experimental physics - Multiple measurements of the same quantity - combining uncertainties

I have a number of measurements of the same quantity (in this case, the speed of sound in a material). Each of these measurements has their own uncertainty.

$$v_{1} \pm \Delta v_{1}$$ $$v_{2} \pm \Delta v_{2}$$ $$v_{3} \pm \Delta v_{3}$$ $$\vdots$$ $$v_{N} \pm \Delta v_{N}$$

Since they're measurements of the same quantity, all the values of $v$ are roughly equal. I can, of course, calculate the mean:

$$v = \frac{\sum_{i=1}^N v_{i}}{N}$$

What would the uncertainty in $v$ be? In the limit that all the $\Delta v_i$ are small, then $\Delta v$ should be the standard deviation of the $v_i$. If the $\Delta v_i$ are large, then $\Delta v$ should be something like $\sqrt{\frac{\sum_i \Delta v_i^2}{N}}$, right?

So what is the formula for combining these uncertainties? I don't think it's the one given in this answer (though I may be wrong) because it doesn't look like it behaves like I'd expect in the above limits (specifically, if the $\Delta v_i$ are zero then that formula gives $\Delta v = 0$, not the standard deviation of the $v_i$).