A data set of 53 H II galaxies and giant H II regions with reliable observational uncertainties reveals a highly significant physical dispersion in the oxygen abundance helium abundance relation commonly used to infer the primordial helium abundance (He/H)p. Consideration of possible sources of scatter indicates that it is almost certainly one-sided, i.e., upward from a minimum value of He/H at each O/H. The presence of an asymmetric physical dispersion violates the assumptions of conventional least-squares fitting, and hence casts doubt on values of (He/H)p obtained by extrapolation of ridge-line fits to the O/H-He/H relation. The loss of oxygen- but not helium in supernova explosions will lead naturally to a positive dispersion in He/H. Such galactic winds must be present in H II galaxies because their masses are significantly below the critical mass needed for prevention. The observed dispersion can be explained by chemical evolution models in which between 0% and 95% of the newly synthesized oxygen is lost. Variations in the time between successive bursts of star formation and incomplete mixing of newly synthesized helium will also introduce upward scatter in He/H. We therefore conclude that (He/H)p can be reliably determined only by fitting to the lower envelope of the O/H-He/H relation. The paucity of galaxies near the envelope makes its position difficult to determine at present, but existing data give (He/H)p = 0.0759 +/- 0.0014, somewhat higher than the recent, very low estimate of 0.0738 +/- 0.0016, obtained via ridge-line fitting. More high-quality H II galaxy data with reliable formal uncertainties are required to establish the position of the lower envelope of the O/H He/H relation and produce an accurate value for (He/H)p.
