# [time-nuts] σ vs s in ADEV

William H. Fite omniryx at gmail.com
Thu Jan 5 12:27:33 EST 2017

```Professional statistician here.

Your explanation is clear and lucid, in contrast to some earlier attempts
here. I agree that with oscillators the distinction between N and N-1 is
not particularly relevant. I must caution you, however, not to be too
dismissive of the difference between the two. There are excellent reasons
for the "pedantic" distinction between samples and populations, especially
in small-sample work and when extrapolations are involved. A mistake by
NASA between N and n might mean putting the lander on Mars or missing it by
100,000km. You may count that distinction "beyond ridiculous" but it isn't
going away because, for mamy applications, it is absolutely critical.

Many thanks for your invaluable contributions in your field over the year.

Bill (PhD, as if that mattered)

On Wednesday, January 4, 2017, Tom Van Baak <tvb at leapsecond.com> wrote:

> Hi Attila,
>
> The plain ADEV calculation is essentially a measure of unexpected or
> unwanted drift in frequency; which is the 1st difference of frequency
> error; the 2nd difference of phase error; the 3rd difference in clock time
> itself.
>
> When measuring the quality of a clock, the key idea is that initial phase
> doesn't matter (you can always manually set the time), and even initial
> frequency doesn't matter (you can often adjust the rate: whether pendulum,
> quartz or atomic clock), and so a more honest measure of intrinsic
> timekeeper stability is its ability to maintain frequency; that is,
> statistically speaking, the lower the change in frequency, tau to tau, the
> better. Change in frequency is frequency drift.
>
> If you have N phase samples, you get N-1 frequency samples and N-2 drift
> samples. The standard ADEV calculation is simply based on the mean of those
> drift samples. (and you know Hadamard takes this one step deeper).
>
> If you look a the code at http://leapsecond.com/tools/adev_lib.c you'll
> see I avoid the confusing issue of N-1, N, N+1 and simply count the number
> of terms in the rms sum. Not only does that give the correct result but
> IMHO it make it clear what is being averaged. The code passes the official
> NBS ADEV sample suite, agrees with Bill's Stable32, is used in John's
> TimeLab, and also Mark's Lady Heather.
>
> I've never quite understood the pedantic separation of "sample" and
> "population" mean that statistic textbooks and academics love to discuss.
> They clearly have never measured oscillators. In my experience if you think
> there's an important difference between N and N-1, then that's nature's way
> of telling you to go back to sleep and wait until tomorrow when you have
> (TimeLab is good at displaying this in real-time) -- meaning that the
> distinction between sample (n-1) and population (n) mean is beyond
> ridiculous; even if there's a "correct" textbook answer.
>
> /tvb
>
> ----- Original Message -----
> From: "Attila Kinali" <attila at kinali.ch <javascript:;>>
> To: "Discussion of precise time and frequency measurement" <
> time-nuts at febo.com <javascript:;>>
> Sent: Wednesday, January 04, 2017 12:12 PM
> Subject: [time-nuts] σ vs s in ADEV
>
>
> 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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--
If you gaze long into an abyss, your coffee will get cold.
```