean absolute deviation around the mean
caption_attribution_description: English: Ratio of Mean Abs. Dev. to Std.Dev. Beta distribution with alpha and beta from 0 to 5 - J. Rodal
context_page_description: In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] parameterized by two positive shape parameters, denoted by α and β, that appear as exponents of the random variable and control the shape of the distribution. The generalization to multiple variables is called a Dirichlet distribution.
  The beta distribution has been applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines.
  In Bayesian inference, the beta distribution is the conjugate prior probability distribution for the Bernoulli, binomial, negative binomial and geometric distributions. The beta distribution is a suitable model for the random behavior of percentages and proportions.
  The formulation of the beta distribution discussed here is also known as the beta distribution of the first kind, whereas beta distribution of the second kind is an alternative name for the beta prime distribution.
context_section_description: The mean absolute deviation around the mean for the beta distribution with shape parameters α and β is:
  The mean absolute deviation around the mean is a more robust estimator of statistical dispersion than the standard deviation for beta distributions with tails and inflection points at each side of the mode, Beta(α, β) distributions with α,β > 2, as it depends on the linear (absolute) deviations rather than the square deviations from the mean.  Therefore, the effect of very large deviations from the mean are not as overly weighted.
  Using Stirling's approximation to the Gamma function, N.L.Johnson and S.Kotz derived the following approximation for values of the shape parameters greater than unity (the relative error for this approximation is only −3.5% for α = β = 1, and it decreases to zero as α → ∞, β → ∞):
  At the limit α → ∞, β → ∞, the ratio of the mean absolute deviation to the standard deviation (for the beta distribution) becomes equal to the ratio of the same measures for the normal distribution: .  For α = β = 1 this ratio equals , so that from α = β = 1 to α, β → ∞ the ratio decreases by 8.5%.  For α = β = 0 the standard deviation is exactly equal to the mean absolute deviation around the mean. Therefore, this ratio decreases by 15% from α = β = 0 to α = β = 1, and by 25% from α = β = 0 to α, β → ∞ . However, for skewed beta distributions such that α → 0 or β → 0, the ratio of the standard deviation to the mean absolute deviation approaches infinity (although each of them, individually, approaches zero) because the mean absolute deviation approaches zero faster than the standard deviation.
  Using the parametrization in terms of mean μ and sample size ν = α + β > 0:
  α = μν, β = (1−μ)ν
  one can express the mean absolute deviation around the mean in terms of the mean μ and the sample size ν as follows:
  For a symmetric distribution, the mean is at the middle of the distribution, μ = 1/2, and therefore:
  Also, the following limits (with only the noted variable approaching the limit) can be obtained from the above expressions:
caption: Ratio of Mean Abs.Dev. to Std.Dev. for Beta distribution with α and β ranging from 0 to 5
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