Results. We find that 80% of the star-forming galaxies in our sample have EW0(Ly alpha) < 10 angstrom, and correspondingly f(esc)(Ly alpha) < 1%. By comparing these results with the literature, we conclude that the bulk of the Ly alpha luminosity at 2 < z < 6 comes from galaxies that are fainter in the UV than those we sample in this work. The strong Ly alpha emitters constitute, at each redshift, the tail of the distribution of the galaxies with extreme EW0(Ly alpha) and f(esc)(Ly alpha). This tail of large EW0(Ly alpha) and f(esc)(Ly alpha) becomes more important as the redshift increases, and causes the fraction of strong Ly alpha with EW0(Ly alpha) > 25 angstrom to increase from similar to 5% at z similar to 2 to similar to 30% at z similar to 6, with the increase being stronger beyond z similar to 4. We observe no difference, for the narrow range of UV luninosities explored in this work, between the fraction of strong Ly alpha emitters among galaxies fainter or brighter than M-FUV*, although the fraction for the faint galaxies evolves faster, at 2 < z < 3.5, than for the bright ones. We do observe an anticorrelation between E(B - V) and f(esc)(Ly alpha): generally galaxies with high f(esc)(Ly alpha) also have small amounts of dust (and vice versa). However, when the dust content is low (E(B - V) < 0.05) We observe a very broad range of f(esc)(Ly alpha), ranging from 10(-3) to 1. This implies that the dust alone is not the only regulator of the amount of escaping Ly alpha photons.