With equipment advances now enabling the measurement and processing of alluvial-bed elevation data at high spatial and temporal resolutions, the moments of measured bed-elevation distributions can be used to characterize riverbed structure. From analyses of a wide range of high-quality data sets for a range of flows, sediments, and types of bed surface (e.g., armoured gravels, uniform sands, ripples, and dunes), at laboratory and field scales, the standard deviation of bed elevations s is found to provide a robust integral measure of bed-roughness height. This approach has the statistical advantage of utilizing all relevant bed-elevation information, and also removes any potential need to subjectively identify valid individual roughness elements within the measured bed data. In addition to s defining the vertical roughness scale for a bed surface, the general shape of the bed surface forms or elements can be characterized by the bed-elevation distribution skewness Sk, with the distribution kurtosis Ku providing a measure of the evenness or intermittency of these elements. Based on the present analyses, a schematic Sk-Ku plane is presented to aid interpretation of the structure of steady state and developing alluvial bed surfaces from measured bed-elevation distributions, where various bed surface types (e.g., artificially smoothed beds, armoured gravel beds, planar beds of mobile uniform sediments, ripples, and dunes) can be differentiated based on this diagram. Relatively large values of Ku that can occur for a measured riverbed surface are associated with intermittent bed-roughness elements, as can arise for supply-limited dunes or widely distributed pebble clusters or bed forms on an otherwise plane bed.