Report the uncertainty, always

Why this rule exists

A measured number without an uncertainty is not yet a measurement; it is a claim with no stated confidence, and it cannot be compared to a prediction or another measurement. The uncertainty says how far the true value may plausibly lie from your figure, and thus how many of your digits mean anything. Two results agree or disagree only relative to their error bars, so without them the central question — do these numbers match? — has no answer.

In practice

Report every result as value ± uncertainty with a unit, and quote only digits the uncertainty supports. Propagate errors through calculations: for sums, add absolute uncertainties in quadrature; for products and ratios, add relative uncertainties in quadrature. Distinguish random error, which shrinks as you average repeated trials, from systematic error, which does not and must be hunted down separately. When you compare to theory, ask whether the gap exceeds the combined uncertainty.

Example

Area of a rectangle, A = L·W:
L = 20.0 ± 0.2 cm  (1% relative)
W = 10.0 ± 0.3 cm  (3% relative)

Relative errors add in quadrature:
δA/A = sqrt(0.01² + 0.03²) ≈ 0.032

A = 200 cm², so δA ≈ 0.032·200 ≈ 6 cm²
Report A = 200 ± 6 cm².

When it doesn't apply

Quadrature addition assumes independent, roughly Gaussian errors; correlated or one-sided errors need fuller treatment. And a stated uncertainty covers only the effects you accounted for — an unknown systematic can leave the true value far outside honest-looking bars.