SC QUALITY GUILDMeasurement uncertainty budget
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List the sources of measurement uncertainty, convert each to a standard uncertainty, combine them by root-sum-square, and compare the expanded uncertainty with the tolerance.
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Uncertainty in one paragraph
Every measurement result is an estimate. Its uncertainty is a parameter that describes the spread of values that could reasonably be attributed to the measurand. The international method is the Guide to the Expression of Uncertainty in Measurement (GUM, JCGM 100:2008): express each source as a standard uncertainty (one standard deviation), combine them, then multiply by a coverage factor.
Type A and Type B
- Type A is evaluated statistically from repeated readings. The standard uncertainty is the standard deviation s, or s/√n if the reported result is the average of n readings.
- Type B comes from any other information: a calibration certificate, the instrument's resolution, a manufacturer's specification, temperature limits. You choose a distribution and divide by its divisor.
- Divisors: certificate stated at k = 2, divide by 2; rectangular (equally likely anywhere within ±a), divide a by √3; triangular, by √6; U-shaped (for example a cycling temperature), by √2. Resolution is usually taken as rectangular with a half-width of half the last digit.
Combining and expanding
A sensitivity coefficient c converts a source into the units of the result (for thermal expansion, c = coefficient of expansion × length). The combined standard uncertainty is uc = √(Σ (c·u)²), valid when the sources are independent. The expanded uncertainty is U = k × uc; k = 2 gives roughly 95 percent coverage when the result is close to normal.
Is the measurement good enough?
The test uncertainty ratio compares the tolerance with the uncertainty: TUR = tolerance half-width divided by U at about 95 percent coverage (k = 2). The common target is 4:1, which ANSI/NCSL Z540.3 accepts when false-accept risk is not calculated. Below that, readings near a limit cannot be trusted to decide conformance. One remedy is a guard band: accept only inside the limits pulled in by U. That lowers the chance of accepting bad parts and raises the chance of rejecting good ones.
Common traps
- Adding uncertainties directly instead of as squares, which overstates the total.
- Using a certificate's expanded U as if it were a standard uncertainty.
- Counting the same effect twice, for example resolution inside repeatability data and again as its own row. Some labs keep both rows on purpose as a conservative choice, as the worked example does; if you do, say so.
- Working hard on small sources. Because they add as squares, only the largest few matter.