Uncertainty Propagation Calculator

About the Uncertainty Propagation Calculator

Type the formula of your result — g = 4π²L/T², R = U/I, ρ = m/V or anything with powers, roots, logarithms and trigonometric functions — and give each quantity its value and what you know about its uncertainty: a standard uncertainty, a set of repeated readings, limits ± a with a rectangular, triangular or U-shaped distribution, or a calibration certificate’s expanded uncertainty with its coverage factor k. The calculator differentiates the formula for the sensitivity coefficients, builds the uncertainty budget with each input’s share, adds correlations where you give them, works out the effective degrees of freedom (Welch–Satterthwaite) and the coverage factor from the t-distribution, and rounds the result the way the GUM recommends: g = (9.801 ± 0.016) m/s².

The first-order formula is an approximation, so the result is checked by the Monte Carlo method of JCGM 101: up to two million random trials drawn from the distributions you chose, run in your browser. When the two coverage intervals agree, the GUM result stands; when one rectangular input dominates or the formula bends strongly, the calculator says so and gives the Monte Carlo interval, its histogram and the shortest interval for a lopsided result.

With a Pro pass, copy the result or download the budget as a CSV, or unlock this one result. Without a pass the free preview shows the result with its expanded uncertainty in full, the first rows of each table, the first working step and a watermarked chart; other figures hidden.

Common uses

• Physics and chemistry lab reports: the uncertainty of g, a resistance, a density or a concentration from your readings
• Calibration and testing labs: an uncertainty budget with Type A and Type B contributions, ν_eff and k
• Engineering: how tolerances and instrument specifications add up through a formula
• Teaching the GUM: every derivative, contribution and step, with a Monte Carlo check of the result

The law of propagation of uncertainty

For a result y = f(x₁, …, x_N), each input’s standard uncertainty u(xᵢ) is multiplied by its sensitivity coefficient cᵢ = ∂f/∂xᵢ, and the products add in quadrature:

u_c²(y) = Σ cᵢ²·u²(xᵢ) + 2 Σ Σ cᵢ·cⱼ·u(xᵢ)·u(xⱼ)·r(xᵢ, xⱼ)

The second sum is there only for correlated inputs. For products and quotients this means relative uncertainties add in quadrature: for R = U/I, (u_R/R)² = (u_U/U)² + (u_I/I)²; for g = 4π²L/T², the period counts twice, because it is squared. The uncertainty budget lists each input’s contribution cᵢ·u(xᵢ) and its share of u_c², which shows where a better measurement pays off.

Type A and Type B uncertainties

• Repeated readings (Type A): the mean, the standard deviation s and u = s/√n, with n − 1 degrees of freedom
• Limits ± a (Type B): rectangular when any value between them is equally likely (u = a/√3; also a digital display’s resolution d, with a = d/2), triangular when values near the middle are more likely (u = a/√6), U-shaped for a quantity that swings between the limits (u = a/√2)
• Certificates: an expanded uncertainty U with coverage factor k gives u = U/k
• Exact values such as defined constants have no uncertainty

Expanded uncertainty and how to report it

The expanded uncertainty U = k·u_c gives an interval y ± U with a stated coverage probability, usually 95 %. With many degrees of freedom k is 1.96 (or 2 by convention); with few readings, the Welch–Satterthwaite formula gives the effective degrees of freedom νeff = u_c⁴ / Σ (cᵢu(xᵢ))⁴/νᵢ, and k comes from the t-distribution, so it grows. Uncertainties are rounded to two significant digits (or one, with the teaching rule or the Particle Data Group rule) and the result to the same decimal place: write g = (9.801 ± 0.016) m/s² with k = 2.18 for 95 %, or the standard uncertainty in the concise form g = 9.8008(73) m/s².

The Monte Carlo check

The calculator draws random values for every input from its distribution — normal, t for repeated readings, rectangular, triangular or U-shaped, jointly normal for correlated inputs — and evaluates the formula for each set, in batches of at least 10 000 until the mean, the standard deviation and both ends of the coverage interval are stable (the adaptive procedure of JCGM 101). The GUM interval is confirmed when both of its ends lie within δ of the Monte Carlo ones, δ being half a unit in the second significant digit of u_c. It fails when the formula is too curved over the spread of the inputs, a derivative is close to zero, or a rectangular input dominates (its 95 % interval is narrower than the normal one). The histogram shows the real shape of the result; for a lopsided one the shortest interval is given too.

Sources

• JCGM 100, Evaluation of measurement data — Guide to the expression of uncertainty in measurement (the GUM) and JCGM 101, Supplement 1: Propagation of distributions using a Monte Carlo method, both from the BIPM Joint Committee for Guides in Metrology: the method, the distributions, Welch–Satterthwaite, the t-table, the adaptive procedure and the random-number generator
• B. N. Taylor and C. E. Kuyatt, NIST Technical Note 1297, Guidelines for evaluating and expressing the uncertainty of NIST measurement results
• The rounding rule of the Particle Data Group, Review of Particle Physics

Limitations

• The GUM formula is first order; strongly curved formulas or derivatives near zero need the Monte Carlo interval, which is shown next to it.
• Units are labels only: they are not converted or checked, so enter every value in consistent units.
• Correlated inputs are drawn as normal distributions in the Monte Carlo method, whatever distribution you chose for them.
• The Monte Carlo check stops after two million trials; for 99 % or more its last digit can still move.
• One result per calculation: models with several outputs or implicit equations are not covered.

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