I intend to write up parts of my recent presentation into a blog post, but here is something on timekeeping from The Virtuosi: Time Keeps On Slippin’

However, there is a subtlety to one of their arguments that requires more detail

The second measure of “good-ness” is precision or, in watch parlance, stability. This is essentially a measure of the consistency of the watch. If I have a watch that is consistently off by 5 minutes from the official time, then it is not accurate but it is still stable.

This is true but does not extend far enough. If a watch is consistently off by 5 minutes (or constant amount) from another source, then both must be running at the same frequency — they are accumulating phase at the same rate. But stability goes one step further. Even if the clocks were running at different rates, and phase was accumulating between them (i.e. one is running fast), you can make the same statement. The fractional frequency stability — given by the Allan deviation — depends on how the difference of the two frequencies, as measured over different intervals, changes. But the difference doesn’t have to be zero. A clock that consistently gains e.g. a second per day is also a very stable device: the frequency difference is always ~1/86400, but it’s constant, and the Allan deviation looks at the difference between subsequent frequency comparisons.

The danger here is in the assumption that two stable clocks will give you the same time readings. That’s not what we mean by stability. Stability is a measure of how the frequency is changing. An analogue would be a common conceptual mistake in using Newton’s second law: an object at rest feels zero force, but a force of zero does not mean the object is at rest — it simply means an object has a constant velocity.