[time-nuts] σ vs s in ADEV

Magnus Danielson magnus at rubidium.dyndns.org
Wed Jan 4 16:13:04 EST 2017

Hi Attila,

On 01/04/2017 09:12 PM, Attila Kinali wrote:
> Hi,
> A small detail caught my eye, when reading a paper that informally
> introduced ADEV. In statistics, when calculating a variance over
> a sample of a population the square-sum is divided by (n-1)(denoted by s in
> statistics) instead of (n) (denoted by σ) in order to account for a small bias
> the "standard" variance introduces
> (c.f. https://en.wikipedia.org/wiki/Unbiased_estimation_of_standard_deviation )
> In almost all literature I have seen, ADEV is defined using an average,
> i.e. dividing by (n) and very few use (n-1).
> 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.

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

No wonders the formulas become like these:

There is nothing magic really.

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


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