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