Error Propagation

When you combine measured quantities (each with its own uncertainty) into a formula, the result also has uncertainty. Error propagation tells you how to estimate it.

Concept

We use lowercase $\Delta a$ for the absolute uncertainty of $a$ , and the convention that uncertainties are small ( $\Delta a \ll a$ ). Two key rules:

Rule 1: Sum and Difference

For $Z = A + B$ or $Z = A - B$ : $\Delta Z = \Delta A + \Delta B.$ Absolute errors add (worst-case bound).

Derivation: $Z + \Delta Z = (A + \Delta A) + (B + \Delta B) = (A+B) + (\Delta A + \Delta B)$ .

Rule 2: Product and Quotient

For $Z = A \cdot B$ or $Z = A/B$ : $\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}.$ Relative errors add.

Derivation (product): $Z + \Delta Z = (A + \Delta A)(B + \Delta B) = AB + A\Delta B + B\Delta A + \Delta A \Delta B.$ Neglect the second-order term $\Delta A \Delta B$ : $\Delta Z \approx A\Delta B + B\Delta A.$ Divide by $Z = AB$ : $\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$ .

Rule 3: Powers

For $Z = A^n B^m / C^p$ : $\frac{\Delta Z}{Z} = n\frac{\Delta A}{A} + m\frac{\Delta B}{B} + p\frac{\Delta C}{C}.$ Powers multiply the relative error by their exponent (absolute value).

Quick Reference

Operation	Rule
$Z = A \pm B$	$\Delta Z = \Delta A + \Delta B$
$Z = AB$ or $Z = A/B$	$\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$
$Z = A^n$	$\frac{\Delta Z}{Z} = n\frac{\Delta A}{A}$
$Z = A^p B^q / C^r$	$\frac{\Delta Z}{Z} = p\frac{\Delta A}{A} + q\frac{\Delta B}{B} + r\frac{\Delta C}{C}$

(Statistical addition in quadrature is more refined but at school level we add directly.)

Worked Example

Q: The acceleration due to gravity is computed from a pendulum: $g = 4\pi^2 l / T^2$ . If $l = 100.0 \pm 0.1$ cm and $T = 2.00 \pm 0.02$ s, find the percent error in $g$ .

Solution: $\frac{\Delta g}{g} = \frac{\Delta l}{l} + 2\frac{\Delta T}{T}$

Relative errors:

$\Delta l / l = 0.1/100.0 = 0.001 = 0.1\%$ .
$\Delta T / T = 0.02/2.00 = 0.01 = 1.0\%$ .

So $\frac{\Delta g}{g} = 0.1\% + 2(1.0\%) = 2.1\%.$

The dominant contribution is the period — it enters squared.

Common Confusions

For sums/differences, absolute errors add; for products/quotients, relative errors add. Don't mix the two.
A constant multiplier (like $4\pi^2$ ) is exact — it contributes zero error.
Doubling $T$ measurement frequency reduces $\Delta T$ random component, but a single systematic error remains regardless.
Subtraction can blow up relative errors: subtracting two near-equal quantities gives a small result with large fractional uncertainty.

Key Takeaways

Sum/diff → add absolute errors.
Product/quot → add relative errors.
Powers → multiply relative error by exponent.
Identify the dominant error source — that's where measurement should be improved.
Avoid subtracting near-equal numbers when precision matters.