Uncertainties and error bars
Uncertainty quantifies the plausible range around a measurement and the confidence in a derived result.
Absolute error bars show plausible ranges; worst acceptable lines estimate gradient uncertainty.
By the end of this lesson, you can:
- estimate absolute uncertainty;
- convert between absolute and percentage uncertainty;
- combine percentage uncertainties;
- use error bars and extreme gradients.
Measurement uncertainty
For an analogue scale, a common estimate is half the smallest division for one reading. A difference between two readings may carry both endpoint uncertainties. Repeated readings can reveal random spread.
Propagation
Add absolute uncertainties for sums and differences. Add percentage uncertainties for products and quotients. Multiply percentage uncertainty by the power for a quantity raised to that power.
WORKED EXAMPLEReveal method
For A = πd²/4 and d has 2% uncertainty, A has approximately 4% uncertainty.
Graphs
Plot error bars where required. Compare the best-fit gradient with the steepest or shallowest acceptable line to estimate gradient uncertainty.
Try another worked problem
Attempt the question yourself, then reveal the examiner-style solution when you are ready.
Gradient uncertainty
mmax=4.8 and mmin=4.2. Find m and absolute uncertainty.
CHECK YOUR ANSWERReveal solution
m=(4.8+4.2)/2=4.5; Δm=(4.8−4.2)/2=0.3, so 4.5±0.3.
Mastery check
If Q = x³ and x has 2% uncertainty, the approximate uncertainty in Q is: