Estimate standard uncertainties and propagate them through a calculation to an expanded uncertainty.
Why it matters
No measurement is exact. A result of "0.995 mg/mL" really means "somewhere close to 0.995, within some range". Measurement uncertainty quantifies that range. It tells a client whether a result is clearly inside a specification, or close enough to the limit that the difference may not be real. Accredited testing and calibration labs (ISO/IEC 17025) must estimate it for their methods, and medical labs accredited to ISO 15189 must too.
Key terms
- Standard uncertainty (u): the uncertainty of one input expressed as a standard deviation.
- Relative standard uncertainty: u divided by the value, for example u(m) / m.
- Combined standard uncertainty (u_c): the uncertainty of the final result, combining all inputs.
- Expanded uncertainty (U): u_c multiplied by a coverage factor k, usually k = 2, which gives a range expected to contain the true value with about 95 percent confidence.
Results are reported as value ± U, with k stated: "0.9950 ± 0.0024 mg/mL (k = 2)".
Where the input uncertainties come from
Type A evaluations use statistics on repeated measurements: the standard deviation of replicates (or of the mean, s / sqrt(n)).
Type B evaluations use other information: certificates, manufacturer specifications, tolerances. A tolerance written as ±a is converted to a standard uncertainty by assuming a distribution:
| Information | Assumed distribution | Standard uncertainty |
|---|---|---|
| Certificate gives U with k = 2 | Normal | U / 2 |
| Tolerance ±a, any value equally likely | Rectangular | a / sqrt(3) |
| Tolerance ±a, values near the centre more likely (often glassware) | Triangular | a / sqrt(6) |
Combining uncertainties
For a result calculated by adding or subtracting inputs, combine the standard uncertainties in quadrature (square, add, square root):
y = a + b - c u(y) = sqrt( u(a)^2 + u(b)^2 + u(c)^2 )
For a result calculated by multiplying or dividing, combine the relative uncertainties in quadrature:
y = a x b / c u(y) / y = sqrt( (u(a)/a)^2 + (u(b)/b)^2 + (u(c)/c)^2 )
Quadrature works because independent errors partly cancel: it is unlikely that every input is at its worst in the same direction at once. This assumes the inputs are independent.
Worked example: a standard solution
You dissolve 100.0 mg of a reference material with purity 0.995 and make it up to 100.00 mL in a Class A volumetric flask.
c = m x P / V = 100.0 x 0.995 / 100.00 = 0.9950 mg/mL
Input uncertainties:
| Input | Value | Standard uncertainty | Relative |
|---|---|---|---|
| Mass m | 100.0 mg | 0.05 mg (from the balance calibration) | 0.00050 |
| Purity P | 0.995 | 0.001 (certificate) | 0.00101 |
| Volume V | 100.00 mL | 0.08 / sqrt(3) = 0.046 mL (rectangular) | 0.00046 |
relative u_c = sqrt(0.00050^2 + 0.00101^2 + 0.00046^2) = sqrt(1.47e-6) = 0.00121
u_c = 0.00121 x 0.9950 = 0.00121 mg/mL
U = 2 x 0.00121 = 0.0024 mg/mL (k = 2)
Result: 0.9950 ± 0.0024 mg/mL (k = 2). The purity contributes most, so a higher-grade reference material would help more than a better balance. An uncertainty budget table like the one above shows exactly where to improve.
Using uncertainty in decisions
A specification is 0.990 to 1.010 mg/mL. A result of 0.9950 ± 0.0024 is clearly inside. A result of 0.9915 ± 0.0024 straddles the lower limit: the true value might be below it. Labs agree a decision rule in advance (often with the client) for these cases, as ISO/IEC 17025 requires.
Common mistakes
- Adding uncertainties directly instead of in quadrature (overestimates).
- Mixing absolute and relative uncertainties in one formula.
- Using a tolerance as if it were a standard uncertainty (forgetting / sqrt(3) or / sqrt(6)).
- Reporting U without saying k.
- Quoting far more digits than the uncertainty justifies. Round U to one or two significant figures and the result to match.
Quick recap
- Standard uncertainty is a standard deviation; expanded U = k x u_c, usually k = 2.
- Type A from repeats; Type B from certificates and tolerances (a / sqrt(3) rectangular).
- Sums: combine absolute uncertainties in quadrature. Products and quotients: combine relative uncertainties in quadrature.
- An uncertainty budget shows the biggest contributor.
NIST's guide to expressing uncertainty: https://www.nist.gov/pml/nist-technical-note-1297 (Checked on 2026-10-23.)
Try it
Open the Do tab and finish a Python uncertainty calculator: you write the quadrature sum of relative uncertainties.