Single-cell RNA sequencing (scRNA-seq) has been widely used to characterize cell types based on their average gene expression profiles. However, most studies do not consider cell type-specific variation across donors. Modelling this cell type-specific inter-individual variation could help elucidate cell type-specific biology and inform genes and cell types underlying complex traits. We therefore develop a new model to detect and quantify cell type-specific variation across individuals called CTMM (Cell Type-specific linear Mixed Model). We use extensive simulations to show that CTMM is powerful and unbiased in realistic settings. We also derive calibrated tests for cell type-specific interindividual variation, which is challenging given the modest sample sizes in scRNA-seq. We apply CTMM to scRNA-seq data from human induced pluripotent stem cells to characterize the transcriptomic variation across donors as cells differentiate into endoderm. We find that almost 100% of transcriptome-wide variability between donors is differentiation stage-specific. CTMM also identifies individual genes with statistically significant stage-specific variability across samples, including 85 genes that do not have significant stage-specific mean expression. Finally, we extend CTMM to partition interindividual covariance between stages, which recapitulates the overall differentiation trajectory. Overall, CTMM is a powerful tool to illuminate cell type-specific biology in scRNA-seq.