It is known that the quality factors (Q) of photonic crystal nanocavities vary from sample to sample due to air-hole fabrication fluctuations. In other words, for the mass production of a cavity with a given design, we need to consider that the Q can vary significantly. So far, we have studied the sample-to-sample variation in Q for symmetric nanocavity designs, that is, nanocavity designs where the positions of the holes maintain mirror symmetry with respect to both symmetry axes of the nanocavity. Here we investigate the variation of Q for a nanocavity design in which the air-hole pattern has no mirror symmetry (a so-called asymmetric cavity design). First, an asymmetric cavity design with a Q of about 250,000 was developed by machine learning using neural networks, and then we fabricated fifty cavities with the same design. We also fabricated fifty symmetric cavities with a design Q of about 250,000 for comparison. The variation of the measured Q values of the asymmetric cavities was 39% smaller than that of the symmetric cavities. This result is consistent with simulations in which the air-hole positions and radii are randomly varied. Asymmetric nanocavity designs may be useful for mass production since the variation in Q is suppressed.