Propagation of Uncertainty from Systematic Error

Atomic mass varies due to differences in the relative proportion of isotopes in each source. Since samples rarely reflect the true global isotopic distribution, the measured atomic mass represents a weighted average specific to that sample. This variation constitutes a systematic error, creating uncertainty in the reported atomic mass value.

Systematic error from isotopic variation produces a rectangular distribution, where values have approximately equal probability across the range from lowest to highest. Random errors follow a Gaussian distribution with higher frequency near the mean and lower frequency at extremes. Standard deviation from either distribution type serves as the measure of uncertainty.

Standard uncertainty is the standard deviation calculated from a rectangular distribution plot. It quantifies the spread of values within the uncertainty range and serves as a numerical measure of how much the reported atomic mass might deviate from the true value due to isotopic variation in the sample.

Individual uncertainties from constituent atoms propagate to create net uncertainty in molecular mass. For identical atoms in a molecule, multiply each atom's standard uncertainty by the number of those atoms. For heteroatomic molecules, sum the squares of total uncertainties from each element type, then take the square root to find total molecular mass uncertainty.

Systematic errors are not reduced by repeated measurements because they consistently bias results in one direction. Unlike random errors that average out over multiple trials, systematic errors accumulate additively through calculations. This fundamental difference means their propagation follows different mathematical rules, requiring direct addition rather than root-sum-square methods.

For heteroatomic molecules, calculate total uncertainty by taking the square root of the sum of squares of all element-specific uncertainties. First, multiply each element's standard uncertainty by its atom count to get total contribution per element. Then apply the root-sum-square formula across all different element types present in the molecule.

The measured atomic mass is a weighted average of isotopic masses present in a sample. Different samples contain different isotopic ratios, producing different weighted averages. Since no single sample perfectly represents Earth's true isotopic composition, each measurement carries sampling error—a type of systematic error that creates uncertainty in the reported atomic mass.