Combine independent errors to reveal true process spread. See contributions, limits, and uncertainty in seconds. Export clean reports and act on data today confidently.
Root sum square combines independent contributors by squaring each value, summing the squares, then taking the square root: RSS = √(x₁² + x₂² + … + xₙ²). This is widely used to combine measurement uncertainty or tolerance stack contributors when correlations are negligible.
| Component | Value (mm) | Squared | Contribution |
|---|---|---|---|
| Gage repeatability | 0.12 | 0.0144 | 45.43% |
| Fixturing | 0.08 | 0.0064 | 20.19% |
| Alignment | 0.10 | 0.0100 | 31.55% |
| Rounding | 0.03 | 0.0009 | 2.84% |
| Example RSS | 0.1780 mm | ||
Use this example to validate your understanding before applying production data.
In quality control, multiple small sources of variation can combine into a meaningful overall uncertainty. Root sum square (RSS) is a practical way to summarize independent contributors such as gage repeatability, fixturing effects, operator technique, rounding, and thermal drift. Instead of adding errors directly, RSS reflects how independent noise accumulates and gives a defensible combined estimate for risk-based decisions.
Each contributor is squared before summing, so larger contributors dominate the result. For example, doubling one contributor increases its squared term by four. This “energy” view is useful when you want to identify what actually drives spread in an inspection process. The table shows each squared term and the percentage contribution to the total.
Contribution percentage ranks where improvement effort will pay back. If a single source contributes 55% of the sum of squares, reducing it by 20% can materially lower the RSS. By contrast, chasing a 3% contributor may not move the needle. Use the chart to visually confirm the “few vital” drivers and focus corrective actions, training, or tooling upgrades.
Many teams use an internal limit for combined uncertainty (for example, a fraction of tolerance or a maximum uncertainty budget). When you enter a limit, the calculator flags whether RSS is within target. This supports consistent release decisions, method validation checks, and auditing. Keep units consistent and document the basis for your limit in your control plan.
Practical inputs come from gage studies, historical inspection results, calibration certificates, or designed experiments. Use standard deviations when working with repeatability and reproducibility, and use half-widths or bounds for certain systematic terms only if your procedure defines them consistently. If contributors are not independent, RSS may understate combined risk.
Exporting CSV supports quick review and integration with corrective-action logs. The PDF report helps with traceability when attaching evidence to deviation records or capability reviews. For best practice, record the measurement method, date, sample size, and any assumptions (like independence) alongside the exported report so reviewers can reproduce the calculation and verify decisions.
Use RSS when contributors are reasonably independent and represent random variation. Simple addition is conservative and can overstate combined uncertainty for independent noise sources.
Enter the relevant standard deviation for the gage component you want to combine (repeatability, reproducibility, or total). Keep the same unit as other contributors.
Not directly. Correlation can increase or decrease the combined result. If strong correlation exists, use a covariance-based method or redesign the model to avoid double-counting.
Because squaring magnifies large contributors. A contributor twice as large has four times the squared effect, so it drives the sum of squares and the final RSS.
Common practice is budgeting uncertainty as a fraction of tolerance or performance requirement. Align the limit with your control plan, customer requirements, and risk tolerance.
It shows which components consume most of the uncertainty budget. Target the highest bars first to reduce RSS efficiently, then reassess after process or measurement changes.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.