adding in quadrature

Also called: in quadrature, add in quadrature, errors add in quadrature

Combining independent errors as the sides of a right triangle: √(a² + b²). SEs of 0.6 and 0.8 pp give an SE of the difference of 1.0 pp, not 1.4.

Independent random errors don’t simply add: when one is too high, the other is as likely to be too low, so they partly cancel. Their variances add, and the combined error is the square root of that sum, the same rule as Pythagoras:

$SE_{diff} = \sqrt{SE_1^2 + SE_2^2}, \qquad \sigma_{total} = \sqrt{\sum \sigma_i^2}$

This is why the standard error of a difference between two groups is smaller than the sum of their SEs, and why the uncertainty of a chain of tasks grows more slowly than the sum of each task’s uncertainty.

The same shape appears in distances (Euclidean distance) and in interval formulas such as Newcombe’s, which combines two Wilson widths with a square root of a sum of squares.

Example

Variant A converts at 10% (SE 0.6 pp), variant B at 12% (SE 0.8 pp).

SE of the difference: √(0.36 + 0.64) = 1.0 pp.
The 2 pp gap is 2 SEs, not 2 ÷ 1.4 ≈ 1.4.

Common mistakes

Adding SEs directly. Add the squares, then take the root.
Using it for dependent errors. Paired or correlated measurements need a different formula.
Forgetting to square the SEs first and taking the root of their plain sum.

Learn it in the course

Pythagoras in statistics: distances and errors in quadrature · Euclidean and Manhattan distance, why independent errors add as squares, the pooled SE of a z-test vs the unpooled SE of an interval, and how to read the Wilson and Newcombe interval formulas.