Measurement Uncertainty and the GUM

A measurement without an uncertainty is an opinion with a decimal point. The GUM, JCGM 100, sets out the standard method for turning the sources of doubt in a measurement into a single defensible number.

Table of Contents

What uncertainty actually is

Uncertainty is the range within which the true value is believed to lie, given everything you know about the measurement. It is not an error, because an error is a specific unknown difference from the truth. Uncertainty is a description of what you do not know, expressed on the same scale as the measurement.

It is also not the same as Gauge R and R. An R and R study estimates the spread contributed by operators and the gauge under your production conditions. An uncertainty budget accounts for everything that could shift the result, including contributions such as the calibration of your master and thermal expansion, which no R and R study observes because they do not vary between the operators and parts in the study.

A repeatability figure from an R and R study is one input to an uncertainty budget, not a substitute for it. Reporting repeatability alone as uncertainty understates the true doubt, sometimes by a large factor.

Type A and Type B evaluation

The GUM classifies contributions by how you evaluated them, not by their nature. This trips people up: Type A and Type B do not mean random and systematic.

Type A

Evaluated by statistical analysis of a series of observations you actually made. You have data, and you compute a standard deviation from it.

The standard deviation of repeated readings on the same part.
A repeatability estimate taken from your own Gauge R and R study.

Type B

Evaluated by any other means: certificates, specifications, published data, or reasoned engineering judgement. You have information, but not a data series you collected.

The expanded uncertainty quoted on your master's calibration certificate.
The resolution of the display, taken from the gauge specification.
Thermal effects estimated from the coefficient of expansion and the temperature range of the room.

Once each contribution has been converted to a standard uncertainty, the distinction stops mattering. Type A and Type B contributions are combined in exactly the same way. The classification exists to document how you arrived at each figure, which is what makes the budget auditable.

Converting each source to a standard uncertainty

Contributions arrive in different forms: a plus or minus limit, an expanded uncertainty, a resolution. Before anything can be combined, every one must be expressed as a standard uncertainty, meaning one standard deviation. The divisor depends on the distribution you assume.

Form of the source	Divide the half-width by	When this applies
Rectangular (uniform) limits, plus or minus a	√3	Any value in the range is equally likely and nothing favours the centre. The usual default for resolution and for specification limits with no further information.
Triangular limits, plus or minus a	√6	Values near the centre are more likely than values near the edges, but you have no stronger basis than that.
Expanded uncertainty U quoted at k = 2	2	A calibration certificate. Divide by the stated k to recover the standard uncertainty. Do not assume k = 2 without checking; the certificate states it.
A standard deviation you calculated	1	Already a standard uncertainty. No conversion needed.

For a digital display with resolution r, the conventional treatment is a rectangular distribution of half-width r divided by 2, giving a standard uncertainty of r divided by the square root of 12. Skipping the conversion step is one of the most common budget errors, and it always understates the result.

Sensitivity coefficients

Not every source of doubt affects the final result on a one-for-one basis. A sensitivity coefficient converts an uncertainty in an input quantity into the uncertainty it causes in the output.

Sensitivity coefficient

c_i = ∂y ÷ ∂x_i

It is the partial derivative of the measurement result with respect to that input: how much the answer moves when that input moves by one unit. Where the input is already in the same units as the result and affects it directly, the coefficient is 1 and can be omitted, which is why simple dimensional budgets often appear not to use them.

Where it matters

Measuring a 100 mm steel part, thermal expansion is about 11.5 micrometres per metre per degree Celsius. At 100 mm that is 1.15 micrometres per degree. The sensitivity coefficient converts your temperature uncertainty in degrees into a length uncertainty in micrometres. A one degree uncertainty is negligible on a 10 mm part and significant on a 1000 mm one, from the same underlying temperature control.

Combining in quadrature

Independent uncertainty contributions combine as the square root of the sum of squares, not by addition. This is the same variance addition rule that underlies Gauge R and R: variances add, standard deviations do not.

Combined standard uncertainty

u_c = √( (c₁u₁)² + (c₂u₂)² + ... + (c_nu_n)² )

The practical consequence is that the largest contribution dominates, and small ones barely register. Squaring makes this dramatic: a contribution one third the size of the largest adds about one ninth as much to the sum.

Worked example

Illustrative data. Three contributions, all already converted to standard uncertainties with sensitivity coefficients of 1.

Contribution	Standard uncertainty	Squared
Repeatability (Type A, from repeated readings)	0.0030 mm	0.00000900
Master calibration (Type B, certificate at k = 2)	0.0015 mm	0.00000225
Display resolution (Type B, rectangular)	0.0010 mm	0.00000100
Sum of squares	—	0.00001225
Combined standard uncertainty (square root)	0.0035 mm	—

Note what happened. Repeatability at 0.0030 contributes about 73 percent of the sum of squares while resolution contributes about 8 percent. If you want to reduce this uncertainty, improving repeatability is the only change worth making. Reducing the resolution contribution to zero would move the combined figure from 0.0035 to about 0.0034.

Effective degrees of freedom

Each contribution carries a confidence of its own. A standard deviation from 5 readings is a much shakier estimate than one from 50, and a Type B figure taken from a specification is often treated as effectively infinite degrees of freedom. The Welch-Satterthwaite formula blends these into a single effective figure for the combined uncertainty.

Welch-Satterthwaite effective degrees of freedom

ν_eff = u_c⁴ ÷ Σ( (c_iu_i)⁴ ÷ ν_i )

The result is dominated by whichever large contribution has few degrees of freedom. Its practical purpose is to determine the coverage factor: with high effective degrees of freedom, k = 2 gives close to 95 percent coverage, but when a dominant contribution rests on only a handful of readings, the effective figure falls and a larger k is needed for the same confidence.

The coverage factor and why k = 2

The combined standard uncertainty is one standard deviation, which covers only about 68 percent of the distribution. That is a weak statement to put on a certificate. Multiplying by a coverage factor k produces an expanded uncertainty covering a stated confidence level.

Expanded uncertainty

U = k × u_c

Using the worked example above: U = 2 x 0.0035 = 0.0070 mm, reported at approximately 95 percent confidence.

k = 2 is used because for a normal distribution with large effective degrees of freedom, plus or minus two standard deviations covers about 95.45 percent of the distribution. The exact multiplier for 95.00 percent is 1.96, and k = 2 is the rounded convention that has become standard practice.

The approximation depends on the effective degrees of freedom being reasonably large. If Welch-Satterthwaite returns a small figure, k must be taken from the Student t distribution at that number of degrees of freedom instead. At 5 effective degrees of freedom, 95 percent coverage requires k of about 2.57, not 2.

Always state k and the coverage level alongside the expanded uncertainty. A bare plus or minus figure with no stated k cannot be compared with anyone else's, and cannot be recombined into a further budget.

Reporting the result

A defensible uncertainty statement includes more than a number. At minimum, record:

The expanded uncertainty U, the coverage factor k, and the coverage level, for example approximately 95 percent.
The full budget, listing every contribution, its evaluation type, its assumed distribution, its divisor and its sensitivity coefficient.
The measurement conditions the budget assumes, particularly temperature, since a budget is only valid under the conditions it was built for.
Any known bias that was not corrected, since an uncorrected bias must be accounted for rather than quietly omitted.

The one thing to remember

Convert every contribution to a standard uncertainty before combining anything, then combine in quadrature. Most flawed budgets fail at the conversion step, and the failure always makes the reported uncertainty look better than it is.

For information and training only

This material, including all formulas, worked examples and interactive tools, is provided for general information and training purposes only. Examples use illustrative data and are not a substitute for professional judgement. Any method, calculation or result should be reviewed and confirmed by a qualified and trained person before it is relied on for product acceptance, process control, or any other quality decision.

No guarantee is made that this content, or any output of these tools, conforms to or satisfies any standard, specification, customer requirement or audit criterion. Where a published standard applies, the current official text of that standard governs. You remain responsible for verifying compliance with the requirements that apply to your organization.

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