# [time-nuts] σ vs s in ADEV

Attila Kinali attila at kinali.ch
Mon Jan 9 13:18:04 EST 2017

```God kväll Magnus,

On Wed, 4 Jan 2017 22:13:04 +0100
Magnus Danielson <magnus at rubidium.dyndns.org> wrote:

> > My question is two-fold: Why is (n) being used even though it's known
> > to be an biased estimator? And why do people not use s when using (n-1)?
>
> First off all, you need keep number of phase samples (N) or number
> frequency samples (M) separate.
>
> As you derivate the phase samples, you loose the phase bias from the
> samples, so the remaining degree of freedom becomes one less. This is
> the same as number of frequency samples, so any average will be (N-1)
> which is the number of frequency samples M, so M=N-1 is motivated both ways.
>
> Now, as you do an Allan Deviation/Variance estimator, you do second
> derivation, so they the also the frequency bias gets derivated out, and
> another degree of freedom is lost, so as you average you have only M-1
> drift estimates which is what you average over, or N-2.

My statistics is still pretty weak, but I think that the degree of freedom,
as you use it here, does not matter.

The sums of the formulas in [1] and [2] are over (M-1) and (N-2) elements,
respectively. The sums are then divided by (M-1) and (N-2) as well.
Which means we are in the case of σ, ie division by (n) and not (n-1) as it
would be the case for s.

> The ADEV core function is just the square of second derivate of phase,
> and then you do an ensemble average over those squares.

Yes.

> A hint for the use of s, consider the frequency stability. See Allan 1966.

I guess you are refering to [3]. Yes Allan does give tables on the expected
difference of variance for some types of noise, but not explicitly on why
σ and not s is being used.

Attila Kinali

[1] https://en.wikipedia.org/wiki/Allan_variance#Fixed_.CF.84_estimators

[2] "Handbook of Frequency Stability Analysis" NIST Special Pub 1065,
by W.J. Riley, 2008
http://tf.nist.gov/timefreq/general/pdf/2220.pdf

[3] "Statistics of Atomic Frequency Standards", by David Allan, 1966

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```