Editing Allan variance (section)

===Time stability estimators===
A time stability (σ<sub>''x''</sub>) statistical measure, which is often called the time deviation (TDEV), can be calculated from the modified Allan deviation (MDEV).  The TDEV is based on the MDEV instead of the original Allan deviation, because the MDEV can discriminate between white and flicker phase modulation (PM).  The following is the time variance estimation based on the modified Allan variance:

:<math>\sigma_x^2(\tau) = \frac{\tau^2}{3}\bmod\sigma_y^2(\tau),</math>

and similarly for modified Allan deviation to [[time deviation]]:

:<math>\sigma_x(\tau) = \frac{\tau}{\sqrt{3}}\bmod\sigma_y(\tau).</math>

The TDEV is normalized so that it is equal to the classical deviation for white PM for time constant ''τ'' =&nbsp;''τ''<sub>0</sub>.  To understand the normalization scale factor between the statistical measures, the following is the relevant statistical rule:  For independent random variables ''X'' and ''Y'', the variance (σ<sub>''z''</sub><sup>2</sup>) of a sum or difference (''z'' = ''x'' − ''y'') is the sum square of their variances (σ<sub>''z''</sub><sup>2</sup> = σ<sub>''x''</sub><sup>2</sup> + σ<sub>''y''</sub><sup>2</sup>).  The variance of the sum or difference (''y'' = ''x''<sub>2''τ''</sub> − ''x''<sub>''τ''</sub>) of two independent samples of a random variable is twice the variance of the random variable (σ<sub>''y''</sub><sup>2</sup> = 2σ<sub>''x''</sub><sup>2</sup>).  The MDEV is the second difference of independent phase measurements (''x'') that have a variance (σ<sub>''x''</sub><sup>2</sup>). Since the calculation is the double difference, which requires three independent phase measurements (''x''<sub>2''τ''</sub> − 2''x''<sub>''τ''</sub> + ''x''), the modified Allan variance (MVAR) is three times the variances of the phase measurements.