The observation that individuals tend to be friends with people who are similar to themselves, commonly known as homophily, is a prominent and well-studied feature of social networks. Many machine learning methods exploit homophily to predict attributes of individuals based on the attributes of their friends. Meanwhile, recent work has shown that gender homophily can be weak or nonexistent in practice, making gender prediction particularly challenging. In this work, we identify another useful structural feature for predicting gender, an overdispersion of gender preferences introduced by individuals who have extreme preferences for a particular gender, regardless of their own gender. We call this property monophily for "love of one," and jointly characterize the statistical structure of homophily and monophily in social networks in terms of preference bias and preference variance. For prediction, we find that this pattern of extreme gender preferences introduces friend-of-friend correlations, where individuals are similar to their friends-of-friends without necessarily being similar to their friends. We analyze a population of online friendship networks in U.S. colleges and offline friendship networks in U.S. high schools and observe a fundamental difference between the success of prediction methods based on friends, "the company you keep," compared to methods based on friends-of-friends, "the company you're kept in." These findings offer an alternative perspective on attribute prediction in general and gender in particular, complicating the already difficult task of protecting attribute privacy.