Propagation of Error

In many instances, the quantity of interest is calculated from a combination of direct measurements. Two questions face us:

1. Given the experimental uncertainty in the directly measured quantities, what is the uncertainty in the final result?
2. In designing our experiment, where is effort best spent in improving the precision of the measurements?

The approach is called propagation of error. The theoretical background may be found in Garland, Nibler & Shoemaker, ???, or the Wikipedia page (particularly the "simplification"). We will present the simplest cases you are likely to see; these must be adapted (obviously) to the specific form of the equations from which you derive your reported values from direct measurements.

Addition and subtraction

Note--$$S=√{S^2}$$

Formula for the result:
$$x=a+b-c$$
x is the target value to report, a, b and c are measured values, each with some variance S²_a, S²_b, S²_c.
$$S_x=√{S^2_a+S^2_b+S^2_c}$$
(S_x can now be translated to a confidence interval by means previously discussed.

Multiplication/division

Formula for the result:
$$x={ab}/c$$
As above, x is the target value to report, a, b and c are measured values, each with some variance S²_a, S²_b, S²_c.
$$S_x=x√{{(S_a/a)}^2+{(S_b/b)}^2+{(S_c/c)}^2}$$

Exponentials (no uncertainty in b)

Formula for the result:
$$x=a^b$$
$$S_x=xb(S_a/a)$$
Special cases:
Antilog, base 10:
$$x=10^a$$
$$S_x=2.303xS_a$$
Antilog, base e:
$$x=e^a$$
$$S_x=xS_a$$

Logarithms

Base 10:
$$x=log{a}$$
$$S_x=0.434(S_a/a)$$
Base e:
$$x=ln{a}$$
$$S_x={S_a/a}$$