here's the thing I've trying to wrap my head around. Please point out any error in reasoning.
Imagine you want to estimate the number of people in a field of known total area ($A_{T}$). People are walking around randomly and the distribution of people in the field is uniform. So, your method consists of taking a snapshot of a smaller area ($A_{S}$), counting the number of people in this smaller square ($S$), then extrapolating for the whole area. Your estimate for the total number of people ($T$) would be: $$ T = S . \frac{A_{T}}{A_{S}} $$
Now, say the counting method isn't perfect. Observers might differ, since people are moving, the uniformity of the distribution of people isn't perfect either, the same observer counting two different samples from the same field will get different results, even the same sample counted twice might give different results. If that's the case, there will be a random error upon the counts of the samples.
Say the total random error in the counts is $\varepsilon$. That translates to an estimate error of about $T = \varepsilon . \frac{A_{T}}{A_{S}}$, right?
So, for the concrete example, if I have the 100 x 100 field, then count $10 \pm 3 $ people in a random 10 x 10 sample square (1/100 of the area), I get an estimate of $T = 10 \pm 3 . \frac{100}{1} = 1000 \pm 300$.
The thing is, the error happens in the counts of the sample. The error of the actual estimate will depend on the size of my sample (that makes sense, larger samples mean better estimates) but will also depend on the total area. So, reducing total area would improve my estimates? That is, in the sense that the error in the counts wouldn't reflect so much on the final estimate error.
I'm trying to get the intuition behind why this happens. Is it simply because in smaller fields a smaller sample is more representative than in larger ones? Because if it's true, it means "diluting" the people over a larger field (or concentrating them on a smaller one) before taking the sample would make my estimate less precise (or more precise). In my example, if we asked all people to move to one half of the field before taking the samples (of same area) we'd get an estimate of about $T = 20 \pm 3 . \frac{50}{1} = 1000 \pm 150$ (assuming the errors in counting are independent of the number of people counted in a sample, which might not be a good assumption). Does that make sense?
Hopefully someone can help or point to where I'd find a formal statistical treatment to problems of this kind. Thank you!