Error Propagation
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover,
الجزء والصفحة:
...
11-2-2021
1637
Error Propagation
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)
|
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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