Measurement 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 uncertaintyThe uncertainty of one input expressed as a standard deviation. (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 uncertaintyThe combined standard uncertainty multiplied by a coverage factor k, usually 2 for about 95 percent confidence. (U): u_c multiplied by a coverage factorThe multiplier k applied to the combined standard uncertainty; it must be stated with the result. 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 BAn uncertainty evaluated from information such as certificates, specifications or tolerances rather than repeated measurements. evaluations use other information: certificates, manufacturer specifications, tolerances. A tolerance written as ±a is converted to a standard uncertainty by assuming a distribution:

Combining uncertainties

For a result calculated by adding or subtracting inputs, combine the standard uncertainties in quadratureCombining uncertainties by squaring them, adding the squares and taking the square root. (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:

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 budgetA table of every input's uncertainty that shows which input contributes most to the result's uncertainty. 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.