Uncertainty Propagation for Addition and Subtraction

For any formula involving addition or subtraction:

The propagated uncertainty is:

This is called adding uncertainties in quadrature - you square each uncertainty, sum them, then take the square root. Notice that both addition and subtraction follow the exact same rule. Whether you add or subtract two quantities, their absolute uncertainties always combine in the same way.

Students often assume that subtracting two quantities reduces uncertainty. It doesn't. The formula is identical for addition and subtraction - Δf = √[(Δx)² + (Δy)²].

In fact, subtraction can dramatically worsen relative uncertainty. If you subtract two nearly equal numbers, the result is small but the absolute uncertainty stays the same - making the relative uncertainty (Δf/f) very large. This is called catastrophic cancellation.

Let's calculate the total length of two rods placed end to end.

Step 1 - Compute the result:

Step 2 - Apply the rule:

Step 3 - Round and report:

The rule extends naturally to any number of terms. For f = x + y + z + ..., the propagated uncertainty is:

Each additional variable contributes its squared uncertainty under the square root. The more variables you add, the larger the combined uncertainty - but it grows more slowly than linear addition would suggest, because of the square root.