The usual formula for clock instability is given as
First off what do each of these symbols really mean? What is $T_c$? The literature mentions it is cycle time. What "cycle" is it? Is it the period of oscillation $1/f_0$? But, that can't be right as it is already present in the formula.
Next, as per my intuition, the more independent atoms $N$ you have more scrambled the phase of the clock would be. So I don't understand why it is in the denominator. Same for the averaging time $\tau$. Why is it in the denominator? Wouldn't the phase of a clock made of independent atoms scramble more and more with time?