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We present the results of a study of the global properties of combined complete samples of GPS and CSS sources. The resulting sample consists of 67 objects which span a range in size of over 3 orders of magnitude. We have used this sample to examine relationships between turnover frequency, redshift, radio power, and linear size. We find that the distribution of rest-frame turnover frequency extends above 10 GHz. This reveals the presence of a previously unsuspected population of sources which peak at high frequencies. There is a strong anticorrelation between turnover frequency and projected linear size nu_m ~ LS(-0.65) . The continuity on this plane suggests that GPS and CSS sources are related either by evolution or by the mechanism for the turnover or both. We compare the the GPS and CSS sources with the LRL 3CR classical doubles in the redshift range 0.2 lae z lae 1.0. We show that the GPS and CSS sources would be members of the LRL 3CR if their spectra did not turnover. There is no trend for radio power in the GPS and CSS sources to vary with projected linear size. The number of GPS and CSS sources per bin of log projected size is constant. This contrasts with the LRL 3CR classical doubles in which the number increases with increasing size as N ~ LS(0.4) . We discuss the constraints the results place on models for the evolution of these sources. Our results suggest that the GPS and CSS sources evolve in a qualitatively different way than the large scale classical doubles. One way of describing the difference is that for a given dependence on velocity with size, the GPS and CSS sources must decline in luminosity at a faster rate than the 3CR classical doubles. The strong decline in luminosity implies that the GPS and CSS sources are not the progenitors of the LRL 3CR classical doubles and that instead they evolve into less luminous radio sources - possibly FR1 radio sources. We suggest that the strong luminosity evolution may be at least partially due to a decline in the efficiency of conversion of the jet thrust into radio luminosity as the sources evolve.