# Propagations of Measurement Uncertainties - Basic Formulae

Listed are some of the most basic rules for uncertainty propagations in
algebraic manipulations. There are two ways of expressing uncertainties:
absolute uncertainties and fractional uncertainties (also known to you as
"percent uncertainties"). These rules apply to simple uncertainty analysis as you
use in most of your data analysis in PHY121L, PHY142L, and PHY162L. Note that
these rules apply when these fractional uncertainties are much less than 1. The
notations used for measurements, absolute uncertainties, and fractional
uncertainties are explained below.

Upper case letters, e.g. X, Y, Z, etc., are used for measured quantities.

Lower case letters, e.g. a, b, c, etc., are used for constants.

"d" of a quantity represents the absolute uncertainty associated with this measurement, e.g.
dX, dY, dZ,
etc.

dX/X represents the fractional uncertainty of measured
quantity X.

The "percent uncertainty" used in the report guideline is simply expressing
dX/X in terms of percentage, i.e. (dX/X)
´ 100%.

You should
also note that if your data require statistical analysis, then the propagations of
statistical uncertainties (standard deviations) do not follow these rules (these
rules set the upper-bound of the uncertainties), and
shall be derived differently.

## Addition and Subtraction

If the derived quantity Q comes through additions and/or subtractions among
measured quantities, it is better to consider absolute uncertainties when we
investigate the propagations of uncertainties. The fractional uncertainty can be
found after obtaining the absolute uncertainty of the derived quantity.

## Productions and Quotients

When the derived quantity Q is obtained through productions and divisions
among measured quantities, it is more
convenient to consider fractional uncertainties than absolute uncertainties.

## Multiplied by a Constant

If the derived quantity Q is simply a measured quantity X multiplied by a
constant, a, the fractional uncertainty of the derived quantity remains the same
as of the measured quantity.

## Power

If the derived quantity Q is the measured quantity X to a constant
power, a, then the fractional uncertainty of the derived quantity is "a" times
of the fractional uncertainty of the measured quantity.