Allan variance and deviation

Allan variance is a measure of frequency stability in clocks, oscillators, and amplifiers, expressed mathematically as ${\sigma }_y^2(\tau )$ and abbreviated as AVAR. 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 its deviated variant called Allan deviation (abbreviated as ADEV), derivable through the square root of AVAR: ${\sigma }_y(\tau )=\sqrt{{\sigma }_y^2(\tau )}$.

ADEV is typically plotted using a log-log linear graph of ADEV against observation time ($\tau$). The plot suggests how the stability of a signal changes over time.