Lecture 3 Summary

We begin with a brief review of the rules for treating significant figures ("sig-figs") in calculations. After discussing the types of errors associated with measurements, and their relation to the concepts of precision and accuracy, we define absolute and relative uncertainty. Then, similar to what we did for significant figures, we show how to properly treat uncertainties asssociated with numbers used in calculations so that we can report the uncertainty in our final answer. In other words, we determine how uncertainties in measured quantities propagate through calculations involving those quantities.

Propagation of uncertainty

Addition and subtraction: The absolute uncertainty of the result is the square root of the sum of the squares of the absolute uncertainties.

Multiplication and division: The percent relative uncertainty of the result is the square root of the sum of the squares of the percent relative uncertainties.

The real rule for significant figures

Now that we know how to quantitatively treat propagation of uncertainty, we can determine the uncertainty of the result of any calculation, given the uncertainties of the values going into the calculation. This, in turn, allows us to define a more rigorous rule for significant figures, which is stated as follows: The first uncertain figure of the result of the calculation is the last significant figure.