Incerteza de medição e o GUM
Indice
O que a incerteza realmente é
Uncertainty is the range within which the valor verdadeiro is believed to lie, given everything you know about the medição. 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 medição.
It is also not the same as Gauge R e R. An R e R study estimates the spread contributed by operadores e the gauge under your production conditions. An orçamento de incerteza accounts for everything that could shift the result, including contribuiçãos such as the calibration of your master e thermal expansion, which no R e R study observes because they do not vary between the operadores e peças in the study.
Avaliação Tipo A e Tipo B
The GUM classifies contribuiçãos by how you evaluated them, not by their nature. This trips people up: Tipo A e Tipo B do not mean reom e systematic.
Tipo A
Evaluated by statistical analysis of a series of observations you actually made. You have data, e you compute a desvio padrão from it.
- The desvio padrão of repeated readings on the same part.
- A repeatability estimate taken from your own Gauge R e 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 incerteza expeida quoted on your master's certificado de calibração.
- The resolução of the display, taken from the gauge specification.
- Thermal effects estimated from the coefficient of expansion e the temperature range of the room.
Once each contribuição has been converted to a incerteza padrão, the distinction stops mattering. Tipo A e Tipo B contribuiçãos 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.
Convertendo cada fonte em incerteza padrão
Contributions arrive in different forms: a plus or minus limit, an incerteza expeida, a resolução. Before anything can be combined, every one must be expressed as a incerteza padrão, meaning one desvio padrão. The divisor depends on the distribuição 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 e nothing favours the centre. The usual default for resolução e 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. |
| Expeed uncertainty U quoted at k = 2 | 2 | A certificado de calibração. Divide by the stated k to recover the incerteza padrão. Do not assume k = 2 without checking; the certificate states it. |
| A desvio padrão you calculated | 1 | Already a incerteza padrão. No conversion needed. |
For a digital display with resolução r, the conventional treatment is a retangular distribuição of half-width r divided by 2, giving a incerteza padrão of r divided by the square root of 12. Skipping the conversion step is one of the most common budget errors, e it always understates the result.
Coeficientes de sensibilidade
Not every source of doubt affects the final result on a one-for-one basis. A sensibilidade coefficient converts an uncertainty in an input quantity into the uncertainty it causes in the output.
Sensitivity coefficient
ci = ∂y ÷ ∂xi
It is the partial derivative of the medição 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 e affects it directly, the coefficient is 1 e 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 sensibilidade coefficient converts your temperature uncertainty in degrees into a length uncertainty in micrometres. A one degree uncertainty is negligible on a 10 mm part e significativo on a 1000 mm one, from the same underlying temperature control.
Combining in quadrature
Independent uncertainty contribuiçãos combine as the square root of the sum of squares, not by addition. This is the same variance addition rule that underlies Gauge R e R: variances add, desvio padrãos do not.
Combined incerteza padrão
uc = √( (c1u1)² + (c2u2)² + ... + (cnun)² )
The practical consequence is that the largest contribuição dominates, e small ones barely register. Squaring makes this dramatic: a contribuição one third the size of the largest adds about one ninth as much to the sum.
Exemplo desenvolvido
Illustrative data. Three contribuiçãos, all already converted to steard uncertainties with sensibilidade coefficients of 1.
| Contribution | Steard uncertainty | Squared |
|---|---|---|
| Repetibilidade (Tipo A, from repeated readings) | 0.0030 mm | 0.00000900 |
| Master calibration (Tipo B, certificate at k = 2) | 0.0015 mm | 0.00000225 |
| Display resolução (Tipo B, retangular) | 0.0010 mm | 0.00000100 |
| Sum of squares | — | 0.00001225 |
| Combined incerteza padrão (square root) | 0.0035 mm | — |
Note what happened. Repetibilidade at 0.0030 contributes about 73 percent of the sum of squares while resolução contributes about 8 percent. If you want to reduce this uncertainty, improving repeatability is the only change worth making. Reducing the resolução contribuição to zero would move the combined figure from 0.0035 to about 0.0034.
Effective graus de liberdade
Each contribuição carries a confidence of its own. A desvio padrão from 5 readings is a much shakier estimate than one from 50, e a Tipo B figure taken from a specification is often treated as effectively infinite graus de liberdade. The Welch-Satterthwaite formula blends these into a single effective figure for the incerteza combinada.
Welch-Satterthwaite effective graus de liberdade
νeff = uc4 ÷ Σ( (ciui)4 ÷ νi )
The result is dominated by whichever large contribuição has few graus de liberdade. Its practical purpose is to determine the fator de abrangência: with high effective graus de liberdade, k = 2 gives close to 95 percent coverage, but when a dominant contribuição rests on only a heful of readings, the effective figure falls e a larger k is needed for the same confidence.
The fator de abrangência e why k = 2
The combined incerteza padrão is one desvio padrão, which covers only about 68 percent of the distribuição. That is a weak statement to put on a certificate. Multiplying by a fator de abrangência k produces an incerteza expeida covering a stated confidence level.
Expeed uncertainty
U = k × uc
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 distribuição with large effective graus de liberdade, plus or minus two desvio padrãos covers about 95.45 percent of the distribuição. The exact multiplier for 95.00 percent is 1.96, e k = 2 is the rounded convention that has become steard practice.
The approximation depends on the effective graus de liberdade being reasonably large. If Welch-Satterthwaite returns a small figure, k must be taken from the Student t distribuição at that number of graus de liberdade instead. At 5 effective graus de liberdade, 95 percent coverage requires k of about 2.57, not 2.
Reporting the result
A defensible uncertainty statement includes more than a number. At minimum, record:
- The incerteza expeida U, the fator de abrangência k, e the coverage level, for example approximately 95 percent.
- The full budget, listing every contribuição, its evaluation type, its assumed distribuição, its divisor e its sensibilidade coefficient.
- The medição 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.