Measurement uncertainty

Often when we use a physical measurement instrument, there is some uncertainty associated with using it. Incorporating and propagating errors is an important part of analysing experimentally gathered data. For example, a micrometer may give us:

$$8.5\pm 0.01\text{mm}$$

The first term is the measured value, and the second is the uncertainty.

We define the percentage error as:

$$\text{\% error}=\frac{\text{uncertainty}}{\text{measured value}}\times 100\%$$

Propagation

When we add/subtract uncertainties, we sum the absolute values of the uncertainties.

$$(X\pm x)-(Y\pm y)+(Z\pm z)=(X-Y+Z)\pm (x+y+z)$$

For the percentage error of a division or multiplication of two measured values, we have:

$$\text{\% error}=\sqrt{{\left(\frac{\Delta A}{A}\right)}^2+{\left(\frac{\Delta B}{B}\right)}^2}\times 100\%$$