The quality of a clock is not dependent upon its error or its rate. It is the

Regarding the performance of pocket or wrist crystal watches, the most important disturbance comes from the temperature variations to which the watch is exposed. As a rule of thumb, crystals have a temperature coefficient of about 1 part per million. That amounts to a rate change of 0.1s per day per degree temperature change.

All measurements of clock performance, or clock stability, start with a set of regularly executed measurements of the clock correction With these measurements a table is constructed with the time of measurement (or the day number in the series), the measurement, and first and second differences. The table looks like this:

n Clock Error First Diff. Second Diff. Sec.Dif.Square ms ms/d ms/d/d --------------------------------------------------------------- 0 325 0 1 350 25 2 377 27 2 4 3 401 24 -3 9 4 430 29 5 25 5 461 31 2 4 6 494 33 2 4 7 529 35 2 4 8 566 37 2 4 9 601 35 -2 4 10 636 35 0 0 11 673 37 2 4 12 710 37 0 0 13 749 39 2 4 14 790 41 2 4 15 835 45 4 16 etc.The squares column will be needed in a moment. We have assumed that we observe a quartz crystal clock which is temperature controlled. The units are milliseconds and the rates are given in ms/day. In the case of atomic clocks, the unit would probably be in nanoseconds because of the much greater stability of these clocks.

Our example clock would be a very good crystal clock because the rate variations as shown in the 4th column are small. Nevertheless, the rate shows a systematic increase of 20ms in 14 days, i.e., the clock has a noticeable drift. This drift is also visible as the average second difference (sum = 20, divide by 14 ---> average drift = 20/14 = 1.43ms/d/d).

The widely adopted and by far the most simple measure of clock stability
is the co called **Allan Variance**, internationally known as *two sample
sigma*. It is computed as follows: Form the squares of the second differences,
add them, divide by 2 times the number of terms and form the square root.
This gives
86 / 28 = 3.07; the square root finally gives 1.75ms/day
as the measure of stability. Such stability measures are also often
expressed in relative terms, i.e., as parts per million, etc.
One finds the translation between these two styles by remembering that
one day has 86400 s. Therefore 1ms rate change per day corresponds with

(1.0E-3) / 8.64E4 = 1.157

parts in 10 to the eight.

Our test clock, therefore, exhibits a frequency instability of 2.02 parts in 10 to the eight from day to day (1.157x1.75).

Remember: A * clock error* is given in units of time (s, ms, ns), whereas
a *rate difference *is given as a relative number or as ms/day, ns/day, etc.

More information is available in literature upon request.