Measurement Uncertainty and Significant Figures

Measure the width of this page with a ruler and you might get 12.4 cm. Measure it again, a touch more carefully, and you get 12.5 cm. A friend reads 12.3 cm. Nobody made a mistake — a real measurement is never a single perfect number. It always comes with a little fog around it: a small band of values the true length could really be.

That band is the measurement's uncertainty. It isn't a blunder and it isn't carelessness — it's an honest statement of how sure you are. Every good scientist writes a result as a best value plus-or-minus a range, like this:

\text{length} = 12.4 \pm 0.2 \ \text{cm}.

Read that out loud: "twelve-point-four, give or take nought-point-two centimetres." It says the true length is almost certainly somewhere between 12.2\,\text{cm} and 12.6\,\text{cm}. The \pm 0.2 is not decoration — it is part of the answer, and a number quoted without it is only half a measurement.

Uncertainty from a single reading

Suppose you only take one reading. How big is the uncertainty then? It comes from the resolution of your instrument — the size of its smallest step. You've met resolution already in precision, accuracy and resolution; here we turn it into a number.

A common rule of thumb: the uncertainty in a single reading is about half the resolution. If the smallest step is one whole unit, the true value is at most half a step away from the nearest mark.

\text{uncertainty} \approx \tfrac{1}{2} \times (\text{resolution}).

Worked example. A ruler is marked in millimetres, so its resolution is 1\,\text{mm}. You read a length as 84\,\text{mm}.

A finer instrument shrinks the uncertainty: a set of digital callipers reading to 0.01\,\text{mm} has a resolution of 0.01\,\text{mm}, so a single reading is uncertain by only about \pm 0.005\,\text{mm} — a hundred times tighter than the ruler.

Uncertainty from repeats: mean and range

One reading is a lonely thing. Take the measurement several times and two good things happen: you get a better best-estimate, and the readings themselves reveal how uncertain you are.

The best estimate is the mean (the average) of your repeats — adding the readings and dividing by how many there are. Averaging lets the little wobbles above and below the truth partly cancel out.

\text{mean} = \frac{\text{sum of readings}}{\text{number of readings}}.

The uncertainty from repeats comes from how spread out they are. Find the range — the biggest reading minus the smallest — and take half of it:

\text{uncertainty} = \frac{\text{range}}{2} = \frac{\text{max} - \text{min}}{2}.

Worked example. A ball's fall time is measured five times, in seconds:

0.62, \quad 0.65, \quad 0.63, \quad 0.66, \quad 0.64.

Imagine the fall times had been 0.62, 0.65, 0.63, \mathbf{0.98}, 0.64. That 0.98 sticks out like a sore thumb — probably you mistimed the stopwatch or fumbled the release. A reading that clearly doesn't belong is called an anomaly, and the honest thing to do is discard it before you average. Cross it out, then take the mean and range of the remaining good readings. One slip shouldn't be allowed to poison the whole result — but be sure it's a genuine mistake, not just a value you didn't expect.

See it move: spread becomes uncertainty

Below are six repeated readings of the same length, scattered along a number line. The mean is the best estimate and the shaded band is the ± range the true value could lie in. Drag the slider to spread the readings out: the more your repeats disagree, the wider the band and the bigger the uncertainty — while the mean, your best guess, stays put in the middle.

Significant figures: don't claim more than you know

A calculator will happily hand you 12.4375000, but your ruler never knew those last digits — they're invented. Significant figures are the digits in a number that actually carry information, and quoting the right number of them is how you tell the truth about your precision.

Count significant figures with three simple rules:

Worked example. Round 3847 to different precisions:

The golden rule: a result should carry no more figures than the measurement justifies. A stopwatch you thumb by hand is good to maybe a tenth of a second, so 7.4\,\text{s} is honest but 7.418\,\text{s} is a fib — those extra digits pretend to a precision the stopwatch simply doesn't have.

Percentage uncertainty: is ±1 big or small?

An uncertainty of \pm 1\,\text{cm} feels tiny when you're measuring a cricket pitch and enormous when you're measuring a matchstick. To judge it fairly, compare the uncertainty with the size of the value itself. That comparison is the percentage uncertainty:

\text{percentage uncertainty} = \frac{\text{uncertainty}}{\text{value}} \times 100\%.

Worked example. The page measured 12.4 \pm 0.2\,\text{cm}:

\frac{0.2}{12.4} \times 100\% = 1.6\% \ \text{(to 2 s.f.)}.

Now compare two lengths, A = 50 \pm 1\,\text{cm} and B = 5 \pm 1\,\text{cm}. They share the same \pm 1\,\text{cm}, yet A is \tfrac{1}{50}\times 100\% = 2\% uncertain while B is \tfrac{1}{5}\times 100\% = 20\% uncertain. The same absolute uncertainty is ten times more damaging on the smaller measurement — which is exactly why we measure small things with finer instruments.

Three classic slips that make an uncertainty meaningless — dodge them all:

Next time a poll reports "45% support, margin of error ± 3 points," you're looking at exactly the idea on this page. The pollsters didn't ask everyone — they measured a sample of a few thousand people, and a sample is like a set of repeat readings: it wobbles around the true figure. That margin of error is their measurement uncertainty, so the real support is somewhere between about 42\% and 48\%.

It also explains why "45% versus 44%" is a non-story: the two numbers overlap inside each other's uncertainty, so the poll genuinely can't tell them apart. And to halve the margin of error a pollster must ask about four times as many people — better precision costs dearly, whether you're wielding a ruler or a clipboard.