In probability theory and statistics, the skew normal distribution is a continuous probability distribution that generalises the normal distribution to allow for non-zero skewness.

denote the standard normal probability density function with the cumulative distribution function given by where "erf" is the error function.

Then the probability density function (pdf) of the skew-normal distribution with parameter

is given by This distribution was first introduced by O'Hagan and Leonard (1976).

[1] Alternative forms to this distribution, with the corresponding quantile function, have been given by Ashour and Abdel-Hamid[2] and by Mudholkar and Hutson.

[3] A stochastic process that underpins the distribution was described by Andel, Netuka and Zvara (1984).

[4] Both the distribution and its stochastic process underpinnings were consequences of the symmetry argument developed in Chan and Tong (1986),[5] which applies to multivariate cases beyond normality, e.g. skew multivariate t distribution and others.

The distribution is a particular case of a general class of distributions with probability density functions of the form

is any CDF whose PDF is symmetric about zero.

[6] To add location and scale parameters to this, one makes the usual transform

One can verify that the normal distribution is recovered when

The probability density function with location

) of the distribution is limited to slightly less than the interval

As has been shown,[7] the mode (maximum)

, but a quite accurate (numerical) approximation is:

Maximum likelihood estimates for

can be computed numerically, but no closed-form expression for the estimates is available unless

In contrast, the method of moments has a closed-form expression since the skewness equation can be inverted with where

ξ = μ − ω δ

As long as the sample skewness

is not too large, these formulas provide method of moments estimates

The maximum (theoretical) skewness is obtained by setting

in the skewness equation, giving

However it is possible that the sample skewness is larger, and then

When using the method of moments in an automatic fashion, for example to give starting values for maximum likelihood iteration, one should therefore let (for example)

Concern has been expressed about the impact of skew normal methods on the reliability of inferences based upon them.

The skew normal still has a normal-like tail in the direction of the skew, with a shorter tail in the other direction; that is, its density is asymptotically proportional to

In contrast, the exponentially modified normal has an exponential tail in the direction of the skew; its density is asymptotically proportional to

In the same terms, it shows "borderline mild randomness".

Thus, the skew normal is useful for modeling skewed distributions which nevertheless have no more outliers than the normal, while the exponentially modified normal is useful for cases with an increased incidence of outliers in (just) one direction.