Propagating Uncertainties and Uncertainty in Graphs

When measured quantities are combined in a calculation, their uncertainties combine too. Learning to propagate uncertainties lets you attach a meaningful uncertainty to any calculated result. Graphs provide a second powerful tool, both for reducing random error and for extracting uncertainties in gradients and intercepts.

Adding and subtracting quantities

When quantities are added or subtracted, their absolute uncertainties add: $$\text{if } y = a \pm b, \quad \text{then } \Delta y = \Delta a + \Delta b.$$

Note that even when you subtract quantities, the uncertainties still add — errors never cancel. For example, if a thermometer reads an initial temperature $T_1 = (20.0 \pm 0.5)\ ^\circ\text{C}$ and a final temperature $T_2 = (85.0 \pm 0.5)\ ^\circ\text{C}$, the temperature change is $$\D