# [time-nuts] Allan deviation vs standard deviation

Magnus Danielson magnus at rubidium.dyndns.org
Thu Sep 8 12:58:31 EDT 2011

```Dear Rick,

On 08/09/11 01:16, Rick Karlquist wrote:
> 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).

Yes.

When you do measures using standard deviation rather than allan
deviation you get statistical biases. These biases is really in the core
of the Allan article of 1966, since he provides means to compare various
M-sample variance measures and various dead-time measures by comparing
them to 2-sample variance with no dead-time. which is what we call Allan
variance.

I've spent some effort to explain this here:
http://en.wikipedia.org/wiki/Allan_variance#Bias_functions

In your case the B1 bias function can be used to convert your M-sample
variance into 2-sample variance. Now, look at the formula and you will
realize that this bias varies with number of samples and which is the
dominant noise form (u).

So, you can vary your number of samples and your time-base time to find

To assist you, I warmly recommend looking at these papers:

Allan, D Statistics of Atomic Frequency Standards, pages 221–230.
Proceedings of IEEE, Vol. 54, No 2, February 1966.
http://tf.boulder.nist.gov/general/pdf/7.pdf

Barnes, J.A.: Tables of Bias Functions, B1 and B2, for Variances Based
On Finite Samples of Processes with Power Law Spectral Densities, NBS
Technical Note 375, 1969
http://tf.boulder.nist.gov/general/pdf/11.pdf

J.A. Barnes and D.W. Allan: Variances Based on Data with Dead Time
Between the Measurements, NIST Technical Note 1318, 1990
http://tf.boulder.nist.gov/general/pdf/878.pdf

So, you can use your standard deviation measures if you also note the
number of measurements used and vary number of measurements to find out