Physics › Measurements › Uncertainty and error
Uncertainty and error
No measurement is a bare number. Every reading carries an interval it might sit inside, and the whole machinery of this topic exists to find that interval, shrink it, and carry it honestly through a calculation.
Builds on SI units and prefixes.
IN THIS TOPIC
- Identify random and systematic errors and choose the treatment that actually reduces each one.
- Use precision, accuracy, resolution, repeatability and reproducibility with their exam meanings.
- Estimate, convert and combine uncertainties, and find the uncertainty in a gradient from error bars.
WHAT YOU PROBABLY THINK
A measurement is a number.
Two kinds of error, treated differently
Random errors scatter readings either side of the true value: electrical noise, a reaction time that varies from one press to the next, air currents nudging a balance. Because the direction is unpredictable, repeating and averaging works, and more repeats pull the mean in tighter.
Systematic errors shift every reading the same way: a ruler whose scale starts at 1 mm, an ammeter showing 0.02 A with nothing connected, a stopwatch always started late. Averaging does nothing here except deliver a beautifully consistent wrong answer. The cure is to find the cause and remove it. The commonest case is a zero error, and its fix is the model for all of them: check the instrument reads zero when it should, and subtract any offset from every reading.
The vocabulary examiners test
Precision is how tightly repeated readings cluster, and it says nothing about being right. Accuracy is how close a result sits to the true value. The two are independent, which is exactly what the pair of error types predicts: a systematic error leaves you precise but inaccurate, while heavy random error leaves you imprecise even when the mean is accurate.
Three more words carry marks. Resolution is the smallest change an instrument can display, one millimetre for an ordinary ruler. Repeatability means the same experimenter with the same equipment gets consistent results; reproducibility means a different experimenter, or different equipment, still agrees, which is the stronger claim.
Putting a number on the doubt
The uncertainty is the interval the true value probably sits inside, written after a ± sign. For a single reading, a working rule is half the instrument's resolution, so a millimetre ruler contributes ±0.5 mm at each end; measuring a length means judging both ends, giving ±1 mm overall. For a set of repeated readings, use half the range: subtract the smallest from the largest and halve it.
One uncertainty, three costumes. The absolute uncertainty keeps the unit, ±0.5 mm. Dividing by the value gives the fractional uncertainty, and multiplying that by 100 gives the percentage:
Combining uncertainties
Calculations mix measured quantities, and their uncertainties combine by three rules. Adding or subtracting quantities: add the absolute uncertainties. Multiplying or dividing: add the percentage uncertainties. Raising to a power n: multiply the percentage uncertainty by n, so a radius known to 2% gives a volume, which goes as r3, known only to 6%.
The subtraction rule hides a trap worth seeing once. Take x = 50 ± 1 mm and y = 20 ± 1 mm: the difference is 30 ± 2 mm. The value shrank while the uncertainty grew, so subtracting two nearly equal quantities can leave a hopeless percentage uncertainty, and good experiments are designed to avoid it.
On a graph, each point's uncertainty becomes an error bar. Draw the best-fit line, then the steepest and the shallowest lines that still pass through every bar. Half the difference between their gradients is the uncertainty in the gradient, and where those two lines cut the axis brackets the intercept in the same way. Finally, let the uncertainty set your significant figures: quoting 12.4783 when the uncertainty is ±0.5 claims a precision the experiment never had.
THE EXAM BIT
- “Repeat and average” earns the mark only for random error. If the question describes a zero error or a mis-set instrument, that answer scores nothing; the fix is to find and remove the offset.
- Percentage-uncertainty questions lean on the power rule. Spot the square or cube first: a cubed quantity carries three times the percentage uncertainty of its base measurement.
- Asked which measurement most needs improving, pick the one with the largest percentage uncertainty. The largest absolute uncertainty is the planted wrong answer.
- The error-bar procedure is mechanical marks: steepest and shallowest lines through every bar, then half the gradient difference is the gradient's uncertainty.
- Match significant figures to the uncertainty. If the doubt is in the first decimal place, the answer stops there too.
CHECK YOURSELF
A wire's diameter is measured as 0.40 ± 0.01 mm. What is the percentage uncertainty in its cross-sectional area?
Show a hint
Area depends on the diameter squared. What does the power rule do to a percentage uncertainty?
Show the answer
Percentage uncertainty in the diameter: (0.01 / 0.40) × 100 = 2.5%.
The area goes as d2, so the power rule doubles it: the area is uncertain by 5%.
Working through the radius changes nothing: halving a value halves its absolute uncertainty too, so the percentage uncertainty is untouched.
Averaging fixes random error.
It does nothing to systematic error.
No animated video for this topic yet; these notes stand alone. InkPhysics on YouTube.