The quality of a clock need not be dependent upon its error or its rate, because these can be corrected for with calibration. 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. Some watches have an adjustment for the cooling when you are sleeping and your watch is presumably not being worn. Others have an adjustment for the slightly different heat exposures expected by wearers of "male-styles" and "female-styles".

There are many statistical ways to measure clock stability, but the most widely used
is the **Allan Variance**, more prosaically known as *two sample,
zero dead-time variance*. It is computed as follows: first compute the second differences and
divide them by tau, the interval between the measurments you use. Form the
squares of the second differences as divided by tau,
add them, divide by 2 times the number of terms and that is the Allan variance.
The Allan deviation is the square root of the Allan variance.
You can compute
the Allan variance for any multiple of tau, say N times tau, by using only every Nth point.
Such stability measures are also often
expressed in relative terms, i.e., as parts per million, etc.
You can convert between these two styles by remembering that
one day has 86400 s. Therefore rate change of 1 ms per day is equivalent to
(1.0E-3) / 8.64E4 = 1.157 parts in 10 to the eight.

All measurements of clock performance, or clock stability, start with a set of regularly executed measurements of the clock correction, spaced by the interval tau. As an example, this table shows sample crystal clock measurements. The columns are the time of measurement expressed as the day number, the measured clock error against the USNO Master Clock, the first and second differences (which are divided by tau so that the units are ms/day), and the square of the second differences divided by tau:

n Clock Error First Diff./tau Second Diff./tau Square(2nd diff/tau) ms ms/d ms/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.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 math for the Allan deviation gives 86 / 28 = 3.07, and the square root finally gives 1.75ms/day as the measure of stability. Converting to dimensionless units, our test clock, exhibits a frequency instability of 2.02 parts in 10 to the eight from day to day.

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.