Measurement Uncertainty Formulas
Measurement uncertainty formulas: the standard uncertainty of a rectangular limit, the combined uncertainty of independent sources and the expanded uncertainty.
The formulas
| To find | Formula |
|---|---|
| Standard uncertainty from a limit ±a (rectangular) | u = a / √3 |
| Combined standard uncertainty | uc = √(u1² + u2² + u3² + ...) |
| Expanded uncertainty | U = k × uc |
The sources must be independent and in the same unit. A tolerance on a data sheet, such as ±a, is treated as a rectangular distribution, which is why it is divided by √3. The coverage factor k = 2 gives about 95% coverage for a normal distribution. This is the simple root-sum-square method; a sensitivity coefficient is needed when a source does not enter the result one for one. The values here are examples.
Worked example: three independent sources
Standard uncertainties of 0.5, 0.3 and 0.2 (all in the same unit).
- uc = √(0.5² + 0.3² + 0.2²) = 0.6164
- U = 2 × 0.6164 = 1.233
Worked example: a tolerance of ±0.5
- u = 0.5 / √3 = 0.2887
Expanded uncertainty for the first example
| Coverage factor | Expanded uncertainty | Approximate coverage (normal) |
|---|---|---|
| k = 1 | 0.6164 | 68% |
| k = 2 | 1.233 | 95% |
| k = 3 | 1.849 | 99.7% |
Frequently asked questions
Why is the tolerance divided by √3?
A data sheet limit means the error can be anywhere in the band with equal chance. The standard deviation of that flat distribution is the half-width divided by √3.
Why add the squares and not the values?
Independent errors seldom all hit their maximum together, so they combine as the root of the sum of squares and the total is less than the plain sum.
What is the coverage factor?
A multiplier that turns the standard uncertainty into a range that holds the true value with a stated chance. k = 2 is about 95% for a normal distribution.
Is this enough for a calibration certificate?
It is the method in outline. A formal budget follows a guide such as the GUM and adds the sensitivity coefficients, the type A terms and the degrees of freedom.
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