Uncertainty Propagation Calculator
Compute uncertainty propagation using validated scientific equations. See step-by-step derivations, unit analysis, and reference values.
Formula
Addition/Subtraction: delta_c = sqrt(delta_a^2 + delta_b^2) | Multiplication/Division: (delta_c/c) = sqrt((delta_a/a)^2 + (delta_b/b)^2)
For addition and subtraction, absolute uncertainties add in quadrature. For multiplication and division, relative uncertainties add in quadrature. These rules assume independent, random uncertainties.
Worked Examples
Example 1: Addition of Two Mass Measurements
Problem: A chemist measures two samples: Sample A = 25.0 +/- 0.3 g and Sample B = 10.0 +/- 0.2 g. What is the total mass and its uncertainty?
Solution: Combined value = 25.0 + 10.0 = 35.0 g\nUncertainty = sqrt(0.3^2 + 0.2^2) = sqrt(0.09 + 0.04) = sqrt(0.13) = 0.3606 g\nRelative uncertainty = 0.3606 / 35.0 x 100 = 1.03%
Result: Total mass = 35.0 +/- 0.36 g (1.03% relative uncertainty)
Example 2: Division for Concentration Calculation
Problem: Calculate concentration: mass = 5.00 +/- 0.05 g, volume = 250.0 +/- 0.5 mL. Concentration = mass / volume.
Solution: Concentration = 5.00 / 250.0 = 0.0200 g/mL\nRel. uncertainty mass = 0.05/5.00 = 1.00%\nRel. uncertainty volume = 0.5/250.0 = 0.20%\nCombined rel. uncertainty = sqrt(1.00^2 + 0.20^2) = sqrt(1.04) = 1.0198%\nAbsolute uncertainty = 0.0200 x 0.010198 = 0.000204 g/mL
Result: Concentration = 0.0200 +/- 0.0002 g/mL (1.02% relative uncertainty)
Frequently Asked Questions
What is uncertainty propagation in analytical chemistry?
Uncertainty propagation is the process of determining how measurement uncertainties in individual variables combine when those variables are used in a mathematical calculation. In analytical chemistry, every measurement carries some degree of uncertainty due to instrument limitations, environmental factors, and human error. When you combine measurements through addition, subtraction, multiplication, or division, the uncertainties propagate through the calculation following specific mathematical rules derived from calculus and statistics. Understanding this propagation is essential for reporting accurate and meaningful results in laboratory work and scientific research.
How does uncertainty propagate through addition and subtraction?
For addition and subtraction operations, the absolute uncertainties combine in quadrature, meaning you take the square root of the sum of squared individual uncertainties. The formula is delta_result equals the square root of (delta_A squared plus delta_B squared). This applies regardless of whether you are adding or subtracting the values. For example, if you measure two masses as 25.0 plus or minus 0.3 grams and 10.0 plus or minus 0.2 grams, the uncertainty in their sum (35.0 g) would be the square root of (0.09 plus 0.04) which equals 0.36 grams. This quadrature rule assumes the uncertainties are independent and random.
How does uncertainty propagate through multiplication and division?
For multiplication and division, the relative (percentage) uncertainties combine in quadrature. The formula is: relative uncertainty of result equals the square root of (relative uncertainty of A squared plus relative uncertainty of B squared). The relative uncertainty is calculated as the absolute uncertainty divided by the measured value. This means that in multiplication and division, it is the fractional uncertainties that add in quadrature rather than the absolute uncertainties. After computing the combined relative uncertainty, you multiply it by the result value to obtain the absolute uncertainty of the final answer for proper reporting.
What is the difference between absolute and relative uncertainty?
Absolute uncertainty is expressed in the same units as the measurement itself, such as 25.0 plus or minus 0.3 grams, where 0.3 grams is the absolute uncertainty. Relative uncertainty is the ratio of the absolute uncertainty to the measured value, typically expressed as a percentage or decimal fraction. For the same example, the relative uncertainty is 0.3 divided by 25.0 equals 0.012 or 1.2 percent. Absolute uncertainty is used when propagating through addition and subtraction, while relative uncertainty is used for multiplication and division. Both forms are important for proper scientific reporting and quality assurance in analytical laboratories.
What is expanded uncertainty and coverage factor?
Expanded uncertainty provides a wider interval around the measurement result that is expected to encompass a larger fraction of the distribution of values that could reasonably be attributed to the measurand. It is calculated by multiplying the combined standard uncertainty by a coverage factor k. A coverage factor of k equals 2 corresponds to approximately a 95 percent confidence interval for a normal distribution, while k equals 3 corresponds to approximately 99.7 percent. Most analytical laboratories report results with expanded uncertainty using k equals 2, following guidelines from the Guide to the Expression of Uncertainty in Measurement published by international standards organizations.
Is my data stored or sent to a server?
No. All calculations run entirely in your browser using JavaScript. No data you enter is ever transmitted to any server or stored anywhere. Your inputs remain completely private.