Measurement Uncertainty Calculator
Measurement Uncertainty Calculator
Combine within-laboratory imprecision and the uncertainty of the bias estimate into a combined standard uncertainty, then expand it to the coverage interval ISO 15189 expects you to be able to quote.
Measurement Uncertainty
u(Rw), u(bias), k → UA method with 2.5% long-term within-laboratory precision and 1.5% uncertainty on the bias, expanded at k = 2
Formula
U = k × uc
k = 2 for approximately 95% coverage
- u(Rw)
- within-laboratory reproducibility, as a relative standard deviation. It must be the long-term figure — months of internal quality control spanning reagent lots, calibrations, operators and maintenance — because repeatability measured within a single run can understate it by a factor of two
- u(bias)
- the uncertainty attached to the trueness of the method: the stated uncertainty of the certified reference material or EQA target value, combined with the scatter of your own recovery or comparison experiments. If a bias has been identified and not corrected, its magnitude is included here as a component
- √( … )
- the two components are independent, so they combine in quadrature rather than by addition. This is the mathematical heart of the difference from total analytical error, and it is why the smaller of the two components often barely matters: 2.5 and 1.5 combine to 2.9, not 4.0
- uc
- the combined standard uncertainty — one standard deviation of the distribution of values that could reasonably be attributed to the measurement
- k
- the coverage factor. k = 2 gives approximately 95% coverage and is the convention; k = 1 gives about 68% and k = 3 about 99.7%. U without a stated k cannot be interpreted
- why it is required
- ISO 15189 requires a medical laboratory to determine the measurement uncertainty of each quantitative measurement procedure, to have performance requirements for it, and to make it available to users on request. The figure is also what allows a result to be compared honestly with a decision limit
Worked example
A method with 2.5% long-term within-laboratory precision and 1.5% uncertainty on the bias, expanded at k = 2
u(Rw)² = 2.5² = 6.25 and u(bias)² = 1.5² = 2.25
6.25 + 2.25 = 8.5, and √8.5 = 2.92% — the combined standard uncertainty
U = 2 × 2.92 = 5.8% at approximately 95% coverage
Applied to a result of 5.00 mmol/L, that is 5.00 ± 0.29 mmol/L (k = 2), so values from about 4.71 to 5.29 are all plausible for this sample
Notice how little the smaller component contributed: dropping u(bias) to zero altogether would only take U from 5.8% to 5.0%, because 1.5 combined in quadrature with 2.5 adds 0.4 rather than 1.5
The same two figures as a total analytical error, with the 1.65 multiplier, give 1.5 + 4.125 = 5.6%. Very nearly the same size, arrived at by a different model, and answering a different question
How the two components combine
| u(Rw) | u(bias) | uc | U at k = 2 |
|---|---|---|---|
| 2.5% | 0% | 2.50% | 5.0% |
| 2.5% | 1.0% | 2.69% | 5.4% |
| 2.5% | 1.5% | 2.92% | 5.8% |
| 2.5% | 2.5% | 3.54% | 7.1% |
| 2.5% | 5.0% | 5.59% | 11.2% |
| 1.0% | 1.5% | 1.80% | 3.6% |
Measurement uncertainty against total analytical error
| Measurement uncertainty | Total analytical error | |
|---|---|---|
| Combines bias and imprecision | In quadrature: √(u(Rw)² + u(bias)²) | By addition: |bias| + k × CV |
| Treats bias as | An uncertainty component, on the assumption that known bias is corrected | A fixed offset that is present in every result |
| Answers | What interval around this result plausibly contains the true value? | How wrong is a single result likely to be, at worst? |
| Required by | ISO 15189, which asks for it to be estimated and available on request | No standard requires it; it underpins sigma metrics and EQA criteria |
| Typical relative size | Smaller, because independent errors partly cancel | Larger, because it assumes they line up |
What ISO 15189 asks for, and why it is not total error
Every measurement is an estimate, and measurement uncertainty is the formal statement of how good an estimate it is: the interval around a reported result that plausibly contains the true value. ISO 15189 requires a medical laboratory to determine it for each quantitative measurement procedure, to set performance requirements for it, and to be able to give it to a user who asks. That last clause is the one that turns it from a metrological exercise into a practical obligation, because it means somebody will eventually ask.
The top-down approach most clinical laboratories use needs two components. The first is within-laboratory reproducibility, u(Rw), which is the long-term relative standard deviation of internal quality control — months of data spanning reagent lots, recalibrations, different operators and maintenance events. Repeatability measured inside a single run is not a substitute and can be half the true figure. The second is the uncertainty of the bias, u(bias): the uncertainty of the certified reference material or EQA target value that trueness was assessed against, together with the scatter of the comparison itself. Where a bias has been demonstrated and not corrected, its magnitude is carried here as a component too.
The two are independent, so they combine in quadrature: the square root of the sum of their squares. With 2.5% precision and 1.5% bias uncertainty the combined standard uncertainty is 2.92%, and multiplying by a coverage factor of 2 gives an expanded uncertainty of 5.8% for approximately 95% coverage. On a result of 5.00 mmol/L that is ±0.29, and a report can honestly say 5.00 ± 0.29 mmol/L (k = 2). Always state k, because an expanded uncertainty without one cannot be read.
The point that deserves care is the relationship with total analytical error. The same two numbers give a total error of 1.5 + 1.65 × 2.5 = 5.6%, which is so close to 5.8% that the two models look interchangeable. They are not. Total error adds a bias that it assumes is present and uncorrected to a large random excursion in the same direction, and answers the worst-case question about a single result. Uncertainty combines independent components in quadrature on the assumption that known bias has been corrected, and answers a question about an interval. The numerical resemblance is a coincidence of ordinary laboratory magnitudes, and it disappears as soon as the bias term grows. Calculate whichever the question calls for, quote the concentration, and say which model produced the figure.
Frequently asked questions
How do I calculate measurement uncertainty for a laboratory method?
The usual top-down approach combines two components in quadrature: long-term within-laboratory precision, u(Rw), from months of internal quality control, and the uncertainty of the bias, u(bias), from the reference material or EQA target used to assess trueness. The combined standard uncertainty is the square root of the sum of their squares, and multiplying by a coverage factor of 2 gives the expanded uncertainty for approximately 95% coverage.
Does ISO 15189 require measurement uncertainty?
Yes. It requires the laboratory to determine measurement uncertainty for each quantitative measurement procedure, to define performance requirements for it, to review it periodically, and to make it available to users on request. It does not mandate any single method of estimating it, which is why the top-down approach from quality control data is so widely used.
What is the difference between measurement uncertainty and total analytical error?
They combine bias and imprecision differently and answer different questions. Uncertainty adds them in quadrature and describes an interval around the reported value, assuming known bias has been corrected. Total error adds bias linearly to a multiple of the CV and describes how wrong one result could be at worst. The figures are often similar in size, which makes treating them as the same thing an easy and common error.
Why is the coverage factor 2?
Because for a roughly normal distribution about 95% of values lie within two standard deviations, and 95% coverage is the reporting convention. k = 1 gives about 68% and k = 3 about 99.7%. An expanded uncertainty is uninterpretable without the k that produced it, so the two must always be quoted together.
Should uncorrected bias go into the uncertainty?
The correct approach is to correct a bias you have demonstrated. Where that is not possible, its magnitude is included as a component of u(bias) so that the interval quoted is honest about it. What you must not do is ignore a known bias and report a narrow uncertainty derived from precision alone.
Related calculators
References
- ISO 15189:2022. Medical laboratories — Requirements for quality and competence. International Organization for Standardization; 2022.
- JCGM 100:2008. Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM). Joint Committee for Guides in Metrology; 2008.
- White GH. Basics of estimating measurement uncertainty. Clin Biochem Rev. 2008;29(Suppl 1):S53–S60.
- Oosterhuis WP, Theodorsson E. Total error vs. measurement uncertainty: revolution or evolution? Clin Chem Lab Med. 2016;54(2):235–239.
Medical Disclaimer: The tools and content provided here are for educational and reference purposes only. They are not intended to substitute for professional medical advice, diagnosis, or treatment. Clinical decisions should always be based on the comprehensive assessment of a qualified healthcare professional.