Error Propagation

Given a Formula $y = f(x)$ with an Absolute Error in $x$ of $dx$, the Absolute Error is $dy$. The Relative Error is ${dy/y}$. If $x=f(u,v)$, then

$$x_i-\bar x = (u_i-\bar u){\partial x\over \partial u}+(v_i-\bar v){\partial x\over \partial v} +\ldots,$$ (1)

so

$${\sigma_x}^2 \equiv {1\over N-1} \sum_i^N (x_i-\bar x)^2$$

$$= {1\over N-1} \sum_i^N \left[{(u_i-\bar u)^2\left({\partial x\over... ...l u}\right)^2+(v_i-\bar v)^2\left({\partial x\over \partial v}\right)^2}\right.$$

$$\phantom{=} \left.{\mathop{+}2(u_i-\bar u)(v_i-\bar v)\left({\partial x\over \partial u}\right)\left({\partial x\over \partial v}\right)+ \ldots}\right].$$ (2)

The definitions of Variance and Covariance then give

$${\sigma_u}^2 \equiv {1\over N-1} \sum_{i=1}^N (u_i-\bar u)^2$$ (3)

$${\sigma_v}^2 \equiv {1\over N-1} \sum_{i=1}^N (v_i-\bar v)^2$$ (4)

$${\sigma_{uv}}^2 \equiv {1\over N-1} \sum_{i=1}^N (u_i-\bar u)(v_i-\bar v),$$ (5)

so

$${\sigma_x}^2 = {\sigma_u}^2\left({\partial x\over\partial u}... ...tial u}\right)\left({\partial x\over\partial v}\right)+\ldots.$$ (6)

If $u$ and $v$ are uncorrelated, then $\sigma_{uv}=0$ so

$${\sigma_x}^2 = {\sigma_u}^2\left({\partial x\over \partial u}\right)^2+{\sigma_v}^2.$$ (7)

Now consider addition of quantities with errors. For $x=au\pm bv$, $\partial x/\partial u=a$ and $\partial x/\partial v=\pm b$, so

$${\sigma_x}^2=a^2{\sigma_u}^2+b^2{\sigma_v}^2\pm 2ab{\sigma_{uv}}^2.$$ (8)

For division of quantities with $x=\pm au/v$, $\partial x/\partial u=\pm a/v$ and $\partial x/\partial v=\mp au/v^2$, so

$${\sigma_x}^2={a^2\over v^2}{\sigma_u}^2+{a^2u^2\over {\sigma_v}^4}-2{a\over v}{au\over v^2}{\sigma_{uv}}^2.$$ (9)

$$\left({\sigma_x\over x}\right)^2 = {a^2\over v^2} {v^2\over a^2u^2} {\sigma_u}^2 +{a^2u^2\over v^4} ... ...over a^2u^2} -2\left({a\over v}\right)\left({au\over v^2}\right){\sigma_{uv}}^2$$

$$= \left({\sigma_u\over u}\right)^2+\left({\sigma_v\over v}\right)^2-2\left({\sigma_{uv}\over u}\right)\left({\sigma_{uv}\over v}\right).$$ (10)

For exponentiation of quantities with

$$x=a^{\pm bu}=(e^{\ln a})^{\pm bu} = e^{\pm b(\ln a)u},$$ (11)

$${\partial x\over \partial u}=\pm b(\ln a)e^{\pm b\ln au} = \pm b(\ln a)x,$$ (12)

so

$${\sigma_x} = {\sigma_u} b(\ln a)x$$ (13)

$${\sigma_x\over x}=b\ln a\sigma_u.$$ (14)

If $a=e$, then

$${\sigma_x\over x}=b\sigma_u.$$ (15)

For Logarithms of quantities with $x=a\ln(\pm bu)$, $\partial x/\partial u=a(\pm b)/(\pm bu)= a/u$, so

$${\sigma_x}^2={\sigma_u}^2 \left({a^2\over u^2}\right)$$ (16)

$$\sigma_x = a{\sigma_u\over u}.$$ (17)

For multiplication with $x=\pm auv$, $\partial x/\partial u=\pm av$ and $\partial x/\partial v=\pm au$, so

$${\sigma_x}^2=a^2v^2{\sigma_u}^2+a^2u^2{\sigma_v}^2+2a^2uv{\sigma_{uv}}^2$$ (18)

$$\left({\sigma_x\over x}\right)^2 = {a^2v^2\over a^2u^2v^2} {\sigma_u}^2+{a^2u^2\over a^2u^2v^2} {\sigma_v}^2+{2a^2uv\over a^2u^2v^2} {\sigma_{uv}}^2$$

$$= \left({\sigma_u\over u}\right)^2+\left({\sigma_v\over v}\right)^2+2\left({\sigma_{uv}\over u}\right)\left({\sigma_{uv}\over v}\right).$$ (19)

For Powers, with $x=au^{\pm b}$, $\partial x/\partial u=\pm abu^{\pm b-1}= \pm bx/u$, so

$${\sigma_x}^2={\sigma_u}^2{b^2x^2\over u^2}$$ (20)

$${\sigma_x\over x}=b{\sigma_u\over u}.$$ (21)

See also Absolute Error, Percentage Error, Relative Error

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 14, 1972.

Bevington, P. R. Data Reduction and Error Analysis for the Physical Sciences. New York: McGraw-Hill, pp. 58-64, 1969.