Stable isotope analyses are often used to quantify the contribution of multiple sources to a mixture, such as proportions of food sources in an animal's diet, or C3 and C4 plant inputs to soil organic carbon. Linear mixing models can be used to partition two sources with a single isotopic signature (e.g., δ(13)C) or three sources with a second isotopic signature (e.g., δ(15)N). Although variability of source and mixture signatures is often reported, confidence interval calculations for source proportions typically use only the mixture variability. We provide examples showing that omission of source variability can lead to underestimation of the variability of source proportion estimates. For both two- and three-source mixing models, we present formulas for calculating variances, standard errors (SE), and confidence intervals for source proportion estimates that account for the observed variability in the isotopic signatures for the sources as well as the mixture. We then performed sensitivity analyses to assess the relative importance of: (1) the isotopic signature difference between the sources, (2) isotopic signature standard deviations (SD) in the source and mixture populations, (3) sample size, (4) analytical SD, and (5) the evenness of the source proportions, for determining the variability (SE) of source proportion estimates. The proportion SEs varied inversely with the signature difference between sources, so doubling the source difference from 2‰ to 4‰ reduced the SEs by half. Source and mixture signature SDs had a substantial linear effect on source proportion SEs. However, the population variability of the sources and the mixture are fixed and the sampling error component can be changed only by increasing sample size. Source proportion SEs varied inversely with the square root of sample size, so an increase from 1 to 4 samples per population cut the SE in half. Analytical SD had little effect over the range examined since it was generally substantially smaller than the population SDs. Proportion SEs were minimized when sources were evenly divided, but increased only slightly as the proportions varied. The variance formulas provided will enable quantification of the precision of source proportion estimates. Graphs are provided to allow rapid assessment of possible combinations of source differences and source and mixture population SDs that will allow source proportion estimates with desired precision. In addition, an Excel spreadsheet to perform the calculations for the source proportions and their variances, SEs, and 95% confidence intervals for the two-source and three-source mixing models can be accessed at http://www.epa.gov/wed/pages/models.htm.
