During a single average-life,

The number of active nuclei remaining after time $t$ is given by the radioactive decay law:

$N = N_0 e^{-\lambda t}$

The average life (or mean life) of a radioactive substance is $\tau = \dfrac{1}{\lambda}$.

Substituting $t = \tau = \dfrac{1}{\lambda}$ in the decay equation:

$N = N_0 e^{-\lambda \left(\dfrac{1}{\lambda}\right)} = N_0 e^{-1} = \dfrac{N_0}{e}$

Since $e \approx 2.718$, the number of remaining active nuclei is:

$N = \dfrac{N_0}{2.718} \approx 0.368 N_0$

The number of nuclei that have decayed during this time is:

$N_{decayed} = N_0 - N = N_0 - 0.368 N_0 = 0.632 N_0$

This means approximately $63.2\%$ of the initial active nuclei decay during one average-life, which is more than half.