What is the difference between Type A and Type B uncertainty?

Type A comes from the repeated measurements themselves: it is the scatter you observe, divided by the square root of the number of readings. Type B comes from knowledge you had before you started, such as an instrument tolerance, a calibration certificate or the resolution of a display, and it is converted to a standard uncertainty using an assumed distribution. The distinction is about where the information comes from, not about which is better.

Why do we divide a tolerance by √3?

Because a tolerance says what is allowed, not how the error is distributed. Assuming the error is equally likely anywhere inside the tolerance, the rectangular distribution has a standard deviation of half-width over √3. Assuming instead that it clusters at the centre gives √6, and assuming it is roughly bell-shaped across the tolerance gives 2.

What does the expanded uncertainty U actually mean?

It is k times the combined standard uncertainty, and it is the number that belongs after the ± sign. U is not a standard deviation: it is the half-width of an interval that a single future measurement would fall inside with the stated probability, 95% of the time by convention.

How many readings do I need?

Two is the minimum, and four or five is usually enough to spot a repeatability problem. Because uA = s/√n, going from five readings to twenty only divides the Type A component by two, so it is rarely worth the effort compared with fixing a dominant Type B term.

Which component should I try to reduce first?

The one with the largest share of the combined variance, not the largest share of the raw numbers. The budget table ranks by share of uC², so improving a component that supplies 2% of the variance cannot move the answer, however large it looks on its own.

Can I use a coverage probability other than 95%?

Yes, and it is worth being explicit about it. Lower coverage gives a narrower interval for a routine screening decision, and higher coverage such as 99% widens it for a safety case. The coverage factor k is read from the t distribution at your effective degrees of freedom, so the calculator widens the interval automatically when the budget rests on few readings.

Uncertainty Calculator — Free Online Calculator

What Is the Uncertainty Calculator?

Measurement uncertainty quantifies the doubt attached to the result of a measurement. The GUM expresses that doubt as a standard uncertainty, uC, being the standard deviation of the quantity being measured, and then as an expanded uncertainty, U = k·uC, which is what gets quoted alongside a result.

The GUM splits the inputs into Type A, evaluated statistically from repeated observations, and Type B, evaluated by other means from knowledge of the measurement process. Both end up as standard uncertainties on the same footing, because the final combination rule does not care where a number came from.

How Does the Uncertainty Calculator Work?

Type A uncertainty is the sample standard deviation s of the readings divided by √n. The Bessel correction divides by n − 1 rather than n, because the mean is estimated from the same readings, and this is the only source in the budget with a finite degrees of freedom.

Type B starts from knowledge rather than data. A tolerance of ±Δ is divided by the divisor for the assumed distribution: √3 rectangular, √6 triangular, 2 for a normal spread across the tolerance. A component already quoted as a standard uncertainty, from a calibration certificate for instance, is used unchanged.

The components are combined in quadrature, uC = √(uA² + Σuᵢ²), never arithmetically, because they are independent. Their variances are fractions of uC², and those fractions are what the budget table ranks: the component with the largest share is the only one worth spending effort on.

The coverage factor k comes from the t distribution, not a lookup table, using the Welch–Satterthwaite effective degrees of freedom νeff = uC⁴ / Σ(uᵢ⁴/νᵢ). A budget resting on few Type A readings has a small νeff and needs a larger k; a budget of only Type B terms has unbounded νeff and reduces to the normal quantile.

The reported result is then written as result ± U at the chosen coverage probability, with U rounded to at most two significant figures and the result to the same decimal place.

Uncertainty Calculator Formula & Variables

The core mathematical equation utilized by this calculator is expressed as:

u_A = \frac{s}{\sqrt{n}} \qquad u_i = \frac{\Delta_i}{\sqrt{3}} \qquad u_C = \sqrt{u_A^2 + \sum_i u_i^2} \qquad \nu_{\mathrm{eff}} = \frac{u_C^4}{\sum_i u_i^4/\nu_i} \qquad U = k\,u_C

Variable Definitions

Symbol	Variable Meaning & Units
s	sample standard deviation of the repeated readings
n	number of repeated readings
Δᵢ	the ± half-width allowed by one Type B source
uC	combined standard uncertainty
νeff	Welch–Satterthwaite effective degrees of freedom
k	coverage factor from the t distribution

Type A uncertainty is the scatter divided by the square root of the number of readings, because averaging reduces scatter by √n. A tolerance is not a standard uncertainty, so it is divided by √3 for a rectangular error. The components then add in quadrature, and the coverage factor k comes from the t distribution at the effective degrees of freedom, which is where the interval U = k·uC comes from.

How to Use the Uncertainty Calculator

Start in combined mode. Enter the measured result in the units you will report, then paste your repeat readings if you have them.
Add one entry per Type B tolerance, each being the ± half-width that source allows. Several entries are combined automatically, and you can mix in standard uncertainties quoted by a calibration laboratory.
Choose the distribution that matches the component. Rectangular is the GUM default and the safe assumption when you know nothing else; a resolution-limited value is often closer to rectangular than to normal.
Set the coverage probability. 0.95 is the convention for reporting, and the calculator then shows k, the effective degrees of freedom and the resulting interval.
Read the budget table before deciding what to improve. It lists every component with its standard uncertainty and its share of the combined variance.

Step-by-Step Example Calculation

Five repeat readings of a 10 unit process plus a ±0.2 tolerance

Input Values:

mode:expanded

result:10

readings:10.02, 9.98, 10.05, 9.95, 10.00

tolerances:0.2

coverage:0.95

Worked Steps: The readings give uA = 0.017 and the tolerance gives uB = 0.115, so uC = 0.117 and the result is quoted as 10 ± 0.228 with about 95% coverage. The tolerance supplies 97.9% of the variance, so repeatability is not the thing to improve.

Understanding Your Result

uC is one standard deviation, so about 68% of future measurements would fall within ±uC if everything else were held fixed. It is a property of the measurement process, not a promise about any single value.

U is an interval, not a scatter. Quoting 10 ± 0.23 at 95% means roughly 95% of repeat measurements would land in that band, and the remaining 5% are outside it by design.

The relative uncertainty, uC as a percentage of the result, is what makes results comparable across quantities of different size. It is the figure to compare when judging whether a measurement is precise enough for its purpose.

The effective degrees of freedom explain why k is not always 2. With only two readings, k at 95% is about 12.7, and quoting ±2uC there would be badly optimistic.

Factors That Affect the Result

The number of readings. uA falls as 1/√n, which is a slow reward: four times the readings buys a factor of two.
The assumption behind each tolerance. Moving from rectangular to triangular cuts that component by 23%, and for a component that dominates the budget that is a large change in the final answer.
Whether components are independent. Two sources that share a calibration are correlated, and treating them as independent overstates uC.
The coverage probability. Moving from 95% to 99% widens the interval by about 30%, and nothing about the measurement itself has changed.
Rounding at the end. Reporting U to too many figures implies a precision the budget does not support and is a common source of disagreement between laboratories.

When Should You Use This Calculator?

Reporting a measurement with a defensible uncertainty, as ISO/IEC 17025 laboratories must for every result they issue.
Deciding whether an instrument is accurate enough for a task, by turning its specification into the uncertainty it actually contributes.
Building an uncertainty budget to show which component to improve first, and what any proposed improvement is worth.
Comparing measurement methods or laboratories, where overlapping intervals with a stated coverage probability are the meaningful comparison.
Any quoted tolerance, specification limit or pass/fail decision, where the measurement uncertainty decides how close to the limit a result really is.

Assumptions & Limitations

The inputs must be independent unless you correct for covariance, which this budget does not do. Shared calibration errors are the usual source of over-combination.
A rectangular assumption is a guess unless you can justify it. Where the real error is one-sided or peaked, the Type B components are biased even though the arithmetic is exact.
Only Type A carries degrees of freedom here. Type B sources are treated as unbounded, which is conservative for k but ignores any real information about them.
The budget covers random uncertainty only. Systematic effects, such as an uncalibrated reference or a temperature offset, must be corrected before they can be budgeted.
The result is a number, not a model of the process. A budget can be internally consistent and still describe a measurement that is systematically wrong.

Frequently Asked Questions

Calculation Accuracy & Reference Note

Every figure comes from closed-form arithmetic on the entered values, with the sample standard deviation, the quadrature combination, the Welch–Satterthwaite effective degrees of freedom and the t quantile all evaluated in double precision. The only iteration is the bisection inside the t quantile, which converges well beyond the digits displayed.

Standard Reference: JCGM 100:2008 (GUM), Evaluation of measurement data — Guide to the expression of uncertainty in measurement. BIPM, Sèvres.