I'll answer this from a physics perspective, rather than statistical, so be wary, and take it with a big grain of salt.
The measurement taken must have an error, which would be + or - half the resolution limit of the instrument. Since you cannot tell any values below that resolution limit, why not assign random values to it, i.e add the result with a random (below resolution) error component. That way the SD will be able to be calculated, but would probably give values around the magnitude of resolution limit. As it is, you don't really know if you are doing that, measurement-wise, in the first place.
To the statistical experts here, is there anything wrong with that approach?