How to propagate uncertainty in a lab report

Every measurement has an uncertainty attached to it, and when two uncertain measurements get combined into a calculated result, that result inherits an uncertainty of its own. Working out how much is "propagation of uncertainty," and it comes down to exactly two formulas depending on what operation combined the measurements.

Addition and subtraction: absolute uncertainties add in quadrature

If z = x + y or z = x - y, the absolute uncertainties combine as:

δz = √(δx² + δy²)

Note that subtraction still adds uncertainty - you don't get to subtract the uncertainties away just because the values themselves are being subtracted. Both measurements contribute noise to the result regardless of the sign of the operation.

Multiplication and division: relative uncertainties add in quadrature

If z = x × y or z = x ÷ y, it's the relative uncertainties (uncertainty divided by the value itself) that combine this way:

δz/z = √((δx/x)² + (δy/y)²)

Then multiply that relative uncertainty back by z to get an absolute uncertainty for the result. This is the step most students skip - reporting a relative uncertainty as if it were the final absolute uncertainty, which silently understates or overstates the actual error depending on the magnitude of z.

Why this matters beyond "showing your work"

A lab report's uncertainty section isn't a formality - it's what determines whether two measured results actually agree with each other, or with a theoretical value. Two numbers that look different can be statistically consistent once their propagated uncertainties are accounted for, and two numbers that look close can actually disagree once uncertainty is factored in. Skipping propagation and just reporting the raw calculated value with no uncertainty (or worse, with an invented one) is one of the most common ways student lab reports lose points on an otherwise-correct calculation.

Try the error propagation calculator to check a homework or lab result quickly, and if the underlying units are giving you trouble, SI vs. imperial units and the engineering unit converter handle the same problem set. More on how study science outside STEM works in studying outside STEM.