Editing Allan variance (section)

==Confidence intervals and equivalent degrees of freedom==
Statistical estimators will calculate an estimated value on the sample series used. The estimates may deviate from the true value and the range of values which for some probability will contain the true value is referred to as the [[confidence interval]].  The confidence interval depends on the number of observations in the sample series, the dominant noise type, and the estimator being used. The width is also dependent on the statistical certainty for which the confidence interval values forms a bounded range, thus the statistical certainty that the true value is within that range of values. For variable-''τ'' estimators, the ''&tau;''<sub>0</sub> multiple ''n'' is also a variable.

===Confidence interval===
The [[confidence interval]] can be established using [[chi-squared distribution]] with df degrees of freedom by using the [[Scaled chi-squared distribution|distribution of the sample variance]]:<ref name=IEEE1139/><ref name=Howe1981>D. A. Howe, D. W. Allan, J. A. Barnes: [http://tf.boulder.nist.gov/general/pdf/554.pdf ''Properties of signal sources and measurement methods''], pages 464–469, Frequency Control Symposium #35, 1981.</ref>

:<math>\chi^2 = \frac{\text{df}\,s^2}{\sigma^2},</math>

where ''s''<sup>''2''</sup> is the sample variance of our estimate, ''σ''<sup>2</sup> is the true variance value, df is the degrees of freedom for the estimator, and ''χ''<sup>2</sup> is calculated based on the inverse cummulative density distribution of a ''χ''<sup>2</sup>  with df degrees of freedom. For a 90% probability, covering the range from the 5% to the 95% range on the probability curve, the upper and lower limits can be found using the inequality

:<math>\chi^2(0.05) \le \frac{\text{df}\,s^2}{\sigma^2} \le \chi^2(0.95),</math>

which after rearrangement for the true variance becomes

:<math>\frac{\text{df}\,s^2}{\chi^2(0.95)} \le \sigma^2 \le \frac{\text{df}\,s^2}{\chi^2(0.05)}.</math>

===Effective degrees of freedom===
The [[Degrees of freedom (statistics)|degrees of freedom]] represents the number of free variables capable of contributing to the estimate. Depending on the estimator and noise type, the effective degrees of freedom varies. Estimator formulas depending on ''N'' (number of total sample points) and ''n'' (integer multiple of ''&tau;''<sub>0</sub>) has been found empirically:<ref name=Howe1981/>
:{| class="wikitable"
|+ Allan variance degrees of freedom
|-
!Noise type
!degrees of freedom
|-
|white phase modulation (WPM)
|<math>\text{df} \cong \frac{(N + 1)(N - 2n)}{2(N - n)}</math>
|-
|flicker phase modulation (FPM)
|<math>\text{df} \cong \exp\left[\left(\ln \frac{N - 1}{2n} \ln \frac{(2n + 1)(N - 1)}{4}\right)^{-1/2}\right]</math>
|-
|white frequency modulation (WFM)
|<math>\text{df} \cong \left[ \frac{3(N - 1)}{2n} - \frac{2(N - 2)}{N}\right] \frac{4n^2}{4n^2 + 5}</math>
|-
|flicker frequency modulation (FFM)
|<math>\text{df} \cong \begin{cases}\frac{2(N - 2)}{2.3N - 4.9} & n = 1 \\ \frac{5N^2}{4n(N + 3n)} & n \ge 2\end{cases}</math>
|-
|random-walk frequency modulation (RWFM)
|<math>\text{df} \cong \frac{N - 2}{n}\frac{(N - 1)^2 - 3n(N - 1) + 4n^2}{(N - 3)^2}</math>
|}