Editing Allan variance (section)

==Formulations==

===''M''-sample variance===
Given a time-series <math>x(t)</math>, for any positive real numbers <math>T, \tau</math>, define the real number sequence<math display="block">\bar y_i = \frac{x(iT + \tau) - x(iT)}{\tau} \quad i = 0, 1, 2, ...</math>Then the <math>M</math>-sample variance is defined<ref name="Allan1966" /> (here in a modernized notation form) as the [[Bessel's correction|Bessel-corrected variance]] of the sequence <math>\bar y_0, ..., \bar y_{M-1}</math>:<math display="block">\sigma_y^2(M, T, \tau) = \frac{M}{M - 1} \left(\frac 1M \sum_{i=0}^{M-1}\bar{y}_i^2 -  \left[\frac{1}{M} \sum_{i=0}^{M-1} \bar{y}_i\right]^2 \right),</math>The interpretation of the symbols is as follows:

* <math>t</math> is the reading on a reference clock (in arbitrary units).
* <math>x(t)</math> is the reading of a clock we are testing (in arbitrary units), as a function of the reference clock's reading. It can also be interpreted as the [[#Average fractional frequency|average fractional frequency]] time series.
* <math>\bar{y}_n</math> is the ''n''th [[#Fractional frequency|fractional frequency]] average over the observation time <math>\tau</math>.
* <math>M</math> is the number of clock reading intervals used in computing the <math>M</math>-sample variance,
* <math>T</math> is the time between each frequency sample,
* <math>\tau</math> is the time length of each frequency estimate, or the observation period.

Dead-time can be accounted for by letting the time <math>T</math> be different from that of <math>\tau</math>.

===Allan variance===
The Allan variance is defined as

:<math>\sigma_y^2(\tau) = \left\langle\sigma_y^2(2, \tau, \tau)\right\rangle = \frac{1}{2} \left\langle\left(\bar{y}_{n+1} - \bar{y}_n\right)^2\right\rangle = \frac{1}{2\tau^2} \left\langle\left(x_{n+2} - 2x_{n+1} + x_n\right)^2\right\rangle</math>

where <math>x_n:=x(n\tau)</math> and <math>\langle\dotsm\rangle</math> denotes the expectation operator.

The condition <math display="inline">T = \tau</math> means the samples are taken with no dead-time between them.

===Allan deviation===
Just as with [[standard deviation]] and [[variance]], the Allan deviation is defined as the square root of the Allan variance:

:<math>\sigma_y(\tau) = \sqrt{\sigma_y^2(\tau)}.</math>