error propagation
/ EHR-ur prop-uh-GAY-shun /
Suppose you build a recipe from two measured ingredients, each known only roughly. The dish you end up with cannot be more certain than the parts that went into it — the wobble in each ingredient flows downstream into the final result. Error propagation is the bookkeeping that tracks how the uncertainties in your individual measurements combine to give the uncertainty in a value you calculate from them.
When you compute a result from several measured quantities — say a concentration from a mass, a volume, and a molar mass — each input carries its own uncertainty, and error propagation gives the rules for combining them. For sums and differences, the absolute uncertainties combine (added in quadrature, meaning as the square root of the sum of squares); for products and quotients, the relative uncertainties combine the same way. The result is a single uncertainty for the final answer.
Error propagation matters because it tells you where your final uncertainty really comes from, so you can improve the right step instead of fussing over a measurement that barely contributes. A useful and humbling lesson it teaches: the largest relative uncertainty usually dominates the answer, so buying a fancier balance is pointless if a sloppy volume measurement is what's actually limiting you. The caveat is that the standard formulas assume the input errors are independent and random; correlated or systematic errors need separate, more careful treatment.
A density is found by dividing a mass (10.00 ± 0.01 g, so 0.1% relative) by a volume (5.0 ± 0.1 mL, so 2% relative). The volume's 2% swamps the mass's 0.1%, so the density's uncertainty is about 2% — improving the balance would change almost nothing.
The biggest relative uncertainty wins; fix that one.
A common shortcut error is to just add percentages directly; the correct rule combines them in quadrature (square, sum, square-root), which always gives a smaller total than naive addition.