Uncertainty

With every experiment, there will be a source of error, whether you like it or not. Since we aren’t perfect, we need to account for the error, which we call uncertainty. A measurement reported without its uncertainty is almost meaningless: “the period is 2.00 s” is a very different claim from “the period is $2.00\pm0.05$ s.”

A quantity $x$ is written $x\pm\delta x$, where $\delta x$ is the absolute uncertainty (same units as $x$). The relative (or fractional) uncertainty is $\delta x/x$, often quoted as a percentage. The central question of error propagation is: if $f$ depends on measured quantities $x,y,\dots$, how big is $\delta f$?

Note

Note

A note on the two methods. There are two conventions. The worst-case (linear) method simply adds uncertainties and gives the largest the error could possibly be. The quadrature (statistical) method adds them in quadrature (root-sum-of-squares) and is correct when the errors are independent and random, since independent errors partly cancel rather than always conspiring. Quadrature is the standard for real lab data; the linear method is a quick conservative bound. Both are shown below — use whichever a problem specifies, defaulting to quadrature. MOST problems require quadrature uncertainty calculations.

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Addition/Subtraction

When you add or subtract quantities, the absolute uncertainties combine. For $f=x\pm y$:

$$\delta f=\delta x+\delta y\quad(\text{worst case}),\qquad \delta f=\sqrt{(\delta x)^2+(\delta y)^2}\quad(\text{quadrature}).$$

Note that this holds for subtraction too since uncertainties never subtract. This is why subtracting two nearly-equal large numbers is dangerous: the absolute uncertainty stays the same size while the result shrinks, so the relative uncertainty can blow up. (For example, measuring a thin film’s thickness as the difference of two large lengths.)

Example. Independent measurements give an empty container’s mass as $48.20\pm0.05\ \text{g}$ and its filled mass as $48.90\pm0.05\ \text{g}$. Find the sample mass and uncertainty using both conventions.

Subtract the central values, but combine the uncertainties:

$$m=48.90-48.20=0.70\ \text{g},$$ $$\delta m_{\mathrm{quad}}=\sqrt{0.05^2+0.05^2}=0.071\ \text{g},\qquad \delta m_{\mathrm{worst}}=0.05+0.05=0.10\ \text{g}.$$

Thus the quadrature result is $0.70\pm0.07\ \text{g}$; the worst-case result is $0.70\pm0.10\ \text{g}$. Although each mass was measured to about $0.1\%$, the sample mass has about $10\%$ statistical uncertainty. This calculation assumes independent errors; a shared balance offset could cancel in the subtraction.

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Multiplication/Division

When you multiply or divide, the relative uncertainties combine. For $f=xy$ or $f=x/y$:

$$\frac{\delta f}{f}=\frac{\delta x}{x}+\frac{\delta y}{y}\quad(\text{worst case}),\qquad \frac{\delta f}{f}=\sqrt{\left(\frac{\delta x}{x}\right)^2+\left(\frac{\delta y}{y}\right)^2}\quad(\text{quadrature}).$$

The rule is the same whether you multiply or divide. The practical takeaway: the least precise factor dominates the result’s precision, so there’s no point measuring one quantity to 0.1% if another enters at 5%.

Example. A cart travels $1.200\pm0.006\ \text{m}$ in $0.800\pm0.008\ \text{s}$. Treat the uncertainties as independent. Find its average speed and identify which measurement limits the precision.

The speed is $v=d/t=1.500\ \text{m/s}$. The relative distance and time uncertainties are $0.005$ and $0.010$, so

$$\frac{\delta v}{v}=\sqrt{0.005^2+0.010^2}=0.01118,\qquad \delta v=1.500(0.01118)=0.0168\ \text{m/s}.$$

Report $v=1.500\pm0.017\ \text{m/s}$. The timing contribution is twice the distance contribution and accounts for $80\%$ of the variance; improving the timer helps most. A worst-case linear estimate would give $\delta v=1.500(0.015)=0.0225\ \text{m/s}$.

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Exponent Rules

A power multiplies the relative uncertainty by the exponent. For $f=x^n$ (where $n$ can be negative or fractional):

$$\frac{\delta f}{f}=\lvert n\rvert\,\frac{\delta x}{x}.$$

So squaring a quantity doubles its relative uncertainty, while a square root halves it. For a combination like $f=\dfrac{x^a y^b}{z^c}$, the relative uncertainties add with their exponents as weights:

$$\frac{\delta f}{f}=\sqrt{\left(a\frac{\delta x}{x}\right)^2+\left(b\frac{\delta y}{y}\right)^2+\left(c\frac{\delta z}{z}\right)^2}.$$

Example. Suppose for a simple pendulum $L=1.000\pm0.005\ \text{m}$ (0.5%) and $T=2.00\pm0.02\ \text{s}$ (1%). Find $g$ with uncertainty.

From $T=2\pi\sqrt{L/g}$ we solve $g=4\pi^2 L/T^2$. Since $g\propto L\,T^{-2}$, the exponents are $1$ for $L$ and $2$ for $T$:

$$\frac{\delta g}{g}=\sqrt{\left(1\cdot 0.005\right)^2+\left(2\cdot 0.01\right)^2}=\sqrt{0.005^2+0.02^2}\approx 0.0206,$$

about 2%. With $g=4\pi^2(1.000)/(2.00)^2=9.87\ \text{m/s}^2$, the result is $g=9.87\pm0.20\ \text{m/s}^2$. Notice the timing error dominates because $T$ enters squared — so to improve the measurement, time many periods at once rather than measuring $L$ more carefully.

Example. A uniform cylindrical rod has mass $m=100.0\pm0.2\ \text{g}$, radius $r=0.500\pm0.005\ \text{cm}$, and length $L=10.00\pm0.02\ \text{cm}$. Assuming independent uncertainties, find its density and uncertainty.

Since $\rho=m/(\pi r^2L)$, the powers are $1,-2,-1$:

$$\rho=\frac{100.0}{\pi(0.500)^2(10.00)} =12.732\ \text{g/cm}^3,$$ $$\frac{\delta\rho}{\rho} =\sqrt{\left(\frac{0.2}{100.0}\right)^2 +\left(2\frac{0.005}{0.500}\right)^2 +\left(\frac{0.02}{10.00}\right)^2} =0.02020.$$

Therefore $\delta\rho=0.257\ \text{g/cm}^3$, and the result is $\rho=12.73\pm0.26\ \text{g/cm}^3$. Squaring the radius doubles its relative-uncertainty contribution; treating the two copies of $r$ as independent measurements would give the wrong answer.

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Calculus-related Uncertainty

All the rules above are special cases of one master formula. If $f$ is any function of independent measured quantities $x_1,x_2,\dots$, then a small error in each propagates through the partial derivatives:

$$\ \delta f=\sqrt{\sum_i\left(\frac{\partial f}{\partial x_i}\,\delta x_i\right)^2}$$

(or, in the worst-case version, $\delta f=\sum_i\left\lvert\dfrac{\partial f}{\partial x_i}\right\rvert\delta x_i$). The intuition is exactly the linear approximation: $\partial f/\partial x_i$ measures how sensitively $f$ responds to $x_i$, so it acts as the “amplification factor” for that input’s error.

This recovers everything above. For a single variable $f(x)$ it reduces to $\delta f=\left\lvert f'(x) \right\rvert\delta x$. Applying it to $f=x^n$ gives $\delta f=\lvert n x^{n-1}\rvert\,\delta x$, i.e. $\delta f/f=\lvert n \rvert\,\delta x/x$ — the exponent rule. A slick shortcut for products and powers is to take the logarithm first: since $\ln f=a\ln x+b\ln y-c\ln z$, differentiating gives $\dfrac{\delta f}{f}=a\dfrac{\delta x}{x}+b\dfrac{\delta y}{y}+c\dfrac{\delta z}{z}$ directly.

Example. Suppose a refraction experiment gives $f=\sin\theta$ with $\theta=30.0^\circ\pm0.5^\circ$. Find the value (with uncertainty) of $f$.

There’s no add/multiply rule for $\sin$, so use the master formula. First convert the angular uncertainty to radians: $\delta\theta=0.5^\circ=0.0087\ \text{rad}$. Then

$$\delta f=\left\lvert \frac{d}{d\theta}\sin\theta \right\rvert\delta\theta=\lvert \cos\theta \rvert\,\delta\theta=\cos(30^\circ)(0.0087)\approx 0.0076.$$

So $f=\sin30^\circ=0.500\pm0.008$. The derivative-based method handles any function — trig, logs, exponentials — where the elementary rules don’t apply.

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Problem-solving strategy

A quick decision tree for picking the right propagation rule (default to quadrature unless told otherwise):

Problem-solving strategy

1. Adding or subtracting measured quantities ($f=x\pm y$): combine absolute uncertainties, $\delta f=\sqrt{(\delta x)^2+(\delta y)^2}$ (or $\delta f=\delta x+\delta y$ worst case). Subtraction never subtracts errors — watch for two nearly-equal numbers blowing up the relative error.
2. Multiplying, dividing, or taking powers ($f=x^a y^b/z^c$): combine relative uncertainties weighted by exponents, $\dfrac{\delta f}{f}=\sqrt{\left(a\dfrac{\delta x}{x}\right)^2+\left(b\dfrac{\delta y}{y}\right)^2+\left(c\dfrac{\delta z}{z}\right)^2}$. For a single power $f=x^n$, $\dfrac{\delta f}{f}=\lvert n\rvert\dfrac{\delta x}{x}$.
3. Any other function (trig, log, exp, mixed): drop to the master formula $\delta f=\sqrt{\sum_i\left(\dfrac{\partial f}{\partial x_i}\,\delta x_i\right)^2}$; for one variable, $\delta f=\lvert f'(x)\rvert\,\delta x$. Convert angles to radians before differentiating.
4. Products and powers, but the algebra is messy: take logs first — $\ln f=a\ln x+b\ln y-c\ln z$ differentiates straight to the weighted relative-error sum.
5. One error source dominates (e.g. an exponent of 2 amplifies it): don’t waste effort on the small terms — quadrature suppresses them anyway, since the largest contribution sets the scale. Improve the dominant measurement instead.
6. Worst-case vs. quadrature: use quadrature for independent random errors (the lab default), linear sum only when a problem asks for the worst case or the errors are correlated.

Common mistakes

• Adding absolute errors where you should add relative ones (or vice versa) — addition/subtraction uses absolute, multiplication/division uses relative.
• Forgetting the exponent weight: squaring doubles relative uncertainty, square root halves it; a factor entering as $T^{-2}$ contributes $2\,\delta T/T$.
• Trying to subtract uncertainties — they always combine additively, even for $f=x-y$.
• Subtracting two nearly-equal large numbers: the absolute error survives while the result shrinks, so the relative error explodes.
• Using radians-vs-degrees inconsistently in the master formula; $\delta\theta$ must be in radians.
• Rounding to significant figures too early — carry extra digits through and round only the final $\delta f$.
• Defaulting to the worst-case linear sum when the problem expects quadrature (most do).