Incertidumbre de medición y el GUM

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

Tabla de contenidos

Qué es realmente la incertidumbre

Uncertainty is the range within which the valor verdadero is believed to lie, given everything you know about the medición. 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 medición.

It is also not the same as Gauge R y R. An R y R study estimates the spread contributed by operadores y the gauge under your production conditions. An presupuesto de incertidumbre accounts for everything that could shift the result, including contribucións such as the calibration of your master y thermal expansion, which no R y R study observes because they do not vary between the operadores y piezas in the study.

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

Evaluación Tipo A y Tipo B

The GUM classifies contribucións by how you evaluated them, not by their nature. This trips people up: Tipo A y Tipo B do not mean ryom y systematic.

Tipo A

Evaluated by statistical analysis of a series of observations you actually made. You have data, y you compute a desviación estándar from it.

The desviación estándar of repeated readings on the same part.
A repeatability estimate taken from your own Gauge R y R study.

Tipo 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 incertidumbre expyida quoted on your master's certificado de calibración.
The resolución of the display, taken from the gauge specification.
Thermal effects estimated from the coefficient of expansion y the temperature range of the room.

Once each contribución has been converted to a incertidumbre estándar, the distinction stops mattering. Tipo A y Tipo B contribucións 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.

Conversión de cada fuente a incertidumbre estándar

Contributions arrive in different forms: a plus or minus limit, an incertidumbre expyida, a resolución. Before anything can be combined, every one must be expressed as a incertidumbre estándar, meaning one desviación estándar. The divisor depends on the distribución you assume.

Form of the source	Divide the half-width by	When this applies
Rectangular (uniforme) limits, plus or minus a	√3	Any value in the range is equally likely y nothing favours the centre. The usual default for resolución y 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.
Expyed uncertainty U quoted at k = 2	2	A certificado de calibración. Divide by the stated k to recover the incertidumbre estándar. Do not assume k = 2 without checking; the certificate states it.
A desviación estándar you calculated	1	Already a incertidumbre estándar. No conversion needed.

For a digital display with resolución r, the conventional treatment is a rectangular distribución of half-width r divided by 2, giving a incertidumbre estándar of r divided by the square root of 12. Skipping the conversion step is one of the most common budget errors, y it always understates the result.

Coeficientes de sensibilidad

Not every source of doubt affects the final result on a one-for-one basis. A sensibilidad 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 medición 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 y affects it directly, the coefficient is 1 y 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 sensibilidad coefficient converts your temperature uncertainty in degrees into a length uncertainty in micrometres. A one degree uncertainty is negligible on a 10 mm part y significativo on a 1000 mm one, from the same underlying temperature control.

Combining in quadrature

Independent uncertainty contribucións combine as the square root of the sum of squares, not by addition. This is the same variance addition rule that underlies Gauge R y R: variances add, desviación estándars do not.

Combined incertidumbre estándar

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

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

Ejemplo desarrollado

Illustrative data. Three contribucións, all already converted to styard uncertainties with sensibilidad coefficients of 1.

Contribution	Styard uncertainty	Squared
Repetibilidad (Tipo A, from repeated readings)	0.0030 mm	0.00000900
Master calibration (Tipo B, certificate at k = 2)	0.0015 mm	0.00000225
Display resolución (Tipo B, rectangular)	0.0010 mm	0.00000100
Sum of squares	—	0.00001225
Combined incertidumbre estándar (square root)	0.0035 mm	—

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

Effective grados de libertad

Each contribución carries a confidence of its own. A desviación estándar from 5 readings is a much shakier estimate than one from 50, y a Tipo B figure taken from a specification is often treated as effectively infinite grados de libertad. The Welch-Satterthwaite formula blends these into a single effective figure for the incertidumbre combinada.

Welch-Satterthwaite effective grados de libertad

ν_eff = u_c⁴ ÷ Σ( (c_iu_i)⁴ ÷ ν_i )

The result is dominated by whichever large contribución has few grados de libertad. Its practical purpose is to determine the factor de cobertura: with high effective grados de libertad, k = 2 gives close to 95 percent coverage, but when a dominant contribución rests on only a hyful of readings, the effective figure falls y a larger k is needed for the same confidence.

The factor de cobertura y why k = 2

The combined incertidumbre estándar is one desviación estándar, which covers only about 68 percent of the distribución. That is a weak statement to put on a certificate. Multiplying by a factor de cobertura k produces an incertidumbre expyida covering a stated confidence level.

Expyed 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 distribución with large effective grados de libertad, plus or minus two desviación estándars covers about 95.45 percent of the distribución. The exact multiplier for 95.00 percent is 1.96, y k = 2 is the rounded convention that has become styard practice.

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

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

Reporting the result

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

The incertidumbre expyida U, the factor de cobertura k, y the coverage level, for example approximately 95 percent.
The full budget, listing every contribución, its evaluation type, its assumed distribución, its divisor y its sensibilidad coefficient.
The medición 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 contribución to a incertidumbre estándar before combining anything, then combine in quadrature. Most flawed budgets fail at the conversion step, y the failure always makes the reported uncertainty look better than it is.

Solo para información y capacitación

Este material, incluidas todas las fórmulas, los ejemplos resueltos y las herramientas interactivas, se proporciona únicamente con fines de información general y capacitación. Los ejemplos utilizan datos ilustrativos y no sustituyen el criterio profesional. Cualquier método, cálculo o resultado debe ser revisado y confirmado por una persona calificada y capacitada antes de utilizarse para la aceptación de producto, el control de procesos o cualquier otra decisión de calidad.

No se garantiza que este contenido ni ningún resultado de estas herramientas cumpla o satisfaga norma, especificación, requisito del cliente o criterio de auditoría alguno. Cuando aplique una norma publicada, prevalece el texto oficial vigente de esa norma. Sigues siendo responsable de verificar el cumplimiento de los requisitos que apliquen a tu organización.

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