I have a radiation source and every particle emitted is registered. And the mean number of produced ionizations in a time interval T is recorded.
Suppose N number of particles were emitted during the time interval T and the number of ionizations recorded is I. Then the mean number of ionization per emitted particle is I/N. What is the uncertainty of the mean number of ionization per emitted particle?
Example: 853 emitted particles and 2182 registered ionizations in 100 seconds. Hence 2182/853 = 2.56 ionizations per emitted particle. But what is the uncertainty (or margin of error?)?
Strictly speaking, you just don't have enough information. As far as we know, there could be anywhere from exactly $2.56$ ionisations per particle (zero variation) to an arbitrarily large variance. Do you have any more information, such as the full dataset of exactly how many ionisations occur for each?
That said, if you can assume some more information about the distribution, it may be possible. If you assume it is an exponential distribution (which seems pretty feasible for these sorts of radiation questions), it happens that knowing the mean is equivalent to knowing the variance; the variance is just the mean squared! So from that you could probably work out all you want to know. But this is not at all robust unless you already know the exact distribution of the data happens to be exponentially distributed, for example.