In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, a symmetric probability distribution is a
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
—an assignment of probabilities to possible occurrences—which is unchanged when its
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
(for continuous probability distribution) or
probability mass function
In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass ...
(for discrete random variables) is
reflected around a vertical line at some value of the
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
represented by the distribution. This vertical line is the line of
symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
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
A probability distribution is said to be symmetric
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
there exists a value
such that
:
for all real numbers
where ''f'' is the probability density function if the distribution is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
or the probability mass function if the distribution is
discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
* Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a ...
.
Multivariate distributions
The degree of symmetry, in the sense of mirror symmetry, can be evaluated quantitatively for multivariate distributions with the chiral index, which takes values in the interval
;1 and which is null if and only if the distribution is mirror symmetric.
Thus, a d-variate distribution is defined to be mirror symmetric when its chiral index is null.
The distribution can be discrete or continuous, and the existence of a density is not required, but the inertia must be finite and non null.
In the univariate case, this index was proposed as a non parametric test of symmetry.
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.
Properties
*The
median
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic fe ...
and the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...
(if it exists) of a symmetric distribution both occur at the point
about which the symmetry occurs.
*If a symmetric distribution is
unimodal
In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object.
Unimodal probability distribution
In statistics, a unimodal pr ...
, the
mode
Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to:
Arts and entertainment
* '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine
* ''Mode'' magazine, a fictional fashion magazine which is ...
coincides with the median and mean.
*All odd
central moment
In probability theory and statistics, a central moment is a moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the deviation of the random ...
s 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
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.
For a unimodal d ...
equals zero for a symmetric distribution.
Unimodal case
Partial list of examples
The following distributions are symmetric for all parametrizations. (Many other distributions are symmetric for a particular parametrization.)
{, class="wikitable"
, +
!Name
!Distribution
, -
,
Arcsine distribution
In probability theory, the arcsine distribution is the probability distribution whose cumulative distribution function involves the arcsine and the square root:
:F(x) = \frac\arcsin\left(\sqrt x\right)=\frac+\frac
for 0 ≤ ''x''  ...
,
for 0 ≤ ''x'' ≤ 1
on (0,1)
, -
,
Bates distribution
In probability and business statistics, the Bates distribution, named after Grace Bates, is a probability distribution of the mean of a number of statistically independent uniformly distributed random variables on the unit interval. This dist ...
,
, -
,
Cauchy distribution
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...
,
, -
,
Champernowne distribution In statistics, the Champernowne distribution is a symmetric, continuous probability distribution, describing random variables that take both positive and negative values. It is a generalization of the logistic distribution that was introduced by ...
,
, -
,
Continuous uniform distribution
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies betw ...
,
, -
,
Degenerate distribution
In mathematics, a degenerate distribution is, according to some, a probability distribution in a space with support only on a manifold of lower dimension, and according to others a distribution with support only at a single point. By the latter ...
,