Learn7408Paper 3A · Paper 3 Section A: Practical skills and data analysisPractical skills and data analysisErrors and uncertainties

Practical skills and data analysis · Subtopic 4 of 5

Errors and Uncertainties

Random errors scatter readings either side of the true value and are reduced by repeating and averaging. Systematic errors shift every reading the same way and aren't reduced by repeating. Precise readings agree with each other; accurate ones are close to the true value. When quantities are added or subtracted, add absolute uncertainties; when multiplied or divided, add percentage uncertainties; for a power, multiply the percentage by the power.

Cambridge 9702 Paper 1Cambridge 9702 Paper 2Edexcel YPH11 WPH13Edexcel YPH11 WPH16AQA 7408 Paper 1AQA 7408 Paper 3A≈ 35 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Distinguish random and systematic errors, and precision and accuracy.
Express uncertainties in absolute, fractional and percentage form.
Combine uncertainties for sums, products, quotients and powers.
Use uncertainties to give a result to a sensible number of significant figures.

Pause and think

A ruler's end is worn, so every length you measure is $2$ mm too short. Would measuring $20$ times and averaging fix it?

Section 2 of 14

Core idea

Start hereHow sure is your g?

Two students measure $g$ : one gets $9.6$ , the other $10.1$ . Who's right? Without uncertainties you can't say. With them ( $9.6\pm0.3$ , $10.1\pm0.1$ ) you can tell which result is consistent with $9.81$ and which isn't.

The big idea

Systematic errors shift readings (accuracy); random errors scatter them (precision). Combine: add absolute uncertainties for $\pm$ ; add percentage uncertainties for $\times\div$ ; multiply by $n$ for powers.

1. Accuracy and precision

Accuracy and precision on a target

A systematic error moves the whole cluster (second target). Random error spreads it out (third).

Accurate: close to the true value. Precise: close together.
Systematic (e.g. zero error): fix by calibration. Random: reduce by repeats.

Repeated readings of a time (s): mean and spread

A reasonable uncertainty is half the range:

2.10\pm0.05

2. Combining uncertainties

Why percentages add for a product

$y=ab\Rightarrow y+\Delta y=(a+\Delta a)(b+\Delta b)$
Worst case.
$=ab+a\Delta b+b\Delta a+\Delta a\Delta b$
Expand.
$\Delta y\approx a\Delta b+b\Delta a$
$\Delta a\Delta b$ is tiny.
$\frac{\Delta y}y=\frac{\Delta a}a+\frac{\Delta b}b$
Divide by $y=ab$ .

For

a^n

this gives

n\frac{\Delta a}a

(multiply

a

by itself

n

times).

\begin{aligned}y=a\pm b&:\ \Delta y=\Delta a+\Delta b\\y=ab,\ \tfrac ab&:\ \tfrac{\Delta y}y=\tfrac{\Delta a}a+\tfrac{\Delta b}b\\y=a^n&:\ \tfrac{\Delta y}y=n\tfrac{\Delta a}a\end{aligned}

$\Delta a$: absolute uncertainty in $a$ .
$\frac{\Delta a}a$: fractional (×100: percentage) uncertainty.

Worked exampleDensity of a cube

$m=50.0\pm0.1$ g; side $2.00\pm0.02$ cm.

Goal: $\rho$ with uncertainty.
Strategy: Percentages; volume is side³.

$\tfrac{0.1}{50}=0.2\%$
Mass.
$3\times\tfrac{0.02}{2}=3\%$
Volume: power 3.
$0.2+3=3.2\%$
Division: add.
$\rho=\tfrac{50.0}{8.00}=6.25\pm0.20\text{ g cm}^{-3}$
$3.2\%$ of $6.25$ .

Answer

6.25\pm0.20

g cm

^{-3}

✓ CheckSide dominates: improve that measurement first ✓.

Worked exampleg from a pendulum

$g=\frac{4\pi^2L}{T^2}$ , $L=1.000\pm0.005$ m, $T=2.01\pm0.01$ s.

Goal: $g\pm\Delta g$ .
Strategy: Percentages; $T$ squared.

$L:\ 0.5\%,\quad T^2:\ 2\times0.5\%=1.0\%$
Individual.
$1.5\%$
Add.
$g=\frac{39.48}{4.040}=9.77\pm0.15$
Value and $1.5\%$ .

Answer

9.77\pm0.15

m s

^{-2}

✓ Check

9.81

lies inside the range ✓.

Common misconceptionSubtracting uncertainties For

a-b

the uncertainties still add.

Pause and think

$12.4\pm0.1$ cm minus $8.2\pm0.1$ cm. Uncertainty?

Summary

Systematic vs random.
Absolute for $\pm$ ; % for $\times\div$ ; × $n$ for powers.

Precision: Closeness of repeated readings.
Accuracy: Closeness to the true value.

HistoryMillikan's "selected" drops. Robert Millikan measured the charge on the electron in

1913

using oil drops. His notebooks later showed he left out some drops he thought were unreliable. His result was close, but historians still debate how he handled uncertainty, a reminder to report all data honestly.

Fun factThe most precise number in physics. The magnetic moment of the electron has been measured to about

13

significant figures, and theory agrees to about the same precision. It's often called the most accurately tested prediction in all of science.

Keep reading for free

Create a free account to read the rest of this lesson: key formulae, worked examples, explore the graph, common mistakes, exam tip, guided practice and more. Your answers are checked and your progress is saved.

Create free account Sign in