Standard normal statistics, chi-squared statistics, Students t statistics and F statistics are used to map quantitative trait nucleotides for both small and large sample sizes. In genome-wide association studies (GWASs) of single-nucleotide polymorphisms (SNPs), the statistical distributions depend on both genetic effects and SNPs but are independent of SNPs under the null hypothesis of no genetic effects. Therefore, hypothesis testing when a nuisance parameter is present only under the alternative was introduced to quickly approximate the critical thresholds of these test statistics for GWASs. When only the statistical probabilities are available for high-throughput SNPs, the approximate critical thresholds can be estimated with chi-squared statistics, formulated by statistical probabilities with a degree of freedom of two. High similarities in the critical thresholds between the accurate and approximate estimations were demonstrated by extensive simulations and real data analysis.