Combine uncertainty for sums, products, quotients, powers, and linear models. Review formulas and worked examples. Export accurate results for reports and better statistical decisions.
| Mode | Inputs | Result | Combined Uncertainty |
|---|---|---|---|
| Addition | x = 12.50 ± 0.20, y = 4.10 ± 0.10, ρ = 0.15 | 16.60 | 0.2366 |
| Product | x = 8.00 ± 0.30, y = 5.00 ± 0.20, ρ = 0.00 | 40.00 | 2.1932 |
| Quotient | x = 20.00 ± 0.40, y = 4.00 ± 0.10, ρ = 0.00 | 5.00 | 0.1601 |
| Power | x = 10.00 ± 0.20, n = 2.00 ± 0.00, ρ = 0.00 | 100.00 | 4.0000 |
| Linear Model | a = 1.50 ± 0.03, x = 20.00 ± 0.40, b = 2.00 ± 0.10 | 32.00 | 0.8544 |
General first-order Gaussian propagation:
uz2 ≈ Σ(∂f/∂qi)2ui2 + 2ΣΣ(∂f/∂qi)(∂f/∂qj)ρijuiuj
z = x + y
uz2 = ux2 + uy2 + 2ρ(x,y)uxuy
z = x − y
uz2 = ux2 + uy2 − 2ρ(x,y)uxuy
z = xy
uz2 = (y ux)2 + (x uy)2 + 2ρ(x,y)xyuxuy
z = x / y
uz2 = (ux/y)2 + (x uy/y2)2 − 2ρ(x,y)(x uxuy/y3)
z = xn
uz2 = (n xn−1ux)2 + (xnln(x)un)2 + 2ρ(x,n)(n xn−1ux)(xnln(x)un)
z = ax + b
uz2 = (x ua)2 + (a ux)2 + ub2 + 2ρ(a,x)axuaux
Gaussian propagation estimates how measurement uncertainty moves through a formula. It is useful when values come from instruments, surveys, calibration steps, or statistical models. Small input changes can affect the final answer. A clear uncertainty estimate improves interpretation. It also helps teams compare methods, defend assumptions, and report results consistently.
This calculator supports common uncertainty workflows. You can test sums, differences, products, quotients, powers, and a linear model. That covers many laboratory, quality control, and data analysis tasks. It is especially helpful when you need a fast estimate before writing a report, validating a process, or checking whether variation is acceptable.
Correlation changes propagated uncertainty. Two inputs may move together because they share the same instrument, method, or data source. Positive correlation can increase uncertainty in some models. In subtraction, it can reduce it. Ignoring dependence may overstate or understate the final interval. This page lets you include correlation directly where it matters most.
The calculator uses first-order partial derivatives. This method is often called linearized uncertainty propagation. It works well when uncertainties are small relative to the measured values and the model is reasonably smooth. The result is a combined standard uncertainty. You can then multiply it by a coverage factor to get an expanded uncertainty interval.
A strong result includes the estimate, uncertainty, interval, and assumptions. You should also note the model used, the meaning of each uncertainty term, and whether correlation was included. Consistent reporting supports quality audits, reproducibility, and decision making. Export tools make it easier to save the numbers for documentation, analysis logs, and client summaries.
Use standard uncertainty values, not rough guesses. Keep units consistent before calculation. Avoid quotient mode when the denominator is close to zero. For power mode, use a positive base when the exponent has uncertainty. Review intervals against real process limits. When the model is highly nonlinear, use simulation or a more advanced method.
It is a method for estimating how uncertainty in inputs affects a calculated result. It usually assumes small errors and near-normal measurement behavior.
Standard uncertainty is the uncertainty expressed as a standard deviation. It is the usual input form for first-order propagation formulas.
Use correlation when two inputs are not independent. Shared instruments, repeated calibrations, or linked data sources often create dependence.
The coverage factor scales the combined standard uncertainty into an expanded uncertainty interval. Many reports use k = 2 for an approximate 95 percent interval.
It is best for smooth models and relatively small uncertainties. Strong nonlinearity can require simulation methods such as Monte Carlo analysis.
If two inputs are positively correlated, subtracting one from the other can cancel some shared variation. That reduces propagated uncertainty in the result.
Quotient calculations become unstable near zero. Small denominator changes can create very large output changes and unreliable uncertainty estimates.
When exponent uncertainty is included, the formula uses ln(x). That requires a positive base. If exponent uncertainty is zero, some negative-base cases can still work.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.