Every measurement in a lab comes with a wobble. You read a length as 12.4 \pm 0.1\ \text{cm}, a time as 2.03 \pm 0.02\ \text{s} — the number after the \pm is the uncertainty, your honest estimate of how far the reading might stray from the truth (typically one standard deviation). But you rarely want the raw measurements themselves. You want a speed, a density, a resistance — some quantity computed from them. So the real question is: if my inputs are fuzzy, how fuzzy is the answer I calculate from them?

That is propagation of uncertainty: the rules for pushing error bars through a formula. Get them right and your final result carries a defensible \pm; get them wrong — say, by naively adding the errors — and you either overstate your uncertainty (throwing away precision you earned) or, worse, understate it (claiming a result is significant when it is noise). Every experimental physics lab lives or dies by these rules, and they rest on one surprising idea: independent errors add like the sides of a right triangle, not like ordinary numbers.

Why errors add in quadrature, not straight

Suppose you measure two lengths and add them, f = a + b, each with its own uncertainty \delta a and \delta b. It is tempting to say the total error is just \delta a + \delta b. That would be the worst case — both measurements happening to err in the same direction by their full amount at the same time. But if the two errors are independent and random, they are just as likely to partly cancel as to reinforce. On average, that cancellation shrinks the combined error below the naive sum.

The correct combination is the sum in quadrature — add the squares and take the root:

This is exactly Pythagoras. If you think of each independent uncertainty as a leg of a right triangle, the combined uncertainty is the hypotenuse. And that is why quadrature always gives less than plain addition: the hypotenuse of a right triangle is shorter than the sum of its two legs. Drag the legs below and watch the total error grow — always shorter than \delta a + \delta b, and dominated by the larger leg.

The two independent uncertainties \delta a and \delta b are the perpendicular legs; the combined uncertainty \delta f = \sqrt{\delta a^2 + \delta b^2} is the hypotenuse. Because it is a right triangle, the total is always less than the straight sum \delta a + \delta b, and a much larger leg dominates a much smaller one.

Absolute errors for sums and differences

The quadrature rule covers both adding and subtracting, because squaring kills the sign. For f = a + b or f = a - b, combine the absolute uncertainties in quadrature:

Worked example. Two masses read m_1 = 50.0 \pm 0.3\ \text{g} and m_2 = 30.0 \pm 0.4\ \text{g}. Their total is 80.0\ \text{g}, with uncertainty

so m_1 + m_2 = 80.0 \pm 0.5\ \text{g}. Note the total error (0.5) is less than the naive sum (0.7) but more than either alone — the classic 3-4-5 triangle. The same 0.5\ \text{g} is the uncertainty on the difference m_1 - m_2 = 20.0\ \text{g} too.

Relative errors for products and quotients

For multiplication and division the same quadrature idea applies, but to the relative (fractional) uncertainties \delta a / a. For f = a\,b or f = a / b:

Worked example — a density. A block has mass m = 240 \pm 5\ \text{g} (a 2.1\% relative error) and volume V = 100 \pm 3\ \text{cm}^3 (a 3.0\% relative error). The density is \rho = m/V = 2.40\ \text{g/cm}^3, and its relative error is

So \delta\rho \approx 0.037 \times 2.40 \approx 0.09\ \text{g/cm}^3, and \rho = 2.40 \pm 0.09\ \text{g/cm}^3. And here is a lab-planning gem: because errors add in quadrature, the biggest relative error dominates. Chasing the 2.1\% mass error down to 1\% would barely move the total; your effort is far better spent on the 3.0\% volume. Quadrature tells you where to aim.

The one formula that rules them all

Sums, differences, products, quotients, powers, logs — every rule above is a special case of a single master formula built from partial derivatives. For any function f(x_1, x_2, \ldots, x_n) of independently measured inputs:

Watch it reproduce the earlier rules. For f = a + b, both partials are 1, so (\delta f)^2 = (\delta a)^2 + (\delta b)^2 — quadrature, as before. For f = ab, \partial f/\partial a = b and \partial f/\partial b = a, giving (\delta f)^2 = b^2(\delta a)^2 + a^2(\delta b)^2; divide through by f^2 = a^2 b^2 and out pops the relative-error rule. One formula, all the cases.

Worked example — the master formula in action

A simple pendulum measures g through g = \dfrac{4\pi^2 L}{T^2}, where L is its length and T its period. Suppose L = 1.000 \pm 0.005\ \text{m} (0.5\%) and T = 2.006 \pm 0.004\ \text{s} (0.2\%). Rather than differentiate the messy expression, use the relative-error shortcut for products and powers: a factor raised to the power k contributes k times its relative error. Here g \propto L^1 T^{-2}, so

That is a 0.64\% uncertainty. The factor of 2 on the period matters enormously: because T appears squared, its error counts double, which is exactly why careful experimenters time many swings to pin T down tightly. The partial-derivative formula makes that sensitivity explicit — the exponent is the weight.