The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution on [−R, R] whose probability density function f is a scaled semicircle (i.e., a semi-ellipse) centered at (0, 0):

${\displaystyle f(x)={2 \over \pi R^{2}}{\sqrt {R^{2}-x^{2}\,}}\,}$

for −R ≤ x ≤ R, and f(x) = 0 if |x| > R.

It is also a scaled beta distribution: if Y is beta-distributed with parameters α = β = 3/2, then X = 2RY – R has the Wigner semicircle distribution.

The distribution arises as the limiting distribution of eigenvalues of many random symmetric matrices as the size of the matrix approaches infinity. The distribution of the spacing between eigenvalues is addressed by the similarly named Wigner surmise.