What is measurement uncertainty?
Every measurement is a number with an honest companion: how confident can we be? That companion has two pieces — a random error, which the act of repeating the measurement averages down, and a systematic error, a bias that persists no matter how many times you measure. The honest report of any quantity is therefore x ± u(x), with u being a well-defined uncertainty; the ± symbol is shorthand for "approximately this range with this much probability". The four calculators on this panel cover the working tools: counting significant figures (the bookkeeping shortcut), propagating uncertainties through sums, differences, products, quotients, and powers (the calculus of mistakes), standard deviation (how spread out the data is), and the GUM-style combined uncertainty of multiple independent components.
Two kinds of error, two opposite behaviors: random error shrinks with the square root of N (σ/√N), so averaging always wins; systematic error does not shrink at all, which is why calibration matters more than repetition. A measurement can have a tiny σ and still be wrong because of bias.
Significant figures — the bookkeeping shortcut
A significant figure is a digit known with confidence; zeros that pad a number to a size convention are not. Five rules: (1) all nonzero digits are significant, (2) zeros between nonzeros are significant, (3) leading zeros are not significant (they just place the decimal), (4) trailing zeros after a decimal point are significant, (5) trailing zeros without a decimal are ambiguous (use scientific notation). The default value 0.0070800 therefore has 5 sig figs (7, 0, 8, 0, 0) and reads as (7.0800 ± 0.00005)×10−3. Reporting 0.0070800 implies the precision is the same as reporting 7.0800×10−3; reporting 0.00708 implies a thousandfold less precision. The shortcut works for textbook reports; for a real measurement the uncertainty u is the honest number, and sig figs are a convenient way to express it without writing the ± every time.
Error propagation — the calculus of mistakes
For z = f(x, y), the rule for combining independent uncertainties is σz² ≈ (∂f/∂x)²σx² + (∂f/∂y)²σy². For the four working operations, this reduces to: addition/subtraction add absolute errors in quadrature σz = √(σx² + σy²); multiplication/division add fractional errors in quadrature σz/|z| = √((σx/x)² + (σy/y)²); power xn scales the fractional error by |n|: σz/z = |n|σx/x. The default sample (x = 10 ± 0.3, y = 5 ± 0.2, addition) yields z = 15 and σz ≈ 0.36, or 15.00 ± 0.36 (about 2.4% relative). Switching to multiplication with the same inputs gives z = 50 ± 2.5 (5% relative) — the same ± values, very different propagation rule, and the relative uncertainty of the product equals the quadrature sum of the relative uncertainties. Absolute uncertainties are the right currency for sums/differences; fractional uncertainties are the right currency for products/quotients.
Standard deviation — spread of the data
The standard deviation σ is the root-mean-square deviation from the mean. Two versions exist depending on what the data represent: population σ = √(Σ(xi − x̄)² / n) describes the spread of every member of a known set; sample s = √(Σ(xi − x̄)² / (n − 1)) estimates the spread of the larger population the sample was drawn from. The default sample (10 values from 12.2 to 12.8 with mean 12.5) gives population σ ≈ 0.1732, sample s ≈ 0.1826, and SEM = s/√N ≈ 0.0577. The uncertainty of the mean itself is σ/√N (or s/√N for the sample version), which is why averaging 25 measurements of a single quantity cuts the uncertainty by a factor of 5 compared to one. Standard deviation also feeds confidence intervals: for a normal distribution, ±1σ contains 68.3%, ±2σ 95.5%, ±3σ 99.7%.
Extended uncertainty — combining all sources
When a measurement has multiple independent error sources (instrument resolution, calibration, repeatability, temperature drift, operator effect), the GUM-style combined uncertainty is uc = √(Σui²). Sources are classified Type A (statistical, from data) and Type B (everything else, with uncertainties estimated by judgment or manufacturer spec); both feed into uc the same way. A coverage factor k then scales the interval to a chosen confidence: k = 1 for 68.3%, k = 2 for 95.5%, k = 2.58 for 99.0%, k = 3 for 99.7%. The default sample (0.15, 0.08, 0.05, 0.03 mm sources, k = 2) gives uc ≈ 0.1797 mm and U = k·uc ≈ 0.3595 mm at 95.5%. The reported value becomes (measurement ± 0.36) mm with that confidence. The GUM (ISO/IEC Guide 98-3) is the international standard for the procedure. Type A components come from statistics of repeated readings, Type B from instrument specs and judgment — both combine through the same quadrature sum.
Common misconceptions
- More significant figures means a more accurate measurement. Only if those digits really came from the measurement. A length of "12.34567 m" written by hand is not 1 part in 10 million accurate — sig figs are a way to communicate precision, not a way to invent it. Reporting ± is more honest than padding with made-up digits.
- Systematic error can be reduced by averaging. The opposite: random errors average down as 1/√N, but a systematic bias (a miscalibrated scale, an off-center zero) stays put no matter how many measurements you take. Identify and average bias. This is the trap of "the data look so reproducible!" in a broken experiment.
- Standard deviation tells you about a single measurement's error. It characterizes the spread of the distribution, not the error of any single trial. The standard error of the mean (s/√N) is the relevant quantity when reporting how well you know x̄ from a sample of N.
- Uncertainty is always ± in the last digit. The standard convention is "round the uncertainty to one (sometimes two) significant figures, then round the value to match". A measurement of 12.345678 ± 0.234 should be reported as 12.35 ± 0.23, not 12.345678 ± 0.234.
Related tools: Statistics for the underlying probability distributions, Mechanics for the experimental setups that produce the numbers, and Calculator for the algebra of propagation.
