Allan variance and derivation

Also abbreviated as AVAR, it is a measure of frequency stability in clocks, oscillators, and amplifiers. Expressed mathematically as ${\sigma }_y^2(\tau )$. It can be expressed like so:

$${\sigma }_y^2(\tau )=\frac{1}{2}⟨({\bar{y}}_{n+1}-{\bar{y}}_n{)}^2⟩=\frac{1}{2\tau }⟨(x_{n+2}-2x_{n+1}+x_n{)}^2⟩$$

Allan deviation

Like standard deviation, Allan variance has Allan deviation (abbreviated as ADEV), derivable through the square root of AVAR: ${\sigma }_y(\tau )=\sqrt{{\sigma }_y^2(\tau )}$