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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-x^_=(u_i-u^_)(partialx)/(partialu)+(v_i-v^_)(partialx)/(partialv)+...,
(1)

where x^_ denotes the mean, so the sample variance is given by

s_x^2=1/(N-1)sum_(i=1)^(N)(x_i-x^_)^2
(2)
=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))+...].
(3)

The definitions of variance and covariance then give

s_u^2=1/(N-1)sum_(i=1)^(N)(u_i-u^_)^2
(4)
s_v^2=1/(N-1)sum_(i=1)^(N)(v_i-v^_)^2
(5)
s_(uv)=1/(N-1)sum_(i=1)^(N)(u_i-u^_)(v_i-v^_)
(6)

(where s_(ii)=s_i^2), so

 s_x^2=s_u^2((partialx)/(partialu))^2+s_v^2((partialx)/(partialv))^2+2s_(uv)((partialx)/(partialu))((partialx)/(partialv))+....
(7)

If u and v are uncorrelated, then s_(uv)=0 so

 s_x^2=s_u^2((partialx)/(partialu))^2+s_v^2((partialx)/(partialv))^2.
(8)

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

 s_x^2=(a^2)/(v^2)s_u^2+(a^2u^2)/(v^4)s_v^2-2a/v(au)/(v^2)s_(uv).
(10)

Dividing through by x^2 and rearranging then gives

 ((s_x)/x)^2=((s_u)/u)^2+((s_v)/v)^2-2((s_(uv))/u)((s_(uv))/v).
(11)

For exponentiation of quantities with

 x=a^(+/-bu)=(e^(lna))^(+/-bu)=e^(+/-b(lna)u),
(12)

and

 (partialx)/(partialu)=+/-b(lna)e^(+/-blnau)=+/-b(lna)x,
(13)

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

 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)
((s_x)/x)^2=(a^2v^2)/(a^2u^2v^2)s_u^2+(a^2u^2)/(a^2u^2v^2)s_v^2+(2a^2uv)/(a^2u^2v^2)s_(uv)
(20)
=((s_u)/u)^2+((s_v)/v)^2+2((s_(uv))/(uv)).
(21)

For powers, with x=au^(+/-b), partialx/partialu=+/-abu^(+/-b-1)=+/-bx/u, so

 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, Variance

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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.

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Error Propagation

Cite this as:

Weisstein, Eric W. "Error Propagation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ErrorPropagation.html

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