  We studied the dynamics of the order parameter and the winding numbers $W$
formation of a quenched normal-to-superconductor state phase transition in a
finite size holographic superconducting ring. There is a critical circumference
$\tilde{C}$ below it no winding number will be formed, then $\tilde{C}$ can be
treated as the Kibble-Zurek mechanism (KZM) correlation length $\xi$ which is
proportional to the fourth root of its quench rate $\tau_Q$, which is also the
average size of independent pieces formed after a quench. When the
circumference $C \geq 10 \xi$, the key KZM scaling between the average value of
absolute winding number and the quench rate $\langle|W|\rangle \propto
\tau_Q^{-1/8}$ is observed. At smaller sizes, the universal scaling will be
modified, there are two regions. The middle size $5\xi<C<10\xi$ result
$\langle|W|\rangle \propto \tau_Q^{-1/5}$ agrees with a finite size experiment
observation. While at $\xi<C\leq 5\xi$ the the average value of absolute
winding number equals to the variance of winding number and there is no well
exponential relationship between the quench rate and the average value of
absolute winding number. The winding number statistics can be derived from a
trinomial distribution with $\tilde{N}=C/ (f \xi)$ trials, $f\simeq 5$ is the
average number of adjacent pieces that are effectively correlated.
