Combined standard uncertainty
For a result f(x₁, x₂, …, xₙ) with small uncertainties and a first-order approximation, the combined standard uncertainty is:
If variables are independent, covariances are zero and only the squared terms remain.
Two-variable correlation shortcut
If you know a correlation coefficient ρ between two quantities, then:
Expanded uncertainty
Many reports use an expanded uncertainty U = k·u with a coverage factor k such as 2 for approximately 95% coverage under common assumptions.
- Select a calculation method that matches your equation.
- Enter the measured values and their standard uncertainties.
- Add correlation ρ only if you have a justified estimate.
- Set k to get expanded uncertainty if needed.
- Press Calculate to see results above the form.
- Use the CSV or PDF buttons to save your output.
These examples assume independent inputs (ρ = 0) and standard uncertainties.
| Case | Inputs | Model | Output |
|---|---|---|---|
| Addition | a=12.0, u(a)=0.3; b=5.0, u(b)=0.2 | f=a+b | f=17.0, u=0.361 |
| Multiplication | a=10.0, u(a)=0.2; b=3.0, u(b)=0.1 | f=a×b | f=30.0, u=1.044 |
| Power | a=4.00, u(a)=0.05; n=2 | f=a² | f=16.0, u=0.400 |
| Single-variable | x=1.00 rad, u(x)=0.02 | y=sin(x) | y=0.842, u=0.011 |
| General sensitivity | c1=2.0, u1=0.10; c2=−1.0, u2=0.20 | u=√((c1u1)²+(c2u2)²) | u=0.224 |
1) Why uncertainty propagation matters
Measurements are never exact, so calculated results inherit uncertainty. Propagation converts instrument limits into a defensible uncertainty for the final quantity, improving comparisons across trials. In physics reporting, a clear uncertainty budget communicates confidence and supports technical decisions.
2) Standard versus expanded uncertainty
This calculator outputs the combined standard uncertainty u, often interpreted as one standard deviation. Many reports also quote expanded uncertainty U = k·u. A common choice is k = 2 for an interval near 95% coverage when assumptions are reasonable.
3) The first-order model and its limits
Propagation typically uses a first-order Taylor expansion about the measured values. It works best when uncertainties are small and the model is smooth near the operating point. If the function is strongly nonlinear, validate using Monte Carlo sampling or direct repeat measurements. Also watch for boundaries and near-zero denominators.
4) Common operations you use every day
For sums and differences, variances add, with an optional correlation term. For products, ratios, and powers, relative uncertainties are practical: percent uncertainty in f relates to percent uncertainties in the inputs. The correlation coefficient ρ spans −1 to 1 for shared systematics.
5) Sensitivity coefficients connect physics to math
In the general method, each variable contributes through a sensitivity coefficient cᵢ = ∂f/∂xᵢ. Coefficients carry units that convert input uncertainty into output uncertainty. When analytic derivatives are hard, approximate numerically with cᵢ ≈ (f(xᵢ+Δ)−f(xᵢ−Δ))/(2Δ) using a small Δ.
6) Correlation can dominate the result
Independent inputs usually give the smallest combined uncertainty for fixed u(xᵢ). Positive correlation increases uncertainty for sums and can reduce it for differences, while negative correlation does the opposite. Use nonzero ρ only when justified by calibration history, shared sensors, or paired data.
7) Building a traceable uncertainty budget
A professional workflow is: define the measurement model, list inputs and standard uncertainties, compute sensitivities, then combine terms. Document assumptions about distributions and correlations. Keep extra digits during intermediate steps to avoid rounding bias. Finally, round uncertainty to one or two significant digits and round the value to the same decimal place.
8) Quality checks before you publish
Check units first; inconsistencies are the most common failure. Identify dominant contributors by comparing |cᵢuᵢ| values and improve that measurement if possible. If relative uncertainty looks inflated, verify you entered a standard deviation, not a full tolerance range, and re-check denominators.
1) What is the difference between error and uncertainty?
Uncertainty is a quantified range that describes dispersion of values. Error is the difference between a measured value and the true value, which is usually unknown in experiments.
2) Should I use k = 1 or k = 2?
Use k = 1 for standard uncertainty (about one sigma). Use k = 2 for expanded uncertainty when you want an interval near 95% coverage under common assumptions.
3) When does the first-order propagation fail?
It can fail for strong nonlinearity, large relative uncertainties, or functions with discontinuities near your input values. In those cases, validate using a Monte Carlo approach or a direct measurement strategy.
4) How do I choose a correlation coefficient ρ?
Use ρ only when inputs share a systematic source, like the same sensor or calibration. Estimate it from repeated paired measurements, covariance from a fit, or documented instrument behavior.
5) What units should I enter for uncertainty?
Uncertainty must use the same units as its corresponding value. For example, if length is in meters, enter u(length) in meters too. Sensitivity coefficients then handle unit conversion to the output.
6) Why is relative uncertainty blank sometimes?
Relative uncertainty requires a nonzero output value. In the general sensitivity method, the calculator may not compute f, only u(f). Enter or compute the value separately to obtain percentages.
7) How should I report the final result?
Report as value ± U, specify the coverage factor and confidence level, and state key assumptions. Round U to one or two significant digits and match the value’s decimal place.