You rarely measure the thing you actually want. You want a density, but you measure a mass and a volume and divide. You want a speed, but you measure a distance and a time. You want the strength of gravity g, but you measure a pendulum's length and its swing time. In every case the final answer is calculated from measurements that each carry their own little fog of uncertainty.

So a natural question: if each measurement is uncertain, how uncertain is the result you build from them? The fog doesn't vanish when you press the buttons on a calculator — it propagates, flowing through the arithmetic into the final number. This page gives you the small set of rules that tell you exactly how much fog comes out the other end.

Two ways to state an uncertainty

Before combining anything, be fluent in the two languages of uncertainty — because the rules that follow switch between them.

• The absolute uncertainty is the raw \pm in the same units as the quantity: a length of 12.4 \pm 0.2\,\text{cm} has an absolute uncertainty of 0.2\,\text{cm}.
• The percentage (fractional) uncertainty compares that \pm with the size of the value itself:

So 12.4 \pm 0.2\,\text{cm} carries a percentage uncertainty of \dfrac{0.2}{12.4}\times100 = 1.6\%. To go the other way — from a percentage back to an absolute — just undo the division:

A resistor quoted as 470\,\Omega \pm 5\% therefore has an absolute uncertainty of \dfrac{5}{100}\times 470 = 23.5\,\Omega, i.e. 470 \pm 23.5\,\Omega. Being able to slide between the two forms in your head is the whole trick, because each combining rule wants one particular form.

Adding or subtracting: add the absolute uncertainties

When your result is a sum or a difference of measured quantities, the worst case is that both errors push the same way, so their absolute uncertainties simply add:

Notice it is the same rule whether you add or subtract — subtracting doesn't subtract the uncertainties, it still adds them. Uncertainty only ever grows.

Worked example — an extension. A spring's length is measured before and after loading: L_1 = 45.0 \pm 0.5\,\text{mm} and L_2 = 68.0 \pm 0.5\,\text{mm}. The extension is e = L_2 - L_1.

• Best value: e = 68.0 - 45.0 = 23.0\,\text{mm}.
• Absolute uncertainty: \Delta e = \Delta L_2 + \Delta L_1 = 0.5 + 0.5 = 1.0\,\text{mm}.
• Result: e = 23.0 \pm 1.0\,\text{mm}, a percentage uncertainty of \dfrac{1.0}{23.0}\times100 = 4.3\%.

Look what just happened to the percentages. Each length was only about 1\% uncertain, yet their difference is 4.3\% uncertain — because subtracting two close numbers keeps the absolute \pm the same size while shrinking the value. Hold that thought; it comes back in the Watch out! below.

Multiplying or dividing: add the percentage uncertainties

When your result is a product or a quotient, absolute uncertainties can't add — they're in different units (you can't add grams to millilitres). What add instead are the percentage uncertainties:

Every factor on top or bottom contributes its percentage uncertainty, all added together — division counts exactly the same as multiplication.

Worked example — a density. A metal block is measured as m = 240 \pm 2\,\text{g} and V = 30.0 \pm 0.6\,\text{cm}^3, and \rho = \dfrac{m}{V}.

• Best value: \rho = \dfrac{240}{30.0} = 8.0\,\text{g/cm}^3.
• Percentage uncertainties of the inputs: \%\Delta m = \dfrac{2}{240}\times100 = 0.83\% and \%\Delta V = \dfrac{0.6}{30.0}\times100 = 2.0\%.
• Add them: \%\Delta \rho = 0.83\% + 2.0\% = 2.83\%.
• Convert back to absolute: \dfrac{2.83}{100}\times 8.0 = 0.23\,\text{g/cm}^3, so \rho = 8.0 \pm 0.2\,\text{g/cm}^3 (uncertainty to 1 s.f.).

A second example — a speed. d = 100 \pm 1\,\text{m} (a 1\% track) covered in t = 20.0 \pm 0.4\,\text{s} (a 2\% timing). Then v = d/t = 5.0\,\text{m/s} with \%\Delta v = 1\% + 2\% = 3\%, i.e. 5.0 \pm 0.15\,\text{m/s}. The routine is always the same: turn each input into a percentage, add, then turn the total back into an absolute \pm if you need one.

Raising to a power: multiply the percentage by n

A power is just repeated multiplication, so raising a quantity to the power n adds its percentage uncertainty to itself n times — which is the same as multiplying it by n:

Squaring doubles the percentage uncertainty; cubing trebles it. This works for any real n, so a square root (n = \tfrac12) halves it.

• Area of a square. A side s is 1.5\% uncertain. Its area A = s^2 is 2 \times 1.5\% = 3.0\% uncertain.
• Volume from a radius. For a sphere V = \tfrac{4}{3}\pi r^3. If r is 2\% uncertain, then \%\Delta V = 3 \times 2\% = 6\% — the constant \tfrac{4}{3}\pi is exact and adds nothing.

Worked example — g from a pendulum. A pendulum of length L and period T gives g = \dfrac{4\pi^2 L}{T^2}. Suppose \%\Delta L = 1\% and \%\Delta T = 0.5\%. Combining the product/quotient rule with the power rule:

The T^2 makes the timing count double — which is exactly why careful experimenters time many swings at once to shrink \%\Delta T.

The rules for combining uncertainties

To propagate uncertainty through a calculation, match the operation to its rule:

• Addition / subtraction Q = a \pm b: add the absolute uncertainties, \Delta Q = \Delta a + \Delta b.
• Multiplication / division Q = ab or Q = a/b: add the percentage uncertainties, \%\Delta Q = \%\Delta a + \%\Delta b.
• Powers Q = a^n: multiply the percentage uncertainty by n, \%\Delta Q = n\,\%\Delta a.
• Exact constants (like 2, \pi, \tfrac{4}{3}) carry no uncertainty and are ignored.

See it: the biggest contributor wins

Here is the density calculation \rho = m/V as a picture. The top two bars are the percentage uncertainties of your two measurements; the bottom bar is the combined percentage uncertainty of the density — and it is exactly the two upper bars laid end to end, because for a quotient the percentages add. Drag the sliders and watch which input pushes the total up hardest.

Hunt the dominant term

Because the percentage uncertainties simply add, the final uncertainty is bossed around by whichever measurement has the biggest percentage uncertainty. In the density example the volume (2.0\%) dwarfed the mass (0.83\%): halve the volume's uncertainty and you barely-improve the mass, but you nearly halve the total.

This is the single most useful idea on the page. To improve an experiment, don't polish every measurement equally — find the term with the largest percentage uncertainty and attack that one. Spending an afternoon refining a measurement that already contributes 0.1\% while another contributes 5\% is wasted effort.