The energy corresponding to that Fermi energy level is called Fermi energy and the statistic which govern the distribution of energy level is known as Fermi-Dirac statistics.
Explanation of Fermi-Dirac statics:
A metal has very large number of free electrons and these electrons are distributed among various energy levels in the energy bands. The statistics which governs how the free electrons are distributed among various energy levels is called Fermi-Dirac statistics. This obeys Pauli’s exclusion principle and is applicable to any indistinguishable particles of spin 1/2. The distribution of electrons among various available energy levels according Fermi-Dirac statistics under thermal equilibrium is called
N(E) dE = f(E) × g(E) dE,
where N(E) dE is the number of electrons in unit volume possessing the energies between E and E dE. The plot of N(E) dE versus E is shown in the following ﬁgure for various temperatures.