Propagation of Uncertainty from Random Error

An experiment often consists of more than a single step. In this case, measurements at each step give rise to uncertainty. Because the measurements occur in successive steps, the uncertainty in one step necessarily contributes to that in the subsequent step. As we perform statistical analysis on these types of experiments, we must learn to account for the propagation of uncertainty from one step to the next. The propagation of uncertainty depends on the type of arithmetic operation performed on the values. For addition and subtraction, propagating the uncertainty requires us to express the absolute uncertainty of the outcome, which is the square root of the sum of absolute uncertainties for all steps. For multiplication and division, propagating uncertainty requires us to find the square root of the sum of the relative uncertainties for all steps, and this square root is equal to the relative uncertainty of the outcome, also known as the ratio between the absolute uncertainty of the outcome and the magnitude of the expected outcome. For exponential functions, we propagate the uncertainty by multiplying the power with the relative uncertainty of the base value, which then equates to the relative uncertainty of the outcome for the whole data set. Knowing how to propagate uncertainty correctly helps us identify the method that yields the least uncertainty, therefore optimizing our experimental protocols.