Say I have 2 Geiger counters that both generate a set of values (the counter measures background radiation). If I combine the two sets, how does the standard deviation of the new set compare to that of the individual (initial) ones?
The numbers are basically random -- is it impossible to generalize the difference between the new stdev and the ones of the initial sets?
If you receive $m$ observations from one counter, and $n$ observations from the second, then their pooled sample standard deviation will be computed as $$\sigma_{\text {pooled}} = \sqrt{\frac{(m-1) \sigma^2_1 + (n-1) \sigma^2_2}{m+n-2}},$$ where $\sigma_1$ and $\sigma_2$ are the sample standard deviations from the first and second counters, respectively. What this tells us is that your estimate of the pooled standard deviation behaves (in the large-sample approximation) like a weighted root mean square of the individual standard deviations, where the weight is proportional the number of observations in the respective sample.
As this is applied, suppose the first counter and second counter have approximately equally good precision, so the standard deviation of their measurements could be assumed to be approximately equal. Then the pooled SD will be approximately the same as well, irrespective of the sample size of each counter. To see this, set $\sigma_1 = \sigma_2$ in the above equation and simplify.
If, however, the two counters are heterogeneous in precision--say, the first is much more precise than the second--then you will find that the estimate of your overall SD will depend on how many observations you receive from each counter; intuitively, the pooled estimate will be "good" if you draw more observations from the more precise counter (i.e., $m > n$).