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Why don't the elevations on your maps agree with those provided by my GPS system? Which are correct?

Elevations provided by your GPS receiver might disagree with elevations on a USGS map, but they could both be correct if they reference different vertical and/or horizontal datums.

The default horizontal datum setting on most GPS receivers is the World Geodetic System of 1984 (WGS84) and the elevations are based on the NAD83 ellipsoid. 

USGS topographic maps published after 2009 are georeferenced to the North American Datum of 1983 (NAD83; which is virtually equivalent to WGS84 at topographic map scales), but uses the North American Vertical Datum of 1988 (NAVD88) for vertical control of elevation. USGS topographic maps published before 2009 are typically referenced to the North American Datum of 1927 (NAD27) and the elevations are based on mean sea level. 

The use of different datums can create horizontal shifts of tens of meters and vertical shifts of hundreds of meters.