The two-stage least squares approach together with quantile regression analysis is adopted here to estimate the educational production function. Such a methodology is able to capture the extreme behaviors of the two tails of students' performance and the estimation outcomes have important policy implications. Our empirical study is applied to the case of students' scores in the Basic Competence Test in Taiwan. The empirical estimation outcomes between traditional OLS and quantile regression on peer-group effects, school characteristics, and family characteristics are diverse and depend on students' ability. Such findings have important implications for parents as well as for government. (Contains 6 tables and 1 figure.)