Evidence, Uncertainty, and Error Bars
Core concepts about measurement uncertainty, error types, and how scientists report confidence honestly.
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Questions Covered in This Set
10 cards to master
Why is a measurement without an uncertainty 'not a measurement at all'?
The uncertainty tells you what the claim can actually be used for. '12 cm' could mean 12 ± 0.5 cm (eyeballed) or 12.0000 ± 0.0001 cm (laser) — those support wildly different conclusions.
Random (statistical) error
Error that scatters results around the true value (shaky hand, flickering thermometer). It shrinks with more data, typically like 1/√N.
Systematic error
Error that shifts all results in the same direction (short ruler, un-zeroed scale, biased survey). More data does NOT help — you just get a very precise wrong answer. Fought with calibration and paranoia.
Precision vs. accuracy
Precision = results cluster tightly. Accuracy = results cluster around the truth. A tight cluster in the corner of a dartboard is precise but inaccurate.
What did the 2011 OPERA neutrino result demonstrate?
A precise result can still be wrong: a loose fibre-optic cable created a 73-nanosecond systematic timing offset, making neutrinos appear faster than light despite small statistical errors.
Standard deviation vs. standard error of the mean
σ describes spread of individual measurements and does not shrink with more data. Standard error = σ/√N and is the uncertainty in the average — that's what you report.
What fraction of probability lies within ±1σ, ±2σ, ±3σ?
About 68%, 95%, and 99.7% respectively for a roughly normal distribution. So roughly 1 time in 3 the true value sits outside a 1σ bar.
Do two points with barely overlapping 1σ error bars disagree?
No — that is not strong evidence of disagreement. Always check whether the bar is a standard deviation, a standard error, or a 95% confidence interval.
What does a 95% confidence interval actually mean?
If the whole procedure were repeated many times, 95% of the constructed intervals would contain the true value. It does NOT mean 'a 95% chance the truth is in this range' — that's a Bayesian credible interval, which requires a prior.
How do independent uncertainties combine ('in quadrature')?
Adding/subtracting: absolute errors combine, σ_z = √(σ_x² + σ_y²). Multiplying/dividing: relative errors combine, σ_z/z = √((σ_x/x)² + (σ_y/y)²).
