A measurement uncertainty calculation answers one question: how far from the truth could this result plausibly be?
The method is settled and international. You write down what you measured and how, list everything that could shift the answer, put a number on each, combine them with one specific piece of arithmetic, and widen the result to a confidence level someone can act on.
This page covers the whole calculation including the two parts most guides skip: what to do when your result comes from a formula rather than a single reading, and what to do when the standard method doesn’t apply.
What the Calculation Involves
Six stages, and every measurement uncertainty calculation follows them in order:
- Write the measurement model — the equation linking what you measured to what you’re reporting.
- List the input quantities and everything that makes each uncertain.
- Calculate a standard uncertainty for each input — statistically (Type A) or from documents (Type B).
- Calculate sensitivity coefficients — how strongly each input affects the result.
- Combine the contributions in quadrature.
- Expand by a coverage factor and report.
The two formulas that carry the whole thing:
u꜀(y) = √[ Σ (cᵢ · u(xᵢ))² ]
U = k × u꜀
Everything else is working out what goes into those.
The Two Calculation Approaches
There are two internationally recognised methods, and most people only know one exists.
The GUM framework — the standard approach. You linearise your measurement model, propagate standard uncertainties through it using sensitivity coefficients, and combine in quadrature. Fast, transparent, and correct for the vast majority of measurements.
Monte Carlo simulation — described in GUM Supplement 1. Instead of propagating uncertainties, you propagate distributions: sample each input at random from its distribution, evaluate the model, and repeat a hundred thousand times. The spread of the results is your answer.
The GUM framework is what you’ll use almost always. Monte Carlo exists for the cases where its assumptions break down, and there’s a section on that below.
Before You Calculate: The Measurement Model
The model is the equation that turns your inputs into your result:
y = f(x₁, x₂, x₃, …)
For a direct measurement — a micrometer on a shaft the model is trivial: the result is the reading, plus corrections.
y = reading + correction from certificate + thermal correction
For anything calculated, the model is your formula:
Density: ρ = m ÷ V Resistance: R = V ÷ I Volume of a cylinder: V = (π/4) d² h
Write it down before you calculate anything. The model determines your input list, and the input list determines your budget. It also gives you the sensitivity coefficients directly, since those are its partial derivatives.
Alongside the model, define the measurand in words: what quantity, under what conditions, measured how. “The diameter of the shaft” is not a measurand. “The diameter at the journal midpoint, at 20 °C, averaged over three angular positions” is.
Calculating Type A Uncertainty
Type A is the part you get from repeated readings.
Mean:
x̄ = Σxᵢ ÷ n
Sample standard deviation:
s = √[ Σ(xᵢ − x̄)² ÷ (n − 1) ]
Standard uncertainty of the mean:
u = s ÷ √n
Two points that decide whether your calculation is right.
Use n − 1, not n. This is the sample standard deviation STDEV.S in Excel, not STDEV.P. With five readings, using the wrong one understates your uncertainty by about 12%.
Don’t forget the ÷√n. You’re reporting the mean of your readings, not a single reading, and the mean is more reliable. Stopping at the standard deviation gives you the spread of individual measurements, which is a different quantity.
The degrees of freedom for a Type A term is ν = n − 1. You’ll need this later if the coverage factor turns out not to be 2.
Calculating Type B Uncertainty
Type B is everything you can’t get by repeating: certificates, specifications, published data, and documented judgement.
Each source arrives in a different format, so each is divided by a factor reflecting its assumed distribution:
| Source | Distribution | Divide by |
|---|---|---|
| Certificate stating k = 2 | Normal | 2 |
| Certificate stating k = 3 | Normal | 3 |
| Instrument resolution | Rectangular | √3 |
| Manufacturer’s ± limits | Rectangular | √3 |
| Temperature between limits | Rectangular | √3 |
| Value most likely near the centre | Triangular | √6 |
| Value most likely near the extremes | U-shaped | √2 |
Resolution needs an extra step. The half-width is half the resolution, then you divide by √3:
u = (resolution ÷ 2) ÷ √3 ≈ resolution × 0.289
A 0.01 mm display gives 0.0029 mm, not 0.0058 mm.
When you have limits and no other information, choose rectangular. It’s the conservative option and what the GUM recommends by default. Type B sources are normally treated as having infinite degrees of freedom.
Calculating Sensitivity Coefficients
A sensitivity coefficient converts an input’s uncertainty into the units of your result and scales it by how strongly that input matters.
cᵢ = ∂f / ∂xᵢ
The partial derivative of your model with respect to each input. Three common cases:
Direct measurement — c = 1. The reading is the result.
Thermal expansion — c = L × α. For a 100 mm steel part, c = 100 × 11.5 × 10⁻⁶ = 0.00115 mm per °C.
A quantity that appears as a power — the coefficient carries that power. If your model contains d², the diameter’s contribution is doubled. If it contains d³, tripled.
That last case matters more than people expect, and the worked example below shows why.
Calculating the Combined Standard Uncertainty
Multiply each input’s standard uncertainty by its sensitivity coefficient, then combine:
u꜀(y) = √[ (c₁u₁)² + (c₂u₂)² + (c₃u₃)² + … ]
Square each contribution, add the squares, take the square root. This is quadrature, and it’s not arbitrary straight addition would assume every source goes wrong in the same direction at the same moment, which independent sources don’t.
The relative shortcut
When your model is a product or quotient of powers which covers most derived quantities you can work entirely in relative terms:
For y = x₁ᵖ · x₂ᑫ ÷ x₃ʳ:
u꜀(y)/y = √[ p²(u₁/x₁)² + q²(u₂/x₂)² + r²(u₃/x₃)² ]
This is often faster, and it avoids differentiating anything. Both routes give the same answer, which makes them a useful check on each other.
Calculating Uncertainty for Indirect Measurements
Most real results are calculated, not read. Density from mass and volume. Resistance from voltage and current. Flow rate from pressure drop.
The procedure is the same, with one addition: each input quantity gets its own mini-budget first.
So for density from mass and dimensions:
- Build a budget for the mass measurement → u(m)
- Build a budget for the diameter → u(d)
- Build a budget for the height → u(h)
- Calculate sensitivity coefficients from the model
- Combine those three contributions into u(ρ)
It’s budgets inside a budget. Each input’s uncertainty comes from its own repeatability, resolution, certificate and environmental terms and only then does it get propagated through the formula.
Calculating With Correlated Inputs
The standard formula assumes your inputs are independent. Sometimes they aren’t.
If two inputs share a common cause two dimensions measured with the same uncalibrated instrument, or two masses weighed on the same drifting balance they’re correlated, and quadrature alone understates the total. The full expression adds a covariance term:
u꜀² = Σ cᵢ²u²(xᵢ) + 2 ΣΣ cᵢcⱼ u(xᵢ, xⱼ)
In routine work this is usually ignored, and usually that’s fine. But be aware of two situations where it isn’t:
When the same instrument measures multiple inputs to one result. Its calibration uncertainty is common to all of them.
When inputs are subtracted. Correlated errors partly cancel under subtraction, so ignoring correlation here makes your uncertainty too large rather than too small — conservative, but wasteful.
If correlation matters and you’d rather not handle covariances, Monte Carlo deals with it naturally.
Calculating the Coverage Factor and Expanded Uncertainty
U = k × u꜀
k = 2 is the default, giving approximately 95.45% confidence. Use it unless something tells you otherwise.
When it isn’t enough: if a Type A term based on few readings dominates your budget, k = 2 doesn’t deliver 95% confidence. Calculate the effective degrees of freedom:
ν_eff = u꜀⁴ ÷ Σ (uᵢ⁴ ÷ νᵢ)
Then take k from the t-distribution:
| ν_eff | 2 | 3 | 4 | 6 | 10 | 20 | 50 | ∞ |
|---|---|---|---|---|---|---|---|---|
| k | 4.53 | 3.31 | 2.87 | 2.52 | 2.28 | 2.13 | 2.05 | 2.00 |
With three readings and repeatability dominating, k can legitimately be 4.53 more than double the default. In most budgets Type B sources dominate, ν_eff comes out large, and k = 2 is correct.
Rounding and Reporting the Calculated Result
Round the uncertainty to one significant figure. 0.003606 becomes 0.004. Keep two figures if the leading digit is 1, since rounding 0.0134 to 0.01 distorts it too much. Many laboratories use two figures throughout either is defensible if applied consistently.
Round the result to match the uncertainty’s decimal place. 7.97702 ± 0.004 becomes 7.977 ± 0.004.
Round the uncertainty up rather than to nearest when in doubt. Overstating doubt is honest; understating it isn’t.
State the coverage factor and confidence level. “±0.004 g/cm³” alone is ambiguous. “±0.004 g/cm³ (k = 2, approximately 95% confidence)” is a complete statement.
Full Worked Calculation
Density of a steel cylinder, calculated from mass and dimensions. This exercises everything above.
The model:
ρ = m ÷ V, where V = (π/4) d² h
so ρ = 4m ÷ (π d² h)
The input quantities, each with its own small budget already completed:
| Input | Value | Standard uncertainty | Main contributors |
|---|---|---|---|
| Mass, m | 125.402 g | 0.0017 g | Repeatability, balance certificate |
| Diameter, d | 20.004 mm | 0.0022 mm | Repeatability, certificate, roundness |
| Height, h | 50.010 mm | 0.0025 mm | Repeatability, certificate, parallelism |
Calculate the result:
V = 0.7854 × 20.004² × 50.010 = 15 720 mm³ = 15.7205 cm³ ρ = 125.402 ÷ 15.7205 = 7.9770 g/cm³
Calculate the sensitivity coefficients:
∂ρ/∂m = 1/V = 1 ÷ 15.7205 = 0.0636 (g/cm³ per g) ∂ρ/∂d = −2ρ/d = −2 × 7.977 ÷ 20.004 = −0.7975 (g/cm³ per mm) ∂ρ/∂h = −ρ/h = −7.977 ÷ 50.010 = −0.1595 (g/cm³ per mm)
Note the −2ρ/d for diameter. That factor of 2 is the d² in the model, and it’s about to matter a great deal.
Calculate each contribution:
| Input | c | u(xᵢ) | Contribution c·u | % of variance |
|---|---|---|---|---|
| Mass | 0.0636 | 0.0017 g | 0.000108 g/cm³ | 0.4% |
| Diameter | 0.7975 | 0.0022 mm | 0.001755 g/cm³ | 94.7% |
| Height | 0.1595 | 0.0025 mm | 0.000399 g/cm³ | 4.9% |
Combine:
u꜀ = √(0.000108² + 0.001755² + 0.000399²) u꜀ = √(3.249 × 10⁻⁶) = 0.0018 g/cm³
Expand:
U = 2 × 0.0018 = 0.0036 ≈ 0.004 g/cm³
Result: ρ = 7.977 ± 0.004 g/cm³ (k = 2)
Check it with the relative method
u(m)/m = 0.0017 ÷ 125.402 = 0.00136% 2 × u(d)/d = 2 × 0.0022 ÷ 20.004 = 0.0220% u(h)/h = 0.0025 ÷ 50.010 = 0.00500%
Combined relative = √(0.00136² + 0.0220² + 0.00500²) = 0.0226% u꜀ = 0.000226 × 7.977 = 0.0018 g/cm³ ✓
Both routes agree, which is a good sign the calculation is sound.
What the calculation reveals
Diameter contributes 94.7% of the variance. Mass measured on an excellent balance to 0.0017 g contributes 0.4%. It is, for practical purposes, irrelevant.
Two things caused that. The diameter’s sensitivity coefficient carries a factor of 2 because d is squared in the model. And its own budget includes a roundness term that mass and height don’t have.
The practical conclusion: improving the mass measurement would achieve nothing at all. Measuring the diameter at more angular positions, or on a rounder specimen, would cut the total uncertainty by close to half.
That conclusion is invisible until you calculate the contributions individually. It’s the reason the percentage column is worth the extra five minutes.
Monte Carlo: When the GUM Formula Doesn’t Work
The GUM framework linearises your model around the operating point. That’s an excellent approximation most of the time and occasionally it isn’t.
Consider Monte Carlo when:
- Your model is strongly non-linear over the range of your uncertainties
- One non-normal contribution dominates, so the output distribution isn’t normal either
- The output is expected to be asymmetric
- Inputs are correlated and you’d rather not compute covariances
- Degrees of freedom are low and Welch–Satterthwaite feels like a stretch
How it works:
- Assign each input a probability distribution rather than a single uncertainty
- Draw a random value from each
- Evaluate the model
- Repeat 100 000 to 1 000 000 times
- Sort the results; the 2.5th and 97.5th percentiles are your 95% interval
It’s feasible in Excel. For a normally distributed input: =NORM.INV(RAND(),mean,sd). For a rectangular one: =mean+(RAND()-0.5)*2*a. Fill 100 000 rows, evaluate your model in each, then use =PERCENTILE.INC(range,0.025) and =PERCENTILE.INC(range,0.975).
What you gain: no linearisation, no sensitivity coefficients, no Welch–Satterthwaite, and asymmetric intervals when the physics is asymmetric.
What you lose: the contribution ranking. Monte Carlo gives you an excellent answer and tells you nothing about which input caused it so in practice, run both. The GUM calculation to find out what to fix, Monte Carlo to validate the number.
Checking Your Calculation
Four tests before you trust a result.
1. Magnitude. Your combined standard uncertainty should be larger than your biggest single contribution and smaller than the plain sum of all of them. Outside that range, something is wrong.
2. Cross-method. Calculate it both ways absolute with sensitivity coefficients, and relative with the power formula. They must agree, as they did above.
3. Contributions sum to 100%. If your percentage column doesn’t, you’ve mixed up variance and standard deviation somewhere.
4. Comparison. Check the result against your laboratory’s CMC, or against a published budget for a similar measurement. An uncertainty that comes out far smaller than anyone else achieves is a warning, not an achievement.
Common Calculation Errors
Using STDEV.P instead of STDEV.S. Understates the Type A term.
Forgetting the ÷√n. Gives the spread of readings rather than the uncertainty of the mean.
Entering a certificate value without dividing by k. Doubles that contribution.
Using full resolution rather than half. Doubles it again.
Adding contributions instead of combining in quadrature.
Omitting sensitivity coefficients in an indirect measurement, or assuming they’re all 1 without checking the model.
Missing the power factor. A squared input carries a coefficient with a 2 in it. As the density example showed, this can be the difference between a source contributing 25% and 95%.
Assuming k = 2 when a Type A term with few readings dominates.
Reporting the calculator’s full display. Round the uncertainty to one significant figure, then match the value.
Treating a known bias as uncertainty. Correct it, then include only the residual doubt about the correction.
Frequently Asked Questions
How do you calculate measurement uncertainty?
Write the measurement model, calculate a standard uncertainty for each input, multiply each by its sensitivity coefficient, combine in quadrature with u꜀ = √[Σ(cᵢuᵢ)²], then multiply by a coverage factor, normally 2.
What is the formula for measurement uncertainty calculation?
u꜀(y) = √[Σ(cᵢ · u(xᵢ))²] for the combined standard uncertainty, and U = k × u꜀ for the expanded uncertainty.
How do you calculate Type A uncertainty?
Take repeated readings, calculate the sample standard deviation using n − 1, then divide by √n.
How do you calculate Type B uncertainty?
Divide the source value by a factor based on its distribution 2 for a k = 2 certificate, √3 for rectangular sources such as resolution and tolerance limits, √6 for triangular.
How do you calculate a sensitivity coefficient?
Take the partial derivative of your measurement model with respect to that input. For direct measurements it’s 1; for thermal effects it’s L × α; for a squared input it carries a factor of 2.
How do you calculate uncertainty for a calculated quantity?
Build a budget for each input first, then propagate those through the model using sensitivity coefficients. For products and quotients you can work in relative terms and multiply each by its power.
Why combine in quadrature rather than adding?
Because independent sources scatter randomly and partly cancel. Simple addition assumes they all go wrong in the same direction simultaneously, which produces an unnecessarily large result.
When should I use Monte Carlo instead?
When your model is strongly non-linear, when a dominant contribution is non-normal, when the output should be asymmetric, or when inputs are correlated and you’d rather not compute covariances.
How do I know my calculation is correct?
The combined value should exceed your largest contribution but fall below their plain sum, the absolute and relative methods should agree, and the contribution percentages should total 100%.
How many significant figures should the answer have?
Round the uncertainty to one significant figure, then round the result to the same decimal place. Two figures throughout is also acceptable if applied consistently.
Conclusion
A measurement uncertainty calculation is a sequence, and once you’ve done it twice it stops feeling like statistics.
The method, condensed:
- Write the model first. It gives you the input list and the sensitivity coefficients.
- Type A is s ÷ √n with n − 1 in the standard deviation. Type B is the source value divided by 2, √3 or √6 depending on its distribution.
- Sensitivity coefficients are partial derivatives 1 for direct measurements, L × α for thermal effects, and carrying the power for squared or cubed inputs.
- Combine in quadrature, then expand with k = 2 unless few readings and a dominant Type A term push it higher.
- Round the uncertainty to one significant figure, match the result to it, and state k.
- Check both ways. Absolute and relative methods must agree.
