Recent studies have revealed the mutational signatures underlying the somatic evolution of cancer, and the prevalences of associated somatic genetic variants. Here we estimate the intensity of positive selection that drives mutations to high frequency in tumors, yielding higher prevalences than expected on the basis of mutation and neutral drift alone. We apply this approach to a sample of 525 head and neck squamous cell carcinoma exomes, producing a rank-ordered list of gene variants by selection intensity. Our results illustrate the complementarity of calculating the intensity of selection on mutations along with tallying the prevalence of individual substitutions in cancer: while many of the most prevalently-altered genes were heavily selected, their relative importance to the cancer phenotype differs from their prevalence and from their P value, with some infrequent variants exhibiting evidence of strong positive selection. Furthermore, we extend our analysis of effect size by quantifying the degree to which mutational processes (such as APOBEC mutagenesis) contributes mutations that are highly selected, driving head and neck squamous cell carcinoma. We calculate the substitutions caused by APOBEC mutagenesis that make the greatest contribution to cancer phenotype among patients. Lastly, we demonstrate via in vitro biochemical experiments that the APOBEC3B protein can deaminate the cytosine bases at two sites whose mutant states are subject to high net realized selection intensities-PIK3CA E545K and E542K. By quantifying the effects of mutations, we deepen the molecular understanding of carcinogenesis in head and neck squamous cell carcinoma.