Symmetric probability distribution

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In statistics, a symmetric probability distribution is a probability distribution—an assignment of probabilities to possible occurrences—which is unchanged when its probability density function or probability mass function is reflected around a vertical line at some value of the random variable represented by the distribution. This vertical line is the line of symmetry of the distribution. Thus the probability of being any given distance on one side of the value about which symmetry occurs is the same as the probability of being the same distance on the other side of that value.

Formal definition[edit]

A probability distribution is said to be symmetric if and only if there exists a value such that

for all real numbers

where f is the probability density function if the distribution is continuous or the probability mass function if the distribution is discrete.

For continuous symmetric spherical, Mir M. Ali gave the following definition. Let denote the class of spherically symmetric distributions of the absolutely continuous type in the n-dimensional Euclidean space having joint density of the form inside a sphere with center at the origin with a prescribed radius which may be finite or infinite and zero elsewhere.[1]


  • The median and the mean (if it exists) of a symmetric distribution both occur at the point about which the symmetry occurs.
  • If a symmetric distribution is unimodal, the mode coincides with the median and mean.
  • All odd central moments of a symmetric distribution equal zero (if they exist), because in the calculation of such moments the negative terms arising from negative deviations from exactly balance the positive terms arising from equal positive deviations from .
  • Every measure of skewness equals zero for a symmetric distribution.

Probability density function[edit]

Typically a symmetric continuous distribution's probability density function contains the index value only in the context of a term where is some positive integer (usually 1). This quadratic or other even-powered term takes on the same value for as for , giving symmetry about . Sometimes the density function contains the term , which also shows symmetry about

Unimodal case[edit]

Partial list of examples[edit]

The following distributions are symmetric for all parametrizations. (Many other distributions are symmetric for a particular parametrization.)


  1. ^ Ali, Mir M. (1980). "Characterization of the Normal Distribution Among the Continuous Symmetric Spherical Class". Journal of the Royal Statistical Society. Series B (Methodological). 42 (2): 162–164. doi:10.1111/j.2517-6161.1980.tb01113.x. JSTOR 2984955.