Root-sum-square (RSS) combines several contributions by squaring each one, adding the squares, and taking the square root. The result grows more slowly than the arithmetic sum when independent contributions are combined. That difference can be useful in tolerance analysis, but it does not mean the smaller result is automatically safe. The method answers a statistical question only when the inputs represent compatible quantities and the assumptions behind their combination are reasonable. The examples in this table are original calculations intended to make that arithmetic visible.

RSS Tolerance Stackup Worked Data

Read the values correctly

Each row lists the input contributions, the calculated RSS result, and the arithmetic worst-case sum. For ±0.10 mm and ±0.20 mm, RSS is approximately 0.2236 mm while the worst-case sum is 0.30 mm. These figures describe two different ways of combining limits; neither row claims that a real manufactured assembly will follow a particular distribution. A nominal dimension, a bilateral drawing tolerance, a measurement uncertainty, and a process standard deviation are not interchangeable inputs. Identify what each value means before calculation.

The statistical assumptions matter

NIST Technical Note 1297 explains root-sum-square combination for standard uncertainties when the inputs are uncorrelated, and its propagation law includes covariance terms for correlated quantities. This is a valuable boundary for engineering use: components measured by the same biased instrument, dimensions controlled by a shared process shift, or features derived from a common datum may move together. Treating such contributions as independent can understate the combined effect. RSS also does not establish normality, process capability, a yield percentage, or a confidence interval unless the statistical model and input distributions support those conclusions.

When worst case is the right check

Use arithmetic worst-case stackup when the design must guarantee interchangeability at the specified limits, when a functional boundary cannot be exceeded, or when the production distributions are unknown. Add the absolute limit contributions in the direction that closes the clearance or increases the interference. If the resulting worst-case condition fails, an RSS estimate by itself does not remove the failure. A team may separately use statistical tolerance allocation to estimate expected performance, but the report should state that it is probabilistic and should name the evidence used to justify the model.

Build a defensible stack

Start with a functional dimension chain and draw the direction of each contribution. Keep units consistent and distinguish dimensions that add from those that subtract. For independent standard uncertainties, the covariance terms are zero and the familiar square-root-of-sum-of-squares expression applies. If dependence is present, include covariance or use an appropriate model instead of silently discarding it. For drawing tolerances used statistically, select and document the distribution and centering assumptions; a symmetric tolerance band alone does not provide these. Confirm the model with process data when the decision has cost or safety consequences.

Use the result within its limits

The values here are examples, not recommended tolerances for a product. They can help check a spreadsheet formula or explain why RSS and worst-case results differ. They cannot decide whether a joint has enough clearance, whether an assembly will pass inspection, or whether a process is capable. Preserve traceability to the source measurement, specification, and datum scheme for every contribution. If the design requires a guaranteed fit, report the worst-case boundary alongside any statistical estimate. Have a qualified engineer review the stack, the inputs, and the consequences of failure before releasing a drawing or changing inspection limits.

Sources and verification

The combination rule and covariance boundary follow NIST Technical Note 1297, combined standard uncertainty and its law of propagation of uncertainty. The numeric rows are independently calculated illustrations.

This table is a reference aid, not a substitute for the governing design code, product specification, qualified engineering review, or measurements from the actual component and service conditions.