[ska]

This property is quite intuitive. Small and big here are relative to the variance of the underlying distributions.

Simple case: think about trying to decide if as normal distribution has mean 0 or mean 1. If the std dev is 0.001, it won't take you very many samples to be fairly confident to this resolution, but of the deviation is 1000, you'll need a lot of samples.

Similarly if the std deviation is only 1, but you are trying to decide if the mean is 0 or 0.001, Far more samples needed.

The intuition generalizes quite well. In the OP case, typically requires sample size estimates will be proportional to the square of the ratio between the size you want to measure and a deviation estimate.