Error Propagation Calculator

Central Tendency and Dispersion Measures

This tool helps determine the uncertainty of a function based on the uncertainties of its independent variables.

Practical Examples

See how the Error Propagation Calculator works with real-world data.

Adding Two Lengths

sumDifference

Calculating the total length and its uncertainty when adding two measured lengths: L1 = (10.5 ± 0.2) cm and L2 = (5.2 ± 0.1) cm. The formula is z = 1*x + 1*y.

A: 1, X: 10.5, Δx: 0.2

B: 1, Y: 5.2, Δy: 0.1

Perimeter of a Rectangle

sumDifference

Calculating the perimeter P = 2L + 2W, where L = (15.0 ± 0.3) m and W = (8.0 ± 0.2) m. The formula is z = 2*x + 2*y.

A: 2, X: 15.0, Δx: 0.3

B: 2, Y: 8.0, Δy: 0.2

Area of a Rectangle

productPower

Finding the area and its uncertainty for a rectangle with width W = (5.0 ± 0.1) m and length L = (10.0 ± 0.2) m. The formula is A = W * L (z = 1 * x^1 * y^1).

k: 1, x: 5.0, Δx: 0.1, a: 1

y: 10.0, Δy: 0.2, b: 1

Calculating Density

productPower

Calculating density (ρ = m/V) from mass m = (100 ± 2) g and volume V = (10 ± 0.5) cm³. The formula is z = 1 * x^1 * y^-1.

k: 1, x: 100, Δx: 2, a: 1

y: 10, Δy: 0.5, b: -1

Other Titles
Understanding Error Propagation: A Comprehensive Guide
Learn the principles behind calculating uncertainty in derived quantities, a fundamental concept in experimental science and engineering.

What is Error Propagation?

  • The Concept of Uncertainty
  • Why Propagation Matters
  • Sources of Error
Error propagation, or propagation of uncertainty, is a statistical method used to determine the uncertainty in a final result calculated from several variables that each have their own measurement uncertainties. Whenever you measure a quantity in an experiment, there is always some degree of uncertainty. When you use these uncertain measurements to calculate something else, the uncertainties from the original measurements 'propagate' or carry through to the final result. This calculator helps you quantify that final uncertainty.
The Importance in Science
In scientific research, stating a result without its associated uncertainty is incomplete. It's like reporting a location without giving any idea of the area you might be in. Propagation of uncertainty is crucial for comparing results with theoretical values or with other experiments, drawing valid conclusions, and understanding the limitations of your measurements.

Step-by-Step Guide to Using the Error Propagation Calculator

  • Selecting the Right Formula
  • Entering Your Data
  • Interpreting the Results
1. Choose Your Formula Type
The calculator offers two primary models for error propagation that cover a wide range of applications: 'Sum/Difference (z = ax + by)' and 'Product/Power (z = k x^a y^b)'. Select the one that matches the mathematical relationship between your variables.
2. Input Values and Uncertainties
Once you've selected a formula, input fields for your measured values (x, y), their absolute uncertainties (Δx, Δy), and any constants (a, b, k) or exponents (a, b) will appear. Ensure that all uncertainties are entered as non-negative values.
3. Calculate and Analyze
Click 'Calculate' to see the results. The calculator provides three key outputs: the final calculated value (Z), the absolute uncertainty in that value (ΔZ), and the relative uncertainty (%ΔZ), which shows the uncertainty as a percentage of the final result. This helps to understand the magnitude of the error.

Mathematical Derivation and Formulas

  • Formula for Addition and Subtraction
  • Formula for Multiplication and Division
  • The General Formula
Sum/Difference: z = ax ± by
For addition and subtraction, the absolute uncertainties are added in quadrature. The formula for the absolute uncertainty Δz is: Δz = sqrt((a Δx)^2 + (b Δy)^2). The uncertainties add up, regardless of whether the quantities are being added or subtracted.
Product/Power: z = k x^a y^b
For multiplication and division, the relative uncertainties are added in quadrature. The formula for the relative uncertainty (Δz / |z|) is: (Δz / |z|) = sqrt((a Δx / x)^2 + (b Δy / y)^2). The absolute uncertainty Δz is then found by multiplying the relative uncertainty by the absolute value of z.

Real-World Applications of Error Propagation

  • Physics Experiments
  • Chemical Analysis
  • Engineering and Manufacturing
Physics: Calculating Velocity
If you measure distance (d) and time (t) to calculate velocity (v = d/t), the uncertainties in your distance and time measurements will propagate to the calculated velocity. This is a classic product/power rule case (v = d^1 * t^-1).
Chemistry: Titration
In a titration, you might measure the initial and final volume of a burette. The uncertainty in the delivered volume is found by propagating the uncertainties from both the initial and final readings using the sum/difference rule.
Engineering: Tolerance Stack-up
Engineers use error propagation to analyze tolerance stack-up in mechanical assemblies. If multiple parts have a dimensional tolerance (uncertainty), the total tolerance of the assembly is calculated to ensure it functions correctly.

Common Misconceptions and Correct Methods

  • Adding vs. Subtracting Errors
  • Absolute vs. Relative Uncertainty
  • Significance of Quadrature Sum
Myth: You simply add uncertainties.
A common mistake is to simply add the absolute uncertainties (e.g., Δz = Δx + Δy). This overestimates the total error. The correct method is to add them in quadrature (i.e., summing the squares and then taking the square root), which accounts for the statistical independence of the errors.
Myth: Subtracting quantities reduces total error.
When subtracting two large numbers that are very close to each other, the resulting value can be small, but the relative uncertainty can become very large. Uncertainties always add, they never cancel out, even when the quantities themselves are being subtracted.