Uncertainty of addition or subtraction
Consider a quantity A = a b where a and b have standard errors σa and σb, then
As subtraction is simply addition with the signs changed, the above also holds for the error in a difference:
Equation should not be confused with equation (2.4) which refers to the mean, not the sum as above.
Uncertainty of a product:
Consider A = (a × b × c) where a, b and c have standard errors σa, σb and σc. The variance
The terms in brackets may be regarded as the relative errors Ra, Rb, Rc giving
Uncertainty of a quotient
Consider A = a/b, then the variance
Uncertainty of powers and roots:
The case for the power of a number must not be confused with the case for multiplication, for example
a3 = a × a × a, with each term being exactly the same.
Thus if A = an, then the variance
Similarly for roots, if the function is A = a1/n, then the variance
The same approach is adapted to general forms which are combinations of the above.