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
where x^_ denotes the mean, so the sample variance is given by
![画像:1/(N-1)sum_(i=1)^(N)[(u_i-u^_)^2((partialx)/(partialu))^2+(v_i-v^_)^2((partialx)/(partialv))^2+2(u_i-u^_)(v_i-v^_)((partialx)/(partialu))((partialx)/(partialv))+...].](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/ErrorPropagation/Inline13.svg){kind=link}
The definitions of variance and covariance then give
(where s_(ii)=s_i^2), so
If u and v are uncorrelated, then s_(uv)=0 so
Now consider addition of quantities with errors. For x=au+/-bv, partialx/partialu=a and partialx/partialv=+/-b, so
| s_x^2=a^2s_u^2+b^2s_v^2+/-2abs_(uv). |
(9)
|
For division of quantities with x=+/-au/v, partialx/partialu=+/-a/v and partialx/partialv=∓au/v^2, so
Dividing through by x^2 and rearranging then gives
For exponentiation of quantities with
| x=a^(+/-bu)=(e^(lna))^(+/-bu)=e^(+/-b(lna)u), |
(12)
|
and
so
| s_x=s_ub(lna)x |
(14)
|
| (s_x)/x=blnas_u. |
(15)
|
If a=e, then
| (s_x)/x=bs_u. |
(16)
|
For logarithms of quantities with x=aln(+/-bu), partialx/partialu=a(+/-b)/(+/-bu)=a/u, so
| [画像: s_x^2=s_u^2((a^2)/(u^2)) ] |
(17)
|
| s_x=a(s_u)/u. |
(18)
|
For multiplication with x=+/-auv, partialx/partialu=+/-av and partialx/partialv=+/-au, so
| s_x^2=a^2v^2s_u^2+a^2u^2s_v^2+2a^2uvs_(uv) |
(19)
|
For powers, with x=au^(+/-b), partialx/partialu=+/-abu^(+/-b-1)=+/-bx/u, so
| [画像: s_x^2=s_u^2(b^2x^2)/(u^2) ] |
(22)
|
| (s_x)/x=b(s_u)/u. |
(23)
|
See also
Absolute Error, Accuracy, Covariance, Percentage Error, Precision, Relative Error, Significant Digits, VarianceExplore with Wolfram|Alpha
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References
Abramowitz, M. and Stegun, I. 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.Referenced on Wolfram|Alpha
Error PropagationCite this as:
Weisstein, Eric W. "Error Propagation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ErrorPropagation.html