Measurement Uncertainty Calculator
Introduction to the uncertainty budget behind a reported measurement
A measurement result is incomplete until it is accompanied by a quantitative statement of how far the reported value could plausibly sit from the quantity actually being measured. The international framework for making that statement is JCGM 100:2008, the Guide to the expression of uncertainty in measurement, universally shortened to the GUM. Its logic is a chain of four steps: write down a model of the measurement, express every recognised effect as a standard uncertainty with the character of a standard deviation, combine those standard uncertainties into one combined standard uncertainty, and finally scale that figure by a coverage factor so the reported interval carries a stated level of confidence.
This calculator implements that chain end to end for the common case of a directly measured quantity. It accepts one Type A component evaluated statistically from repeated readings and up to three Type B components evaluated from certificates, specifications, resolution limits or physical reasoning. It converts each entry to a standard uncertainty using the divisor appropriate to its assumed distribution, combines them by root sum of squares, computes the effective degrees of freedom from the Welch-Satterthwaite formula, looks up the matching Student factor, and reports the expanded uncertainty together with a full budget breakdown showing which effect dominates.
The measurement model that underpins all of it is the functional relationship between the measurand and the input quantities:
Formula: y = f(x_1, x_2, …, x_N)
When you are simply reading a length from a micrometer or a mass from a balance, that function is close to the identity and every sensitivity coefficient is one, which is why a direct root sum of squares of the component uncertainties is legitimate. When the measurand is derived from several quantities of different units, the sensitivity coefficients matter, and this page expects you to fold them in before entering a value.
Type A and Type B evaluations, and the divisors that standardise them
The GUM classifies evaluations, not the components themselves, into two kinds. A Type A evaluation uses the statistical analysis of a series of observations. A Type B evaluation uses any other information: a calibration certificate, a manufacturer specification, handbook data, previous measurements or the experimenter's judgement. Both kinds yield a standard uncertainty and both are treated identically once they have been standardised, so the distinction is bookkeeping rather than hierarchy.
For a Type A evaluation from independent repeated observations, the best estimate is the arithmetic mean
Formula: q ¯ = 1 / n ∑ k = 1 n q_k
and the experimental variance of those observations is
Formula: s^2(q_k) = 1 / (n − 1) ∑ j = 1 n (q_j−q¯)^2
The uncertainty of the mean is smaller than the scatter of the individual readings by the square root of the number of readings, which is why the calculator asks for both the standard deviation and the count:
Formula: u(x_i) = s(q ¯) = s(q_k) / sqrt(n), ν_i = n − 1
Type B components arrive in units you have to normalise. The most common case is a bound rather than a spread. When you know only that the value lies somewhere in an interval of half-width and have no reason to prefer any point inside it, the GUM assigns a rectangular distribution with variance
Formula: u^2(x_i) = a^2 / 3
and if values near the centre of the interval are more credible than values near its edges, the triangular case applies instead:
Formula: u^2(x_i) = a^2 / 6
Divisor comparison table for standardising a Type B entry
The dropdown beside each Type B field applies one of the divisors below. Getting this choice right matters more than any other decision in the budget, because the divisors differ by up to a factor of three.
| What you actually know | Assumed distribution | Divisor | Standard uncertainty |
|---|---|---|---|
| A standard deviation, or a supplier figure already labelled a standard uncertainty | Not applicable | 1 | the value as entered |
| A calibration certificate quoting an expanded uncertainty at k = 2 | Normal | 2 | U / 2 |
| An expanded uncertainty at k = 3, or bounds described as 99.73 percent limits | Normal | 3 | U / 3 |
| A hard bound of half-width a with no information inside it | Rectangular (uniform) | 1.7321 | a / sqrt(3) |
| A bound of half-width a where central values are more likely | Triangular | 2.4495 | a / sqrt(6) |
| A cyclic influence of amplitude a, such as a sinusoidal temperature swing | U-shaped (arcsine) | 1.4142 | a / sqrt(2) |
The first five rows come straight from the GUM. The U-shaped row is the standard arcsine result used across calibration practice for influences that spend most of their time near their extremes; it is not one of the cases tabulated in JCGM 100:2008, so record your reasoning when you use it.
The GUM formula chain from component to expanded uncertainty
The law of propagation of uncertainty is the engine of the whole method. For uncorrelated inputs it reads
Formula: u_c^2(y) = ∑ i = 1 N ((∂f)/(∂x_i))^2 u^2(x_i)
Writing the partial derivative as a sensitivity coefficient turns each term into a contribution already expressed in the units of the measurand:
Formula: u_i(y) = | c_i | u(x_i), c_i ≡ (∂ f) / (∂ x_i)
which collapses the combination into the familiar root sum of squares that this calculator performs:
Formula: u_c(y) = sqrt(∑ i = 1 N u_i^2(y))
Because the CIPM recommendation behind the GUM asks for an interval in commercial, industrial and regulatory work, the combined standard uncertainty is then multiplied by a coverage factor to give the expanded uncertainty:
Formula: U = k u_c(y), Y = y ± U
Whenever the inputs are not independent the simple sum is wrong and the covariance form must be used, with cross terms that can either inflate or shrink the result:
Formula: u_c^2(y) = ∑ i = 1 N ∑ j = 1 N c_i c_j u(x_i, x_j)
Effective degrees of freedom and the Welch-Satterthwaite equation
Choosing out of habit is the second most common defect in an uncertainty statement. The factor 2 is only justified when the combined standard uncertainty is itself reasonably well known. If a budget leans on a repeatability term from five readings, is a noisy estimate and the interval it defines is optimistic. The GUM handles this by approximating the distribution of the standardised deviation with a Student distribution whose effective degrees of freedom come from the Welch-Satterthwaite formula, equation G.2b:
Formula: ν_eff = u_c^4(y) / (∑ i = 1 N u_i^4(y) / ν_i)
The coverage factor is then the quantile of that distribution at the level of confidence you want:
Formula: k_p = t_p(ν_eff), U_p = k_p u_c(y)
A Type A component contributes . A Type B component whose value is treated as exactly known contributes infinite degrees of freedom, which the formula absorbs harmlessly because its term in the denominator is zero; leave the degrees-of-freedom box blank to get that behaviour. If you genuinely doubt a Type B figure, GUM equation G.3 turns that doubt into a number:
Formula: ν_i ≈ 1 / 2 [(Δu(x_i))/u(x_i)]^−2
Believing a component to about 25 percent gives 8 degrees of freedom; believing it to about 50 percent gives 2. The calculator reads its values from Table G.2 of JCGM 100:2008 and interpolates linearly in between tabulated rows, which is the natural variable because the GUM itself notes the large-sample behaviour
Formula: t_p(ν) ≈ (1+2/ν)^1/2 k_p
At infinite degrees of freedom the table gives 1.960 for a 95 percent interval and exactly 2.000 for 95.45 percent, which is the real meaning of the conventional .
How to use the uncertainty budget worksheet on this page
Work down the form in the order the GUM works through a measurement.
- Measured value and unit. Enter the mean of your readings, not a single reading, and the unit symbol you want carried through the report. Negative values and zero are accepted, because a deviation, an offset or a temperature difference is a perfectly ordinary measurand.
- Type A repeatability. Enter the experimental standard deviation of the individual readings and how many readings you took. The calculator divides by the square root of the count to get the standard uncertainty of the mean and assigns degrees of freedom. Leave the standard deviation at zero if repeatability was not evaluated.
- Type B components. Three slots are provided, labelled instrument, calibration and environment, but the labels are only a prompt: use them for whatever three effects dominate your measurement. For each one, enter the magnitude, pick the divisor that matches the kind of information you have, and leave the degrees of freedom blank unless you have a specific reason not to.
- Coverage. Leave the mode on the Welch-Satterthwaite setting to have the coverage factor derived from your own degrees of freedom at the level of confidence you select. Switch to the fixed setting when a client, a standard or an accreditation scope obliges you to report at a nominated .
- Read the budget. Below the headline result, the breakdown panel lists every component with its standard uncertainty, its degrees of freedom and its share of the combined variance, and draws the same shares as a bar chart. The dominant bar tells you where to spend effort if the uncertainty needs to come down.
The copy and download buttons produce a plain-text report and a CSV file respectively, both formatted so they can be pasted straight into a calibration record or an uncertainty budget spreadsheet.
Worked example: a 50 mm gauge block on a comparator
A workshop checks a nominally 50 mm gauge block against a calibrated reference. Ten independent readings of the difference give a mean corrected length of 50.00042 mm with an experimental standard deviation of 0.00042 mm. Three Type B effects are recognised.
| Component | Entered value (mm) | Divisor | u (mm) | Degrees of freedom | Share of variance |
|---|---|---|---|---|---|
| Repeatability, 10 readings | 0.00042 | sqrt(10) | 0.0001328 | 9 | 32.5 percent |
| Reference standard certificate, U at k = 2 | 0.00030 | 2 | 0.0001500 | infinite | 41.4 percent |
| Comparator resolution, half-width | 0.00005 | sqrt(3) | 0.00002887 | infinite | 1.5 percent |
| Thermal effects, half-width | 0.00020 | sqrt(3) | 0.0001155 | infinite | 24.6 percent |
The combined standard uncertainty follows from the root sum of squares:
Formula: u_c = sqrt(0.0001328^2 + 0.00015^2 + 0.00002887^2 + 0.0001155^2) = 0.0002330 mm
Only the repeatability term has finite degrees of freedom, so the Welch-Satterthwaite denominator has a single surviving entry:
Formula: ν_eff = 0.0002330^4 / (0.0001328^4 /9) = 85.3
Interpolating Table G.2 between 50 and 100 degrees of freedom at the 95 percent level gives , so the expanded uncertainty is
Formula: U = 1.988 × 0.0002330 = 0.00046 mm
and the result is reported as (50.00042 ± 0.00046) mm, where the expanded uncertainty is obtained from a combined standard uncertainty of 0.000233 mm multiplied by a coverage factor of 1.99 based on 85 effective degrees of freedom, defining an interval with a level of confidence of approximately 95 percent. Note how close the derived factor is to 2 here: with 85 effective degrees of freedom the habitual shortcut would have been almost harmless. Repeat the example with three readings instead of ten and the effective degrees of freedom collapse, the factor rises above 2.5, and the shortcut becomes a real understatement.
Reading the result and turning it into a decision
The interval means that values from 50.00042 minus 0.00046 to 50.00042 plus 0.00046 millimetres can reasonably be attributed to the block, with roughly 95 percent of the plausible distribution inside those limits. It does not mean the block is faulty, and it does not mean any single reading was wrong. The GUM is careful with vocabulary here: it uses level of confidence rather than confidence level, because a strict statistical confidence interval requires every component to come from a Type A evaluation, which almost never happens in practice.
Two comparisons follow immediately from a reported interval. First, conformity: if a drawing tolerance is plus or minus 0.001 mm, a measured deviation of 0.00042 mm with an expanded uncertainty of 0.00046 mm sits inside the tolerance but its interval does not, so a decision rule such as a guard band is needed before you can declare a pass. Second, agreement between laboratories: two results are compatible when the difference between them is smaller than the expanded uncertainty of that difference, which is itself a root sum of squares of the two individual values when the measurements are independent.
The share-of-variance column is where the budget earns its keep. Because contributions combine as squares, a component carrying 5 percent of the variance is carrying only about 22 percent of the uncertainty in linear terms, and halving it would change the total by roughly one percent. Effort belongs on the top one or two rows. In the worked example, buying a better comparator would be pointless while the reference standard certificate and the thermal environment together carry two thirds of the variance.
Limitations and assumptions you are accepting when you use this tool
The root sum of squares assumes the components are uncorrelated. Shared influences such as a common ambient temperature, a common reference standard or a common operator create covariance, and the GUM handles those with the double-sum form given above. Where correlation cannot be avoided, the usual practical fix is to reintroduce the shared influence as its own independent input quantity rather than to patch the arithmetic.
The method also assumes the model is sufficiently linear over the range of the input uncertainties, because the law of propagation is a first-order Taylor expansion. Strongly nonlinear models need higher-order terms or a Monte Carlo treatment of the kind described in JCGM 101:2008, the supplement to the GUM.
Two further limitations apply to the coverage factor. The Student approximation assumes the output distribution is approximately normal, which the central limit theorem supports only when several comparable components combine. And the GUM warns that when a single rectangular component with exactly known bounds dominates, an interval built with a factor greater than 1.732 can extend beyond the physical limits of that distribution. The breakdown panel flags that situation for you.
Finally, this page assumes you have already applied every known correction. Uncertainty is not a substitute for a correction: a recognised systematic offset should be subtracted from the reading, and only the uncertainty of that correction belongs in the budget. Accreditation to ISO/IEC 17025:2017 requires laboratories to identify all contributions to measurement uncertainty, which means the completeness of your component list is judged separately from the arithmetic this calculator performs.
Questions metrologists ask about uncertainty budgets
Should I divide a manufacturer tolerance by the square root of three?
Only if the tolerance is a hard limit with no information about where inside it the true value sits. That is exactly the situation the GUM describes in 4.3.7: bounds of half-width a, no specific knowledge inside them, so the standard uncertainty is a divided by the square root of 3. If instead the figure is an expanded uncertainty from a calibration certificate, divide by the stated coverage factor, normally 2. If it is already a standard deviation, use a divisor of 1. Choosing the wrong divisor is the single most common error in an uncertainty budget, and it changes the answer by up to a factor of two.
Why does the calculator sometimes return a coverage factor of 2.3 instead of 2?
Because the effective degrees of freedom are small. The habit of using k equal to 2 is only justified when the combined standard uncertainty is itself well known, which the GUM describes as an effective degrees of freedom of significant magnitude, say greater than 10. When a budget is dominated by a repeatability term from a handful of readings, the Welch-Satterthwaite value drops into single figures and the Student t factor rises. NIST Technical Note 1297 gives the same warning with a worked figure: at 8 effective degrees of freedom the 95 percent factor is 2.3, not 2.0.
What degrees of freedom should I give a Type B component?
Leave the field blank whenever the component can be treated as exactly known, which the GUM says is often the case in practice. A rectangular half-width chosen so that the quantity is essentially certain to lie inside it implies infinite degrees of freedom, and that causes no difficulty in the Welch-Satterthwaite calculation. If you genuinely doubt the size of a Type B component, GUM equation G.3 converts your judgement into a number: degrees of freedom are approximately one half of the inverse square of the relative uncertainty of the component. Judging a value reliable to 25 percent gives 8 degrees of freedom, and to 50 percent gives 2.
How many digits should I quote in the final uncertainty?
Two significant digits in the uncertainty is the normal convention, and the measured value is then rounded to the same decimal place. This calculator does that automatically for the headline result while keeping four significant digits in the budget table so you can audit the arithmetic. Quoting more digits implies a precision that the uncertainty estimate does not have, and quoting fewer throws away information that matters when results are compared or combined.
Does this calculator handle correlated inputs or sensitivity coefficients?
It assumes the components you enter are uncorrelated, which is the root sum of squares case of GUM equation 10. Correlation needs the covariance form, GUM equation 13, which adds cross terms and can raise or lower the combined value. Sensitivity coefficients are handled by convention rather than by an extra field: enter every component already expressed in the units of the measurand, that is the product of the coefficient and the input standard uncertainty. The budget table then reports exactly the quantities the GUM calls the contributions to the combined variance.
Can the expanded uncertainty ever be larger than the physical range of the measurand?
Yes, and the GUM flags it in G.6.5. If one rectangular component with exactly known bounds dominates the budget, an interval built from a t factor greater than the square root of 3 can extend past the limits of that distribution. When the calculator reports that a single component carries most of the variance and the coverage factor is above 1.732, treat the interval as an approximation and consider modelling the dominant term directly instead of leaning on the central limit theorem.
Sources behind the equations on this page
Sources. Equations, divisors and coverage factors on this page are taken from the following primary documents.
- Joint Committee for Guides in Metrology, JCGM 100:2008 — Evaluation of measurement data: Guide to the expression of uncertainty in measurement (GUM 1995 with minor corrections). Equations 3 to 5 (Type A), 7 and 9b (rectangular and triangular divisors), 10 and 11 (law of propagation), 13 (correlated inputs), 18 (expanded uncertainty), G.2b (Welch-Satterthwaite), G.3 (degrees of freedom for Type B) and Table G.2 (Student t factors). BIPM JCGM publications
- B. N. Taylor and C. E. Kuyatt, NIST Technical Note 1297 (1994 edition) — Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, sections 5 and 6 and Appendix B on coverage factors. NIST TN 1297
- ISO/IEC 17025:2017, General requirements for the competence of testing and calibration laboratories, clause 7.6 on the evaluation of measurement uncertainty.
- Joint Committee for Guides in Metrology, JCGM 101:2008 — Supplement 1 to the GUM: Propagation of distributions using a Monte Carlo method, cited above for strongly nonlinear models.
Uncertainty budget breakdown
Every component expressed as a standard uncertainty in the units of the measurand, with its share of the combined variance.
| Component | Evaluation | Divisor | Standard uncertainty | Degrees of freedom | Share of variance |
|---|
The chart lists each component in descending order of its contribution to the combined variance.
Arcade Mini-Game: Measurement Uncertainty Calculator Calibration Run
Use this quick arcade run to practice separating useful scenario inputs from common planning mistakes before you rely on the calculator output.
Start the game, then use your pointer or arrow keys to catch useful inputs and avoid bad assumptions.