Physics/Lessons/Working as a Physicist/Combining Uncertainties
Physics · Working as a Physicist · AS Level

Combining Uncertainties

A single measured quantity carries its own uncertainty, but almost nothing in physics is used on its own — you multiply lengths together, divide voltages by currents, square velocities. So once you combine several uncertain quantities into a calculation, what happens to their uncertainties? The rules are refreshingly simple, and worth understanding rather than just memorising.

9 min read

Three rules for propagating uncertainty

Suppose you've measured two quantities, each with its own uncertainty, and need to combine them into a calculated result. There are three situations you'll meet constantly, each with its own rule for how the uncertainty carries through.

When you add or subtract two measured quantities, their absolute uncertainties add — regardless of whether you're adding or subtracting the quantities themselves. If $x = (5.0\pm0.2)\,\text{cm}$ and $y=(3.0\pm0.1)\,\text{cm}$, then whether finding $x+y$ or $x-y$, the uncertainty in the result is $0.2+0.1=0.3\,\text{cm}$. This might feel odd for subtraction, but uncertainty represents the worst case in either direction, and subtracting doesn't make you more certain about either original measurement.

When you multiply or divide two measured quantities, their percentage (or fractional) uncertainties add. If a current $I$ has a percentage uncertainty of $2\%$ and a resistance $R$ has $3\%$, then $V=IR$ carries a percentage uncertainty of $2\%+3\%=5\%$. The derivation below shows exactly where this comes from.

When a measured quantity is raised to a power $n$, its percentage uncertainty is multiplied by $n$. If a radius $r$ has a percentage uncertainty of $1\%$, a volume proportional to $r^3$ has $3\times1\%=3\%$, since $r^3$ is really $r\times r\times r$, and the multiplication rule applies three times over.

DerivationWhy percentage uncertainties add when multiplying
1
Suppose $z = xy$, where $x$ and $y$ are measured with absolute uncertainties $\Delta x$ and $\Delta y$.
We want to find the resulting uncertainty $\Delta z$ in $z$.
2
The worst-case value of $z$ comes from taking both $x$ and $y$ at their largest plausible values: $z+\Delta z \approx (x+\Delta x)(y+\Delta y)$.
3
Expanding the right-hand side gives $(x+\Delta x)(y+\Delta y) = xy + x\Delta y + y\Delta x + \Delta x\Delta y$.
Multiply out the brackets term by term, exactly as you would for any two binomials.
4
The last term, $\Delta x\Delta y$, is a product of two small uncertainties, so it's tiny compared with the others — a "second-order" term — and can safely be dropped.
If $\Delta x$ and $\Delta y$ are each a small fraction of $x$ and $y$, their product is a much smaller fraction still.
5
So $z+\Delta z \approx xy + x\Delta y + y\Delta x$. Since $z=xy$, subtracting $z$ from both sides gives $\Delta z \approx x\Delta y + y\Delta x$.
6
Divide both sides by $z=xy$: $\dfrac{\Delta z}{z} \approx \dfrac{x\Delta y}{xy} + \dfrac{y\Delta x}{xy} = \dfrac{\Delta y}{y} + \dfrac{\Delta x}{x}$.
This is exactly the claim: the fractional uncertainty in $z$ is the sum of the fractional uncertainties in $x$ and $y$.
7
Multiplying through by 100% converts this into the percentage form: percentage uncertainty in $z$ = percentage uncertainty in $x$ + percentage uncertainty in $y$.
▪ derived
Why this matters beyond the formula

Fractional uncertainties add because uncertainty in a product comes from both factors being uncertain at once, and to first order those two contributions just stack. This also explains the power rule — raising to the power $n$ is multiplying a quantity by itself $n$ times, so its fractional uncertainty gets added to itself $n$ times, the same as multiplying by $n$.

Showing uncertainty on a graph: error bars

Once you've got uncertain values for several data points, plotting them needs a way of showing that uncertainty visually rather than just as a single dot. This is what an error bar does — a short line through each plotted point, extending above and below (uncertainty in $y$) or left and right (uncertainty in $x$), its length representing the range within which the true value could plausibly lie. A large uncertainty gets a long error bar; a confidently measured point gets a short one, or none if the uncertainty is too small to show at the graph's scale.

The sizing rule is entirely literal: the bar runs from (value − uncertainty) to (value + uncertainty). A reading of $(5.0\pm0.2)\,\text{cm}$ gets an error bar from $4.8\,\text{cm}$ to $5.2\,\text{cm}$, centred on the plotted point — no scaling needed, just the endpoints given directly by the measurement and its stated uncertainty.

Error bars turn a graph from a simple connect-the-dots exercise into something that honestly represents the quality of your data — and, as the next section shows, they're the basis for working out the uncertainty in a line's gradient and intercept.

Uncertainty in a gradient and intercept

When you fit a straight line through data points, the standard approach is to draw the line of best fit — the single straight line passing as close as possible to every point overall. But that best-fit line alone tells you nothing about how uncertain its gradient or intercept are. For that, you need the error bars.

The method is to draw two further lines, both still consistent with the data: the steepest straight line that can still pass through every error bar, and the shallowest line that can likewise still pass through every error bar. Both are lines the data cannot rule out — the extremes of what's plausible given the scatter and size of your error bars, with the best-fit line sitting somewhere between them.

Find the gradient of each of these two extreme lines; the uncertainty in the gradient is half the difference between them:

$\Delta(\text{gradient}) = \dfrac{\text{gradient}_{\text{steepest}} - \text{gradient}_{\text{shallowest}}}{2}$

The same idea applies to the $y$-intercept: read off the intercept of each extreme line and take half the difference as the uncertainty in the intercept. Half the spread between the two extreme, still-plausible lines is a sensible measure of how much the true gradient (or intercept) could reasonably differ from the one read off the best-fit line alone.

A scatter graph on labelled x and y axes showing around seven data points, each drawn as a small dot with a short vertical error bar through it (a thin vertical line extending a little above and below the point, indicating the uncertainty range in the y-value). Three straight lines are drawn across the graph: a solid best-fit line running centrally through the middle of the scatter of points, and two dashed lines — one steeper line and one shallower line — that each still pass through every single error bar on the graph, showing the maximum and minimum plausible gradients consistent with the data. The steepest dashed line and shallowest dashed line visibly diverge from the solid best-fit line, fanning out on either side of it, and a small legend identifies the three lines as 'best fit', 'steepest', and 'shallowest'.
The steepest and shallowest lines that still pass through every error bar bracket the plausible range of gradients — half their difference gives the uncertainty in the gradient.
A common slip to avoid

The steepest and shallowest lines must still pass through every error bar, not just through the first and last data points. It's tempting to just pick the two most extreme-looking points and draw between them, but a genuinely valid steepest or shallowest line has to remain consistent with the full set of error bars across the whole graph.

Worked exampleGradient uncertainty from the worst-line method
1
From a graph of data with error bars, the best-fit line has gradient $2.4$.
2
The steepest line still passing through every error bar has gradient $2.7$; the shallowest such line has gradient $2.1$.
3
Uncertainty in the gradient is half the difference: $\Delta(\text{gradient}) = \dfrac{2.7-2.1}{2} = 0.3$.
4
The gradient is quoted as $2.4 \pm 0.3$.
▪ answer
Worked exampleIntercept uncertainty from the same worst-line method
1
Using the same graph, the best-fit line crosses the $y$-axis at an intercept of $1.5$.
2
The steepest line crosses the $y$-axis at $1.1$; the shallowest line crosses it at $1.9$.
The steepest and shallowest lines don't need to give the larger or smaller intercept in any fixed order — just read off whichever value each line actually gives.
3
Uncertainty in the intercept is again half the difference between the two extreme values: $\Delta(\text{intercept}) = \dfrac{1.9-1.1}{2} = 0.4$.
4
The intercept is quoted as $1.5 \pm 0.4$.
▪ answer

Key takeaways

  • When adding or subtracting measured quantities, their absolute uncertainties add, even when the quantities themselves are being subtracted.
  • When multiplying or dividing measured quantities, their percentage (or fractional) uncertainties add.
  • When a quantity is raised to a power $n$, its percentage uncertainty is multiplied by $n$.
  • The multiplication rule can be derived directly from expanding $(x+\Delta x)(y+\Delta y)$ and discarding the negligible second-order term $\Delta x\Delta y$.
  • Error bars on a graph show the plausible range of a plotted value, horizontally and/or vertically.
  • The uncertainty in a best-fit line's gradient is half the difference between the gradients of the steepest and shallowest lines that still pass through every error bar; the same method gives the uncertainty in the intercept.
  • An error bar for a reading of $(5.0\pm0.2)\,\text{cm}$ runs literally from $4.8\,\text{cm}$ to $5.2\,\text{cm}$ — the endpoints are just the value minus and plus its stated uncertainty.
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