The accurate extraction of species-abundance information from DNA-based data (metabarcoding, metagenomics) could contribute usefully to diet reconstruction and quantitative food webs, the inference of species interactions, the modelling of population dynamics and species distributions, the biomonitoring of environmental state and change, and the inference of false positives and negatives. However, capture bias, capture noise, species pipeline biases, and pipeline noise all combine to inject error into DNA-based datasets. We focus on methods for correcting the latter two error sources, as the first two are addressed extensively in the ecological literature. To extract abundance information, it is useful to distinguish two concepts. (1) Across-species quantification describes relative species abundances within one sample. (2) In contrast, within-species quantification describes how the abundance of each individual species varies from sample to sample, as in a time series, an environmental gradient, or different experimental treatments. Firstly, we review methods to remove species pipeline biases and pipeline noise. Secondly, we demonstrate experimentally (with a detailed protocol) how to use a 'DNA spike-in' to remove pipeline noise and recover within-species abundance information. We also introduce a statistical estimator that can partially remove pipeline noise from datasets that lack a physical DNA spike-in.