Our society is increasingly relying on digitalized, aggregated opinions of individuals to make decisions (e.g., product recommendation based on collective ratings). One key requirement of harnessing this "wisdom of crowd" is the independency of individuals' opinions; yet, in real settings, collective opinions are rarely simple aggregations of independent minds. Recent experimental studies document that disclosing prior collective ratings distorts individuals' decision making as well as their perceptions of quality and value, highlighting a fundamental discrepancy between our perceived values from collective ratings and products' intrinsic values. Here we present a mechanistic framework to describe herding effects of prior collective ratings on subsequent individual decision making. Using large-scale longitudinal customer rating datasets, we find that our method successfully captures the dynamics of ratings growth, helping us separate social influence bias from inherent values. Leveraging the proposed framework, we quantitatively characterize the herding effects existing in product rating systems and promote strategies to untangle manipulations and social biases.