Propagation of errors resulting from algebraic manipulations Guide, Meaning , Facts, Information and Description
In statistics propagation of errors means to calculate the error on a quantity f from (usually measured) other quantities xi and the knowledge of how to compute f from the xi.
Given a function f(x1,x2,...) which depends on N uncorrelated variables xj with known errors Δxj one can compute the error Δf in f:
Δf(x1, x2, ..., Δx1, Δx2, ...) = [∑1 ≤ i ≤ N (Δxi
∂i f(x1,x2,...))² ]1/2
where ∂i f designates the partial derivative of f for the i-th variable and evaluated for the values of the xj.
If the xj are correlated then the covariance between pairs Ci,k := cov(xi,xk) enters the formula through a double sum over all pairs (i,k) (where Ci,i = var(xi) = Δxi²):
Δf(x1, x2, ..., C1,1,C1,2, ...) = [
∑1 ≤ i ≤ N, 1 ≤ k ≤N
Ci,k
∂i f ∂k f]1/2General formula
| relationship | error in the result, ΔX |
| X = A ± B | (ΔX)² = (ΔA)² + (ΔB)² |
| X = cA | (ΔX) = c(ΔA) |
| X = c(A×B) or X = c(A/B) | (ΔX/X)² = (ΔA/A)² + (ΔB/B)² |
| X = c(A×B×C) or X = c(A/B)×C | (ΔX/X)² = (ΔA/A)² + (ΔB/B)² + (ΔC/C)² |
| X = cAn | (ΔX/X) = |n| (ΔA/A) |
| X = ln cA | ΔX = (ΔA/A) |
| X = exp A | (ΔX/X) = ΔA |
A practical application is an experiment in which one measures current I and voltage V on a resistor in order to determine the resistance R using Ohm's law, R = V/I.
Given the measured variables with uncertainties, I±ΔI and V±ΔV, the uncertainty in the computed quantity, ΔR is
This is an Article on Propagation of errors resulting from algebraic manipulations. Page Contains Information, Facts Details or Explanation Guide About Propagation of errors resulting from algebraic manipulations Example application: Resistance measurement
Thus, in this simple case, the relative error ΔR/R is simply the geometric mean of the two relative errors of the measured variables.