Any operations that we perform on a measured value that has
uncertainty will require us to propagate the variance associated
with the measurements. Propagation of Variance (POV) is technically
defined as: "the determination of the value to be assigned as the
uncertainty of a given quantity using mathematical formulas for the
combination of errors from constituent contributors". This occurs
when we perform any of the following operations:

- adding two measurements
- dividing by a constant
- subtracting off a background
- multiplication or division of measurements
- producing a mean value of multiple individual measurements
- combining independent measurements with unequal errors

The general error propagation formula when a quantity *u* is calculated using a set of
variables ( *x, y, z,* etc.) is
given by

This formula is fairly complicated; however, for the standard
operations that we commonly perform, it readily simplifies. Below
is a set of simplified formulas for common operations.

#### Adding and Subtracting

When adding (or subtracting) two measurements, each with a given
uncertainty, the uncertainty of the final quantity will be produced
by summing the standard deviations of the individual parts in
quadrature. Thus,

#### Multiplying and Dividing by a Constant

If we divide (or multiply) a quantity with an uncertainty
by a constant (i.e., a number with no uncertainty), then the
standard deviation of the product is simply equal to the standard
deviation of the original value multiplied (or divided) by the
constant. Thus,

#### Multiplying and Dividing Two Quantities with Uncertainty

If we multiply two quantities together, the fractional standard
deviation of the product is determine by summing the fractional
standard deviations of the original quantities in quadrature.
Thus,

#### Summary

So using the POV formulas, we can calculate the uncertainty to
any measured quantity. This uncertainty then tells us something
about the quality of the measured quantity and might tell us
something about how that quantity differs from what we expected.
For example, whether what we have measured has a mass significantly
smaller than what was declared. We might also use this to determine
if we need to measure a sample again or measure more samples.