The Allan variance (AVAR), also known as two-sample variance, is a measure of frequency stability in clocks, oscillators and amplifiers. It is named after David W. Allan and expressed mathematically as . The Allan deviation (ADEV), also known as sigma-tau, is the square root of the Allan variance, . The M-sample variance is a measure of frequency stability using M samples, time T between measurements and observation time . M-sample variance is expressed as The Allan variance is intended to estimate stability due to noise processes and not that of systematic errors or imperfections such as frequency drift or temperature effects. The Allan variance and Allan deviation describe frequency stability. See also the section Interpretation of value below. There are also different adaptations or alterations of Allan variance, notably the modified Allan variance MAVAR or MVAR, the total variance, and the Hadamard variance. There also exist time-stability variants such as time deviation (TDEV) or time variance (TVAR). Allan variance and its variants have proven useful outside the scope of timekeeping and are a set of improved statistical tools to use whenever the noise processes are not unconditionally stable, thus a derivative exists. The general M-sample variance remains important, since it allows dead time in measurements, and bias functions allow conversion into Allan variance values. Nevertheless, for most applications the special case of 2-sample, or "Allan variance" with is of greatest interest. Background When investigating the stability of crystal oscillators and atomic clocks, it was found that they did not have a phase noise consisting only of white noise, but also of flicker frequency noise. These noise forms become a challenge for traditional statistical tools such as standard deviation, as the estimator will not converge. The noise is thus said to be divergent. Early efforts in analysing the stability included both theoretical analysis and practical measurements. An important side consequence of having these types of noise was that, since the various methods of measurements did not agree with each other, the key aspect of repeatability of a measurement could not be achieved. This limits the possibility to compare sources and make meaningful specifications to require from suppliers. Essentially all forms of scientific and commercial uses were then limited to dedicated measurements, which hopefully would capture the need for that application. To address these problems, David Allan introduced the M-sample variance and (indirectly) the two-sample variance. While the two-sample variance did not