Federated learning is a distributed machine learning approach where multiple clients collaboratively train a model without sharing their local data, which contributes to preserving privacy. A challenge in federated learning is managing heterogeneous data distributions across clients, which can hinder model convergence and performance due to the need for the global model to generalize well across diverse local datasets. We propose to use local characteristic statistics, by which we mean some statistical properties calculated independently by each client using only their local training dataset. These statistics, such as means, covariances, and higher moments, are used to capture the characteristics of the local data distribution. They are not shared with other clients or a central node. During training, these local statistics help the model learn how to condition on the local data distribution, and during inference, they guide the client's predictions. Our experiments show that this approach allows for efficient handling of heterogeneous data across the federation, has favorable scaling compared to approaches that directly try to identify peer nodes that share distribution characteristics, and maintains privacy as no additional information needs to be communicated.