A sharp upper bound is derived for the exponential entropy in the class of absolutely continuous distributions with specific standard deviation and an exact description of the extremal distributions. This result is interpreted as determining the least favorable cases for certain methods of quantization of analog sources. It is known that for a large class of quantizers (both zero-memory and vector) the rth power distortion, as well as some other distortion criteria, are bounded below by a constant, depending on r, multiplied by a certain integral of the source's probability density. It is pointed out that this bound can be rewritten in terms of the exponential entropy. The exponential entropy measures the quantitative extent or range of the source distribution. This fact gives a physical interpretation of the indicated limits of quantizer performance, further elucidated by the main result.>