[time-nuts] Allan deviation vs standard deviation

[John Miles]

> I was playing with an Agilent 53132 counter, and noticed that
> it measures "standard deviation" but doesn't seem to offer
> what everyone really wants, ie, Allan deviation.  According
> to the textbooks, standard deviation won't work for oscillators
> because the mean is not fixed and the deviation goes to infinity.
> However, I tried it anyway on a high quality oscillator for
> 100 measurements of one second each (N=100) and it seemed to
> basically work, giving 2E-11 for the deviation.  The drift
> over 100 seconds may be small enough that the mean didn't
> move significantly.  I have a 53230 on order that does
> actually measure Allan deviation, but am trying to get some
> work done in the mean time with what I currently have.
> 
> Can anyone comment on the relationship between the two
> types of measurements in the lab?  (We know how they
> differ mathematically, but what is the practical implication).

W. J. Riley's 2003 IFCS tutorial slide deck at
http://www.wriley.com/Techniques.pdf has a good graph on page 10 that shows
what can go wrong when using STDEV for timing work. 

The standard deviation relies on a consistent mean value over time, if you
think about it, because it's all about the "average difference from the
mean."  To the extent the mean value changes over time, say from cumulative
random-walk behavior, then you can't say anything about the deviation of any
one data point from it... which means you can't say anything about the
deviation of *any* data points from it.  The graph will not converge on a
useful result.

White noise, of the sort your 53132A is emitting at short timescales when
you use it to measure a crystal oscillator that is cleaner than its own
noise floor, has a constant mean (presumably zero), so it's not surprising
that ADEV and STDEV agree.  But that won't be the case with anything that is
drifting, random-walking, or that otherwise exhibits non-white noise above
the counter's own floor.  

Also have a look at pages 56-65 of
http://www.allanstime.com/Publications/DWA/Science_Timekeeping/TheScienceOfT
imekeeping.pdf for a better explanation from the horse's mouth.

-- john, KE5FX