A metallurgical balance drawn around a mineral processing plant or smelter must involve all process streams crossing the plant boundary. Modern sampling theory, as formulated by Gy, and sound statistical method can be used to determine the uncertainties in the observed assays for the set of balance analytes and in the mass flow rates of the process streams for the balance period. These uncertainties can then be used together with the material balance equations to arrive at adjusted component mass flows that satisfy the material balances exactly. To determine whether or not the data is consistent with the level of uncertainty in the data and with the material balance equations, a goodness of fit test is applied. This goodness of fit criterion is the principal tool to quantify the quality of the material balances._x000D_
The mathematical structure of the statistical material balance problem poses a number of challenges. For example, while the variances of the adjusted component mass flows do not vanish, the variances of the sums of the adjusted component mass flows about each balance node do vanish, simply because the balancing computation demands that the balances close exactly. Even though it can be shown that the variances of the adjusted component mass flows are always lower than the estimates for the corresponding unadjusted component flows, it is still not possible to construct a confidence interval on the component mass flow into (and out of) a balance node using the adjusted component flows. On the face of it, such a confidence interval is exactly what is most desired from the balance calculations. The problem arises because the balance problem is a data adjustment rather than a parameter estimation problem._x000D_
It is possible, however, to calculate the recovery of a component to a product stream and determine a confidence interval for the recovery value. It is also possible to calculate an uncertainty in the mass flow of any balance species in any process stream, before and after balancing.