This paper develops a framework to analyze partial population experiments, a generalization of the cluster experimental design where clusters are assigned to different treatment intensities. The framework allows for heterogeneity in cluster sizes and outcome distributions. The paper studies the large-sample behavior of OLS estimators and cluster-robust variance estimators and shows that (i) ignoring cluster heterogeneity may result in severely underpowered experiments and (ii) the clusterrobust variance estimator may be upward-biased when clusters are heterogeneous. The paper derives formulas for power, minimum detectable effects, and optimal cluster assignment probabilities. All the results apply to cluster experiments, a particular case of the framework. The paper sets up a potential outcomes framework to interpret the OLS estimands as causal effects. It implements the methods in a large-scale experiment to estimate the direct and spillover effects of a communication campaign on property tax compliance. The analysis reveals an increase in tax compliance among individuals directly targeted with the mailing, as well as compliance spillovers on untreated individuals in clusters with a high proportion of treated taxpayers.