The uncertainty analysis guidelines currently published by several engineering societies and standards organizations all use variations of the same traditional uncertainty propagation methodology. Under this methodology, a Taylor series approximation is used to estimate how the uncertainty in measured variables propagates through data reduction equations to produce uncertainty in experimental results. Prior to calculation, the elemental uncertainties for each measurement are combined into one or two aggregate uncertainty values for that measurement. The aggregate uncertainty values are then used in the calculations to estimate the uncertainty in the final results. If some of the measurements have correlated or shared elemental uncertainties, then a matrix of uncertainty covariance terms is used to correct the uncorrelated uncertainty estimates. For complex uncertainty analyses, the generation and maintenance of the covariance matrix is burdensome and often confusing. This paper describes an alternative methodology wherein the uncertainty in each measurement is represented as a vector of elemental uncertainties and these vectors are used to calculate the uncertainty in the final results. This procedure simplifies the propagation equation and eliminates the need for a covariance matrix. This method is mathematically equivalent to the traditional method, but it is easier to automate, and it makes complex uncertainty correlations easier to understand.