Given a formula 
with an absolute error
in 
of 
,
the absolute error is 
. The relative error is
. If 
, then
 |
(1) |
where 
denotes the mean, so the sample
variance is given by
The definitions of variance and covariance then give
(where 
),
so
 |
(7) |
If 
and 
are uncorrelated, then 
so
 |
(8) |
Now consider addition of quantities with errors. For 
, 
and 
, so
 |
(9) |
For division of quantities with 
, 
and 
, so
 |
(10) |
Dividing through by 
and rearranging then gives
 |
(11) |
For exponentiation of quantities with
 |
(12) |
and
 |
(13) |
so
 |
(14) |
 |
(15) |
If 
,
then
 |
(16) |
For logarithms of quantities with 
, 
, so
 |
(17) |
 |
(18) |
For multiplication with 
, 
and 
, so
 |
(19) |
For powers, with 
, 
, so
 |
(22) |
 |
(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.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Error Propagation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ErrorPropagation.html