Every measured value carries an uncertainty: a ruler can be read to the nearest millimetre, a stopwatch is limited by reaction time, a balance by its last digit. Writing a result as 2.00 ± 0.02 m says the true value very likely lies between 1.98 and 2.02 m. When you calculate something from such values, the uncertainties carry through to the answer, and this calculator works out by how much.
The method is the standard first-order propagation used in physics and chemistry labs and described in the international Guide to the Expression of Uncertainty in Measurement (GUM). For each input, multiply its uncertainty by how sensitive the result is to that input (the partial derivative), then combine the contributions in quadrature. For sums this reduces to combining absolute uncertainties; for products and quotients to combining relative ones; for functions of one variable to multiplying ΔX by the slope of the function.
The rules assume the inputs are independent and the uncertainties small compared with the values, so that the function is nearly straight over the uncertainty range. For very large relative uncertainties or strongly curved functions such as logarithms near zero, a Monte Carlo simulation gives a better answer.
How to use the Uncertainty Calculator
- Operation
- X
- ΔX (uncertainty of X)
- Y
- ΔY (uncertainty of Y)
- Base b
- Base a
- Root index n
Results update as you type, and everything is calculated in your browser.
How it works
Choose the operation, then enter X with its uncertainty ΔX (and Y with ΔY, or the base or root index where needed). The calculator evaluates Z and applies the propagation formula for that operation.
Worked example
A rectangle measures 4.0 ± 0.1 cm by 2.5 ± 0.1 cm. Its area is 10.0 cm². The contributions are 2.5 × 0.1 = 0.25 and 4.0 × 0.1 = 0.40, so ΔZ = √(0.25² + 0.40²) = 0.47 cm² – the area is 10.0 ± 0.5 cm², a relative uncertainty of 4.7%.
Propagation formulas
| Function Z | Uncertainty ΔZ |
|---|---|
| X ± Y | √(ΔX² + ΔY²) |
| X·Y, X/Y | |Z|·√((ΔX/X)² + (ΔY/Y)²) |
| Xʸ | |Z|·√((yΔX/X)² + (ln X·ΔY)²) |
| X² | 2|X|ΔX |
| 1/X | ΔX/X² |
| ln X | ΔX/|X| |
| log_b X | ΔX/(|X|·ln b) |
| eˣ | eˣ·ΔX |
| aˣ | aˣ·ln a·ΔX |
| √X | ΔX/(2√X) |
| ⁿ√X | |Z|·ΔX/(n|X|) |
Inputs explained
| Field | What to enter |
|---|---|
| Operation | Choose one: Addition: X + Y, Subtraction: X − Y, Multiplication: X × Y, Division: X ÷ Y … |
| X | A number |
| ΔX (uncertainty of X) | A number |
| Y | A number |
| ΔY (uncertainty of Y) | A number |
| Base b | A number |
| Base a | A number |
| Root index n | A number |
What each result means
- Relative uncertainty ΔZ ÷ |Z| = ([Uncertainty ΔZ]÷abs([Result Z]))×100
- Lower value Z − ΔZ = [Result Z] – [Uncertainty ΔZ]
- Upper value Z + ΔZ = [Result Z] + [Uncertainty ΔZ]
- Relative uncertainty of X = ([ΔX (uncertainty of X)]÷abs([X]))×100
Formula
General rule: ΔZ = √((∂Z/∂X · ΔX)² + (∂Z/∂Y · ΔY)²) · X ± Y: √(ΔX² + ΔY²) · X^n: |n|·|Z|·ΔX/|X| · ln X: ΔX/|X| · eˣ: eˣ·ΔX · √X: ΔX/(2√X)
Worked example
Inputs — Operation: Multiplication: X × Y; X: 4; ΔX (uncertainty of X): 0.1; Y: 2.5; ΔY (uncertainty of Y): 0.1.
Results — Result Z = 10; Uncertainty ΔZ = 0.471699; Relative uncertainty ΔZ ÷ |Z| = 4.717 %; Lower value Z − ΔZ = 9.528301; Upper value Z + ΔZ = 10.471699; Relative uncertainty of X = 2.5 %.
Key terms
- Uncertainty
- A ± range around a measured value within which the true value is expected to lie.
- Partial derivative
- How much the result changes per unit change in one input, holding the others fixed.
- Combined standard uncertainty
- The quadrature sum of all input contributions.
Frequently asked questions
How do I calculate the uncertainty of a sum?
Combine the absolute uncertainties in quadrature: ΔZ = √(ΔX² + ΔY²). 1.20 ± 0.02 m + 0.80 ± 0.01 m = 2.00 ± 0.022 m, usually written 2.00 ± 0.02 m.
How do I calculate the uncertainty of a product?
Combine relative uncertainties: for a 4.0 ± 0.1 by 2.5 ± 0.1 rectangle the area is 10 with ΔZ = √((2.5 × 0.1)² + (4 × 0.1)²) = 0.47.
What is the uncertainty of X²?
ΔZ = 2|X|ΔX – the relative uncertainty doubles. 3.0 ± 0.1 squared is 9.0 ± 0.6.
What is the uncertainty of a logarithm?
Δ(ln X) = ΔX ÷ X, an absolute uncertainty equal to the relative uncertainty of X. Δ(log₁₀ X) = ΔX ÷ (X ln 10).
What is the difference between absolute and percentage uncertainty?
Absolute uncertainty is in the units of the measurement (±0.02 m); percentage uncertainty divides it by the value (±1%).
Why can uncertainty never be zero?
Every instrument has finite resolution and every measurement is affected by small random influences, so some doubt always remains.
Are the inputs assumed independent?
Yes. If X and Y are correlated, a covariance term must be added, and the worst case is simply adding the contributions.