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))+...].](https://mathworld.wolfram.com/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) |
