Propagating Uncertainty in Predicting Individuals and Means Illustrated with Foliar Chemistry and Forest Biomass

Quantifying uncertainty is important to establishing the significance of comparisons, to making predictions with known confidence, and to identifying priorities for investment. However, uncertainty can be difficult to quantify correctly. While sampling error is commonly reported based on replicate measurements, the uncertainty in regression models used to estimate forest biomass from tree dimensions is commonly ignored and has sometimes been reported incorrectly, due either to lack of clarity in recommended procedures or to incentives to underestimate uncertainties. Even more rarely are the uncertainty in predicting individuals and the uncertainty in the mean both recognized for their contributions to overall uncertainty. In this paper, we demonstrate the effect of propagating these two sources of uncertainty using a simple example of calcium concentration of sugar maple foliage, which does not require regression, then the mass of foliage and calcium content of foliage, and finally an entire forest with multiple species and tissue types. The uncertainty due to predicting individuals is greater than the uncertainty in the mean for studies with few trees-up to 30 trees for foliar calcium concentration and 50 trees for foliar mass and calcium content in the data set we analyzed from the Hubbard Brook Experimental Forest. The most correct analysis will take both sources of uncertainty into account, but for practical purposes, country-level reports of uncertainty in carbon stocks can safely ignore the uncertainty in individuals, which becomes negligible with large enough numbers of trees. Ignoring the uncertainty in the mean will result in exaggerated confidence in estimates of forest biomass and carbon and nutrient contents.