We investigate the statistical properties of a three-dimensional simple and versatile model for weakly nonlinear gravity waves in infinite depth, referred to as the "choppy wave model" (CWM). This model is analytically tractable, numerically efficient, and robust to the inclusion of high frequencies. It is based on horizontal rather than vertical local displacement of a linear surface and is a priori not restricted to large wavelengths. Under the assumption of space and time stationarity, we establish the complete first- and second-order statistical properties of surface random elevations and slopes for long-crested as well as fully two-dimensional surfaces, and we provide some characteristics of the surface variation rate and frequency spectrum. We establish a relationship between the so-called "dressed spectrum," which is the enriched wave number spectrum of the nonlinear surface, and the "undressed" one, which is the spectrum of the underlying linear surface. The obtained results compare favorably with other classical analytical nonlinear theories. The slope statistics are further found to exhibit non-Gaussian peakedness characteristics. Compared to observations, the measured non-Gaussian omnidirectional slope statistics can only be explained by non-Gaussian effects and are consistently approached by the CWM.