Research in many fields has become increasingly reliant on large and complex datasets. "Big Data" holds untold promise to rapidly advance science by tackling new questions that cannot be answered with smaller datasets. While powerful, research with Big Data poses unique challenges, as many standard lab protocols rely on experts examining each one of the samples. This is not feasible for large-scale datasets because manual approaches are time-consuming and hence difficult to scale. Meanwhile, automated approaches lack the accuracy of examination by highly trained scientists and this may introduce major errors, sources of noise, and unforeseen biases into these large and complex datasets. Our proposed solution is to 1) start with a small, expertly labelled dataset, 2) amplify labels through web-based tools that engage citizen scientists, and 3) train machine learning on amplified labels to emulate expert decision making. As a proof of concept, we developed a system to quality control a large dataset of three-dimensional magnetic resonance images (MRI) of human brains. An initial dataset of 200 brain images labeled by experts were amplified by citizen scientists to label 722 brains, with over 80,000 ratings done through a simple web interface. A deep learning algorithm was then trained to predict data quality, based on a combination of the citizen scientist labels that accounts for differences in the quality of classification by different citizen scientists. In an ROC analysis (on left out test data), the deep learning network performed as well as a state-of-the-art, specialized algorithm (MRIQC) for quality control of T1-weighted images, each with an area under the curve of 0.99. Finally, as a specific practical application of the method, we explore how brain image quality relates to the replicability of a well established relationship between brain volume and age over development. Combining citizen science and deep learning can generalize and scale expert decision making; this is particularly important in emerging disciplines where specialized, automated tools do not already exist.