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
 |  |  |
(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))+...].](/images/equations/ErrorPropagation/Inline13.svg) |
(3)
|
The definitions of variance and covariance then give
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
(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)
|
 |  |  |
(20)
|
 |  |  |
(21)
|
For powers, with 
, 
, so
 |
(22)
|
 |
(23)
|
