Unidirectional random waves generated by a wavemaker in a 300-m-long wave tank are investigated experimentally. Spatial evolution of numerous statistical wavefield parameters is studied. Three series of experiments are carried out for different values of the nonlinear parameter ε. It is found that the frequency spectrum of the wavefield undergoes significant variation in the course of the wavefield evolution along the tank. The initially narrow Gaussian spectrum becomes wider at the early stages of the evolution and then narrower again, although it still remains wider than the initial spectrum at the most distant measuring location. It is found that the values of all the statistical wave parameters are strongly related to the local spectral width. The deviations of various statistical parameters from the Gaussian statistics increase with the width of the spectrum so that the probability of extremely large (the so-called freak) waves is highest when the local spectral width attains maximum. The deviations from the Rayleigh distribution also become more pronounced when the nonlinearity parameter ε is higher. It is found that the Tayfun and Fedele 3rd order random wavefield model provides an appropriate description of the observed phenomena. An attempt is made to relate the spatial variations of the wavefield statistics reported here to the wavefield recurrence, as suggested recently.