Amorphous materials are being described by increasingly powerful computer simulations, but new approaches are still needed to fully understand their intricate atomic structures. Here, we show how machine-learning-based techniques can give new, quantitative chemical insight into the atomic-scale structure of amorphous silicon (a-Si). We com- bine a quantitative description of the nearest- and next-nearest- neighbor structure with a quantitative description of local stability. The analysis is applied to an ensemble of a-Si networks in which we tailor the degree of ordering by varying the quench rates down to 10^10 K/s. Our approach associates coordination defects in a-Si with distinct stability regions and it has also been applied to liquid Si, where it traces a clear-cut transition in local energies during vitrification. The method is straightforward and inexpensive to apply, and therefore expected to have more general significance for developing a quantitative understanding of liquid and amorphous states of