# [time-nuts] σ vs s in ADEV

Magnus Danielson magnus at rubidium.dyndns.org
Mon Jan 9 16:49:12 EST 2017

```Scott,

On 01/09/2017 07:41 PM, Scott Stobbe wrote:
> I could be wrong here, but it is my understanding that Allan's pioneering
> work was in response to finding a statistic which is convergent to 1/f
> noise. Ordinary standard deviation is not convergent to 1/f processes. So I
> don't know that trying to compare the two is wise. Disclaimer: I could be
> totally wrong, if someone has better grasp on how the allan deviation came
> to be, please correct me.

There where precursor work to Allans Feb 1966 article, but essentially
that where he amalgamed several properties into one to rule them all
(almost). It is indeed the non-convergent properties which motivates a
stronger method. Standard statistics is relevant for many of the basic
blocks, bit things work differently with the non-convergent noise.
Another aspect which was important then was the fact that it was a
counter-based measure. Some of the assumptions is due to the fact that
they used counters. I asked David some questions about why the integral
looks the way it does, and well, it reflects the hardware at the time.

What drives Allan vs. standard deviation is that extra derive function
before squaring
The bias functions that Allan derives for M-sample is really the
behavior of the s-deviation. See Allan variance wikipedia article as
there is good references there for the bias function. That bias function
is really illustrating the lack of convergence for M-sample standard
deviation. The Allan is really a power-average over the 2-sample
standard deviation.

Cheers,
Magnus

> On Wed, Jan 4, 2017 at 3:12 PM, Attila Kinali <attila at kinali.ch> 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)?
>>
>>                         Attila Kinali
>>
>> --
>> It is upon moral qualities that a society is ultimately founded. All
>> the prosperity and technological sophistication in the world is of no
>> use without that foundation.
>>                  -- Miss Matheson, The Diamond Age, Neil Stephenson
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```