Propagation of Uncertainty from Systematic Error

The atomic mass of an element varies due to the relative ratio of its isotopes. A sample's relative proportion of oxygen isotopes influences its average atomic mass. For instance, if we were to measure the atomic mass of oxygen from a sample, the mass would be a weighted average of the isotopic masses of oxygen in that sample. Since a single sample is not likely to perfectly reflect the true atomic mass of oxygen for all the molecules of oxygen on Earth, the mass we obtain from this particular sample will give rise to sampling error or a type of systematic error.This means that there is an uncertainty in the atomic mass of oxygen. It is not a random error but specific to the system or a systematic error. The probability of finding a particular value is the same within the uncertainty range. The distribution plot of values has a rectangular shape. With a sharp cut in the frequency of occurrence at the minimum and maximum reported values, the values in between have roughly the same frequency of appearance. This differs from the Gaussian distribution of values with random errors, where the frequency of occurrence is high near the central or mean value and tapers down as the value deviates outwards from the mean.

The standard deviation or standard uncertainty obtained from the rectangular distribution plot can be used to measure uncertainty. The individual uncertainties from constituent atoms in a molecule propagate to the uncertainty of its molecular mass. The propagation of systematic errors does not follow the same rules as that of random errors. Because systematic errors are not reduced by repeated measurements, their propagation is much more additive. Consider the following example: In a molecule containing multiple identical atoms, the uncertainty in the mass of one atom is multiplied by the number of similar atoms to get the total contribution of uncertainty toward the molecular mass of those particular atoms. If heteroatoms are present in a molecule, the uncertainty in the molecular mass is the square root of the sum of the squares of total uncertainties from each type of atom present.