propagation of uncertainty calculator - Aaron Graves, PhDude Replica

Calculator

Estimate the uncertainty in a derived quantity using first-order error propagation (independent inputs, standard uncertainty, k = 1).

Model / Operation

a (measured value)

u(a) (absolute uncertainty of a)

b (measured value)

u(b) (absolute uncertainty of b)

n (exponent)

For power mode only: z = a^n

What is propagation of uncertainty?

Whenever you calculate a result from measured inputs, your result inherits uncertainty from those inputs. That transfer is called propagation of uncertainty (also called error propagation).

Example: If you measure length and width to compute area, the area uncertainty depends on both measurement uncertainties—not just one. This matters in physics labs, chemistry, engineering, calibration work, data science, and financial modeling with measured variables.

Core idea (first-order method)

For a function z = f(x, y, ...), the first-order (linearized) combined standard uncertainty is:

u(z) = sqrt( (∂f/∂x · u(x))² + (∂f/∂y · u(y))² + ... )

This calculator applies that approach for common operations. It assumes:

Input uncertainties are standard uncertainties (1σ).
Inputs are independent (no covariance terms).
A linear approximation is reasonable near the measured values.

Formulas used in this calculator

1) Addition and subtraction

z = a ± b u(z) = sqrt( u(a)² + u(b)² )

Even for subtraction, uncertainties still add in quadrature when variables are independent.

2) Multiplication

z = a · b u(z) = sqrt( (b·u(a))² + (a·u(b))² )

This derivative form is robust, even when one input is zero.

3) Division

z = a / b u(z) = sqrt( (u(a)/b)² + (a·u(b)/b²)² ), b ≠ 0

4) Power function

z = a^n u(z) = |n·a^(n-1)| · u(a)

This is the one-variable propagation rule using the derivative of a^n.

How to use this tool correctly

Enter values and absolute uncertainties in consistent units.
Choose the operation that matches your model.
Interpret the output as standard uncertainty (k=1).
If you need expanded uncertainty, multiply by a coverage factor (often k = 2 for ~95% in many contexts).

Worked mini example

Suppose you compute density from mass and volume using division:

m = 25.0 ± 0.2 g
V = 10.0 ± 0.1 mL
ρ = m/V = 2.5 g/mL

Plugging into the division formula gives a density uncertainty of about 0.032 g/mL, so:

ρ = 2.50 ± 0.03 g/mL (k = 1, rounded)

Best practices and limitations

Best practices

Track units carefully; uncertainty uses the same unit as the quantity.
Report uncertainty with sensible significant figures.
Use relative uncertainty (%) to compare measurement quality across scales.

Limitations of first-order propagation

Less accurate for highly nonlinear models with large uncertainties.
Does not include covariance between inputs.
Can break near singular points (for example, dividing by values near zero).

For advanced use cases, consider Monte Carlo uncertainty propagation or full covariance-matrix methods.

Final takeaway

A propagation of uncertainty calculator helps convert “input measurement quality” into “output confidence.” Use it whenever computed results are based on measured quantities, especially in labs, engineering calculations, and technical reports.