## Error Propagation

Given a Formula with an Absolute Error in of , the Absolute Error is . The Relative Error is . If , then

 (1)

so
 (2)

The definitions of Variance and Covariance then give
 (3) (4) (5)

so

 (6)

If and are uncorrelated, then so
 (7)

Now consider addition of quantities with errors. For , and , so

 (8)

For division of quantities with , and , so

 (9)

 (10)

For exponentiation of quantities with

 (11)

 (12)

so
 (13)

 (14)

If , then
 (15)

For Logarithms of quantities with , , so

 (16)

 (17)

For multiplication with , and , so

 (18)

 (19)

For Powers, with , , so

 (20)

 (21)