[quantumet]

The Poisson distribution is just one random number distribution; there are several others for situations where the events are correlated, or have other properties. Half the fun of probability is figuring out which distribution is the right one to apply to the question at hand. So if your measurements don't match up to Poisson, it doesn't mean they're not random - they could just be interdependent.

So yes, for a Poisson process, the spread (standard deviation) is equal to the square root of the mean; as the number of events gets large, the Poisson distribution approaches the normal distribution, but the relationship between the standard deviation and the mean continues to hold.