Propagation of errors resulting from algebraic manipulations
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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.
General formula
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]1/2
Example formulas
A, B ... are uncorrelated variables with errors ΔA, ΔB ...; c is a precisely known constant.
| 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 |
Example application: Resistance measurement
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
- ΔR = √[(ΔV I-1)2 + (ΔI V I-2)2] = R√[(ΔV/V)2 + (ΔI/I)2]
Thus, in this simple case, the relative error ΔR/R is simply the geometric mean of the two relative errors of the measured variables.