A determination of the surface gravity and an abundance analysis of helium in a sample of 17 He-rich and 5 normal, reference stars is presented. These results are derived from low resolution CCD spectra, but each star was measured at least 6 times in order to obtain a significant average spectrum for the spectroscopic variables. The helium abundances derived from the models used are very close to 0.1 for normal, reference stars are larger for the others, clearly indicating the He-rich phenomenon in them. NLTE effects, errors on the microturbulence value or on the surface gravity do not influence the estimated helium abundances. Nevertheless, synthesized Geneva colours are affected by the He-rich peculiarity, especially the [U-B] index which systematically changes by -0.025 mag per 0.1 of He abundance for the coolest stars in the sample. We cannot confirm the correlation between the evolutionary state and the helium abundance reported previously (Zboril et al. 1994), although we used a more reliable technique of log g determination. All He-rich objects lie within the main sequence: their surface gravities are all inside the range 4.1 < log g < 4.5, with no more than three objects having log g < 4.25. We find a significant spread of helium abundances in this range of surface gravities, from the solar value ~ 0.1 up to about 0.4. Some of the programme stars (including reference stars) present emission in their Balmer lines and therefore some kind of stellar activity. Strong helium overabundance often coexists with emission and stellar activity.