Errors & Propagation of Errors
No measurement is a number — it is a number with a doubt attached. When you write a length as 2.53 ± 0.01 cm, the "± 0.01" states how far the truth might sit from your reading. Every result you compute inherits that doubt from its inputs. Error analysis answers one question: given the doubt in what I measured, how much doubt is there in what I calculated?
Accuracy and precision are different virtues. Accuracy is closeness to the true value, spoiled by systematic error. Precision is repeatability — how tightly readings cluster — limited by random error and the least count. A screw gauge with a zero error gives readings that agree beautifully but are all wrong: precise, not accurate. A shaky hand on a fine scale scatters around the true value: accurate on average, not precise. A finer least count buys precision but never fixes a zero error; removing the zero error buys accuracy but does not tighten the spread.
Systematic errors are one-signed; random errors scatter both ways. Systematic errors (zero error, an uncalibrated worn scale, personal bias, environmental drift) push every reading the same direction, so averaging cannot remove them — only calibration or a correction can. Random errors (parallax judged afresh, small fluctuations) fall + and − alike, so they shrink when you average many readings. Gross errors are blunders — discard them.
From repeated readings, build the reported value. Take n readings; the mean is the most probable value. The absolute error of each reading is |mean − reading|. Their average is the mean absolute error Δa, and you report a = mean ± Δa. The relative (fractional) error is Δa/a, and the percentage error is (Δa/a) × 100 — the two differ only by the factor 100.
A larger measured value carries a smaller percentage error for the same absolute error. The "best" measurement is not the one with the smallest Δa, but the one with the smallest Δa/a. Timing many oscillations instead of one spreads a fixed absolute error over a bigger denominator.
The propagation rules
Sum or difference — absolute errors add. For Z = A ± B,
ΔZ = ΔA + ΔB (add the magnitudes, whether you added or subtracted).
You add even for a difference, because in the worst case both errors conspire the same way.
Product, quotient or power — fractional errors add, weighted by the exponent. For a general power law
Q = Aᵃ Bᵇ / Cᶜ,
the maximum permissible (worst-case) result is
%Q = a·%A + b·%B + c·%C.
A squared quantity contributes twice its own percentage error, a cube three times, a square-root half. The sign of the exponent is irrelevant — the c in the denominator still adds, because errors never cancel in the worst case. The same rule reads the exponents straight off a dimensional formula: for a quantity of dimensions Mᵃ Lᵇ Tᶜ (force MLT⁻², energy ML²T⁻², pressure ML⁻¹T⁻², density ML⁻³), %Q = |a|·%M + |b|·%L + |c|·%T. …