Survival rates of cancer patients vary widely within and between malignancies. While genetic aberrations are at the root of all cancers, individual genomic features cannot explain these distinct disease outcomes. In contrast, intra-tumour heterogeneity (ITH) has the potential to elucidate pan-cancer survival rates and the biology that drives cancer prognosis. Unfortunately, a comprehensive and effective framework to measure ITH across cancers is missing. Here, we introduce a scalable measure of chromosomal copy number heterogeneity (CNH) that predicts patient survival across cancers. We show that the level of ITH can be derived from a single-sample copy number profile. Using gene-expression data and live cell imaging we demonstrate that ongoing chromosomal instability underlies the observed heterogeneity. Analysing 11,534 primary cancer samples from 37 different malignancies, we find that copy number heterogeneity can be accurately deduced and predicts cancer survival across tissues of origin and stages of disease. Our results provide a unifying molecular explanation for the different survival rates observed between cancer types.