This probably has a straightforward answer that I'm too tired to figure out right now, but here goes.
Clark County, Nevada -- my home county -- contains the vast majority (almost three-quarters) of the state's residents. For those who may not be aware, and as the map in the Wikipedia article shows, it is located in the southernmost tip of the state. It also extends further east than any other county in Nevada and is approximately tied for being the county with the easternmost geographic center in the state.
Based on this, if there is a parallel of latitude such that half of Nevadans live north of it and the other half south of it, it is bound to pass through Clark County. Similarly, a meridian of longitude with half of Nevadans east of it and half west of it will also pass through Clark County. Thus, assuming the county borders don't become concave at any point that we care about (note that the border does go concave in the southeast, at the Arizona state line, but this is not the side we're concerned about -- see below), the intersection of this parallel and this meridian must be in Clark County, and thus there must be some point in Clark County that has equal amounts of Nevada population "mass" on each side of it (well, when considering the north-south and east-west axes, anyway).
Despite this, however, Wikipedia, as well as several other sources that I've looked at, state that Nevada's center of population is in Nye County. Specifically, it is close to Yucca Mountain, which is located both entirely north and entirely west of Clark County.
Where does the discrepancy lie here? How can the center of population not be in the county that contains 75% of the residents? Am I misinterpreting the definition of center of population? Is my approach of considering the balance only in the north-south and east-west directions incorrect (i.e., could the center of population be different if I did the "halving" with northwest-southeast and northeast-southwest lines instead)?