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

			(2)
			(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)

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.