We investigate what can be learned about a population of distant KBOs by studying the statistical properties of their light curves. Whereas others have successfully inferred the properties of individual, highly variable KBOs, we show that the fraction of KBOs with low amplitudes also provides fundamental information about a population. Each light curve is primarily the result of two factors: shape and orientation. We consider contact binaries and ellipsoidal shapes, with and without flattening. After developing the mathematical framework, we apply it to the existing body of KBO light curve data. Principal conclusions are as follows. (1) When using absolute magnitude H as a proxy for size, it is more accurate to use the maximum of the light curve rather than the mean. (2) Previous investigators have noted that smaller KBOs have higher-amplitude light curves, and have interpreted this as evidence that they are systematically more irregular in shape than larger KBOs; we show that a population of flattened bodies with uniform proportions could also explain this result. (3) Our analysis indicates that prior assessments of the fraction of contact binaries in the Kuiper Belt may be artificially low. (4) The pole orientations of some KBOs can be inferred from observed changes in their light curves; however, these KBOs constitute a biased sample, whose pole orientations are not representative of the population overall. (5) Although surface topography, albedo patterns, limb darkening, and other surface properties can affect individual light curves, they do not have a strong influence on the statistics overall. (6) Photometry from the OSSOS survey is incompatible with previous results and its statistical properties defy easy interpretation. We also discuss the promise of this approach for the analysis of future, much larger data sets such as the one anticipated from the Rubin Observatory.
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