Propagation of Uncertainties in Calculations

A measured quantity is rarely the answer. We measure a mass and a volume in order to report a density; we measure a length and a time in order to report a speed; we measure a diameter in order to report a cross-sectional area. Each measured input carries an uncertainty, and those uncertainties must be carried through the calculation to the final result. That process is propagation of uncertainty, and it is what this chapter is about.

The previous chapter established how to obtain $\Delta x$ for a single measured quantity. Here we answer the next question: given $x \pm \Delta x$ and $y \pm \Delta y$, what is the uncertainty in $x+y$, in $xy$, in $\dfrac{x}{y}$, in $x^{n}$? Three short rules cover everything the IB examines, and — more usefully — the same three rules tell you in advance which measurement in an experiment is worth improving. That is the practical payoff: a quantity raised to the third power contributes three times its percentage uncertainty to the result, so it is the one to measure carefully, whatever the apparatus cupboard happens to offer.

The governing principle. Absolute uncertainties add when quantities are added or subtracted; fractional uncertainties add when quantities are multiplied or divided. Everything below is these two sentences, worked out.

1. Rule 1: Addition and Subtraction

If $z = x + y$ or $z = x - y$, then the absolute uncertainties add:

The reason is worst-case reasoning. Take $z = x + y$: the largest plausible value of $z$ occurs when both $x$ and $y$ sit at the top of their intervals, giving $z_{\max} = (x + \Delta x) + (y + \Delta y) = (x+y) + (\Delta x + \Delta y)$, and the smallest occurs symmetrically. The half-width of the interval for $z$ is therefore $\Delta x + \Delta y$.

The subtraction case is the one that surprises students. If $z = x - y$, the largest value of $z$ occurs when $x$ is at its maximum and $y$ at its minimum: $z_{\max} = (x + \Delta x) - (y - \Delta y) = (x - y) + (\Delta x + \Delta y)$. The uncertainties still add; they never subtract. Uncertainty only ever grows.

Never subtract uncertainties. A difference of two quantities is less certain than either of them, in fractional terms — often dramatically so. Measuring a beaker at $(152 \pm 1)~\text{g}$ full and $(148 \pm 1)~\text{g}$ empty gives a contents mass of $(4 \pm 2)~\text{g}$: two measurements each precise to $0.7\%$ have produced a result uncertain to $50\%$. This is why an experiment should be designed to measure a quantity directly wherever possible, rather than as the small difference of two large numbers.

2. Rule 2: Multiplication and Division

If $z = xy$ or $z = \dfrac{x}{y}$, then the fractional uncertainties add:

To see why for a product, write the extreme value and expand:

The final term is a product of two small quantities and is negligible — if $\Delta x$ is $1\%$ of $x$ and $\Delta y$ is $1\%$ of $y$, that term is $0.01\%$ of $xy$. Dropping it and dividing through by $z = xy$:

which is the rule. The division case follows from the same expansion applied to $\dfrac{x}{y}$, with the same conclusion: fractional uncertainties add, whether you multiply or divide.

Because percentages are fractions scaled by $100$, the rule holds identically for percentage uncertainties, and in practice that is how it is used: convert every input to a percentage uncertainty, add the percentages, then convert the total back to an absolute uncertainty at the end.

3. Rule 3: Powers and Roots

If $z = x^{n}$ (with $n$ positive, negative, integer or fractional), then

For a positive integer $n$ this follows immediately from Rule 2: $x^{3} = x \cdot x \cdot x$ is a product of three factors, each contributing $\dfrac{\Delta x}{x}$, so the total is $3\dfrac{\Delta x}{x}$. The general statement follows from calculus. Writing $z = x^{n}$ and differentiating,

and dividing by $z = x^{n}$ gives $\dfrac{\Delta z}{z} = |n|\dfrac{\Delta x}{x}$ as claimed. This calculus route is the general method behind all three rules — for any function $z(x)$, $\Delta z \approx \left|\dfrac{dz}{dx}\right|\Delta x$ — and it is worth knowing, because it handles cases the three rules do not, such as $z = \sin\theta$ or $z = \ln x$.

Two consequences are examined constantly:

• A square root halves the percentage uncertainty, since $n = \tfrac{1}{2}$. A period found from $T \propto \sqrt{l}$ inherits only half the percentage uncertainty of the length.
• A reciprocal preserves it, since $|n| = 1$. Converting a gradient to its reciprocal (as when a spring constant is read off a graph of extension against load) leaves the percentage uncertainty unchanged.

Worked example. A sphere's radius is measured as $r = (2.50 \pm 0.05)~\text{cm}$, a percentage uncertainty of $2.0\%$. Its volume is $V = \tfrac{4}{3}\pi r^{3}$, so

With $V = \tfrac{4}{3}\pi (2.50)^{3} = 65.4~\text{cm}^{3}$, the absolute uncertainty is $\Delta V = 0.060 \times 65.4 = 3.9~\text{cm}^{3}$, and the result is $V = (65 \pm 4)~\text{cm}^{3}$. Notice that the constants $\tfrac{4}{3}$ and $\pi$ are exact and contribute nothing: pure numbers carry no uncertainty.

4. Combining the Rules: A Full Worked Example

Real calculations mix the rules, and the order of operations matters.

Problem. The density of a cylinder is to be found from $\rho = \dfrac{m}{\pi r^{2} h}$. The measurements are $m = (248 \pm 1)~\text{g}$, $r = (1.40 \pm 0.05)~\text{cm}$ and $h = (12.0 \pm 0.1)~\text{cm}$. Find $\rho$ with its uncertainty.

Step 1 — the value. Work in SI: $m = 0.248~\text{kg}$, $r = 1.40\times10^{-2}~\text{m}$, $h = 0.120~\text{m}$.

Step 2 — percentage uncertainties of the inputs.

Step 3 — combine. The formula is a quotient of products and powers, so all contributions add, with the radius weighted by its exponent of $2$:

Step 4 — quote. $\Delta \rho = 0.084 \times 3.36\times 10^{3} = 2.8\times 10^{2}~\dfrac{\text{kg}}{\text{m}^{3}}$, so

Step 5 — read the result as advice. Of the $8.4\%$ total, the radius alone contributes $7.2\%$ — over $85\%$ of the whole. Improving the mass measurement would be pointless; measuring the radius with a micrometer instead of a rule, or averaging several diameter measurements, is the only change that would materially improve the experiment. Uncertainty propagation is not merely bookkeeping: it identifies the weakest link.

5. Which Term Dominates, and Why That Is the Useful Part

Because contributions add, the total is dominated by the largest one, and a quantity raised to a power contributes $|n|$ times over. Three practical consequences follow.

• Rank the contributions before improving an experiment. A term contributing $0.4\%$ to a total of $8.4\%$ cannot be improved into significance; halving the dominant $7.2\%$ term nearly halves the total.
• Prefer direct measurement to small differences. As Section 1 showed, subtracting two nearly equal numbers inflates the fractional uncertainty catastrophically.
• Watch cubes and squares. In $V = \tfrac{4}{3}\pi r^{3}$ a $2\%$ radius becomes a $6\%$ volume; in the inverse-square law $F \propto \dfrac{1}{r^{2}}$ a $3\%$ separation becomes a $6\%$ force.

A note on rigour: adding the contributions is a worst-case estimate, assuming every input happens to err in the direction that maximises the error. Genuine random uncertainties partially cancel, and professional practice therefore adds them in quadrature, $\dfrac{\Delta z}{z} = \sqrt{\left(\dfrac{\Delta x}{x}\right)^{2} + \left(\dfrac{\Delta y}{y}\right)^{2}}$, which gives a smaller and more realistic figure. The IB requires the simple sum, and quadrature will not be credited in Paper 1B or the IA — but knowing that the simple sum is an over-estimate explains why experimental results so often agree with theory more closely than their quoted uncertainty demands.

6. Uncertainties in Gradients and Intercepts

Most IB experiments extract a constant from the gradient of a linearised graph rather than from a single calculation, and for good reason: a gradient uses every data point, and an additive systematic error shifts only the intercept.

The technique from the previous chapter gives $m_{\max}$ and $m_{\min}$ as the steepest and shallowest lines passing through all uncertainty bars, whence

The propagation rules then carry that uncertainty into the physical constant. If a pendulum experiment plots $T^{2}$ against $l$, theory gives $T^{2} = \dfrac{4\pi^{2}}{g}l$, so the gradient is $\dfrac{4\pi^{2}}{g}$ and

since $4\pi^{2}$ is an exact constant and $g$ depends on $m$ to the power $-1$. A gradient known to $4\%$ therefore gives $g$ to $4\%$ — and if the resulting interval contains $9.81~\dfrac{\text{m}}{\text{s}^{2}}$, the experiment has succeeded.

7. Common Exam Pitfalls

• Subtracting uncertainties when the quantities are subtracted. They always add. A difference is less certain than either term, never more.
• Adding absolute uncertainties in a multiplication. Absolute uncertainties add only for $+$ and $-$; for $\times$ and $\div$, convert to percentages first. Adding $0.05~\text{cm}$ to $1~\text{g}$ is not even dimensionally meaningful.
• Forgetting the factor $|n|$ for a power. A cube triples the percentage uncertainty; a square root halves it.
• Attaching an uncertainty to an exact constant. $\pi$, $\tfrac{4}{3}$, $2$ in $2\pi f$ and a counted number of oscillations are all exact and contribute nothing.
• Rounding percentage uncertainties too early. Keep two significant figures through the combination and round only the final absolute uncertainty to one.
• Quoting the final value to more decimal places than its uncertainty. $(3.362 \pm 0.3)\times 10^{3}$ should be $(3.4 \pm 0.3)\times 10^{3}$.
• Using quadrature in an IB answer. It is the better estimate in professional work, but the syllabus requires the simple sum.

Summary

• Addition and subtraction: absolute uncertainties add, $\Delta z = \Delta x + \Delta y$ — in both cases. Uncertainty never cancels.
• Multiplication and division: fractional (or percentage) uncertainties add, $\dfrac{\Delta z}{z} = \dfrac{\Delta x}{x} + \dfrac{\Delta y}{y}$, because the cross-term $\Delta x\,\Delta y$ is negligible.
• Powers and roots: $z = x^{n}$ gives $\dfrac{\Delta z}{z} = |n|\dfrac{\Delta x}{x}$ — a cube triples, a square root halves, a reciprocal preserves the percentage uncertainty.
• The general calculus form, $\Delta z \approx \left|\dfrac{dz}{dx}\right|\Delta x$, underlies all three rules and extends to functions such as $\sin\theta$ and $\ln x$.
• Exact numbers carry no uncertainty: $\pi$, $\tfrac{1}{2}$, and counted quantities contribute nothing to the total.
• Convert to percentages, add, then convert back to an absolute uncertainty and round it to one significant figure, rounding the value to match.
• The largest contribution dominates, and identifying it is the practical purpose of the whole calculation: it tells you which measurement to improve.
• Constants extracted from a gradient inherit the gradient's percentage uncertainty through the same rules, e.g. $g = \dfrac{4\pi^{2}}{m}$ gives $\dfrac{\Delta g}{g} = \dfrac{\Delta m}{m}$.