# Measurement uncertainty

A measurement result as an approximation for the true value of a measured variable should always include the specification of a measurement uncertainty . This limits a range of values within which the true value of the measured variable lies with a probability to be specified (ranges for approximately 68% and approximately 95% are common). The estimated value or individual measured value used as the measurement result should already be corrected for known systematic deviations .

The measurement uncertainty is positive and is given without a sign. Measurement uncertainties are themselves also estimated values. The measurement uncertainty can also be called uncertainty for short . The term error , which used to be used in similar contexts, is not synonymous with the concept of measurement uncertainty .

As a rule, there is a normal distribution , and the measurement uncertainty defines a value range that is symmetrical to the estimated value of the measured variable . It is usually given as the standard uncertainty u or as the expanded uncertainty 2 u .

## Determination of the measurement uncertainty

### Analytical-computational method according to ISO / IEC Guide 98-3

A measurement uncertainty results from the combination of individual contributions (components) of the input variables of a measurement. According to ISO / IEC Guide 98-3 ( GUM ), a component of the measurement uncertainty can be determined in two ways:

• Type-A: Determination from the statistical analysis of several statistically independent measured values ​​from a repeated measurement.
• Type-B: Determination without statistical methods, for example by taking the values ​​from a calibration certificate , from the accuracy class of a measuring device or based on personal experience and previous measurements. The error limit can also be used to determine the type B measurement uncertainty, assuming a rectangular distribution. It is an a priori distribution.

Both methods are based on probability distributions. With type A, the variance is determined by repeated measurements, and with type B other sources are used. The type A determination method follows the frequentistic and type B the Bayesian interpretation of probability. The type B determination method is based on the Bayes-Laplace theory.

### Determination using interlaboratory test data according to ISO 21748

In an interlaboratory comparison , several laboratories ideally analyze identical samples with the same measuring method. The evaluation of the results leads to two parameters that are of great importance for determining the measurement uncertainty:

• Repeatability standard deviation s r (characterizes the mean spread of the values ​​within the laboratories)
• Standard deviation between laboratories s L (characterizes the variation between laboratories)

The two standard deviations contain all or at least most of the uncertainty components that must be considered individually according to the ISO / IEC 98-3 method. This also applies to components of type 2, which cannot be recorded by multiple measurements in the individual laboratory. If the conditions specified in ISO 21748 are met, the standard uncertainty u in the simplest case results from the following relationship:

${\ displaystyle u = s_ {R} = {\ sqrt {s_ {L} ^ {2} + s_ {r} ^ {2}}}}$

s R is the comparative standard deviation. In certain cases, additional components such as sampling , sample preparation or sample heterogeneity must be taken into account. Proficiency test data can depend on the value of the measurand.

## Metrological importance

The measurement uncertainties in science and technology should fulfill three tasks.

• They are intended to objectify measurement results by specifying the interval in which the true value of the measured variable can be expected. According to classic diction , these were confidence intervals , the size of which depended on the level of confidence. The classic error calculation must be expanded to include so-called unknown systematic measurement errors. Therefore, the measurement uncertainty cannot be assigned a probability in the same way as it is possible with only statistical deviations.
• The network of physical constants created in this way must in itself be free of contradictions; H. If one calculated another, numerically already known constant from a subset of constants on the basis of a given linkage function , then the measurement uncertainty resulting from the uncertainty propagation must in turn localize the true value of this constant. Measurement uncertainties must therefore meet the requirement for “traceability of true values”.

## Quantitative information

Another characteristic is the expanded uncertainty . This characteristic value identifies a range of values ​​that contains the true value of the measured variable with a certain probability. For the expansion factor contained therein should preferably be used. With the probability is about 95%. ${\ displaystyle U = k \ cdot u}$${\ displaystyle k}$${\ displaystyle k = 2}$${\ displaystyle k = 2}$

In special cases , one speaks of a standard uncertainty (based on the term standard deviation) . Here the probability is about 68%. ${\ displaystyle k = 1}$

using the example of a measurement result with a standard measurement uncertainty : ${\ displaystyle l = 23 {,} 478 \, 2 \; \ mathrm {m}}$${\ displaystyle u = 0 {,} 003 \, 2 \; \ mathrm {m}}$

• The information is summarized in what a range from to means.${\ displaystyle l = (23 {,} 478 \, 2 \ pm 0 {,} 003 \, 2) \; \ mathrm {m}}$${\ displaystyle 23 {,} 475 \, 0 \; \ mathrm {m}}$${\ displaystyle 23 {,} 481 \, 4 \; \ mathrm {m}}$
The notation with ± should be avoided whenever possible in the event of uncertainties.
• if it is not made clear which parameter of the measurement uncertainty or which coverage factor it stands for,
• because the notation with ± is also used for other information such as the confidence interval or tolerances .
• The notation can be used to express that the deviations are different upwards and downwards - for example with a logarithmic value scale .${\ displaystyle l = {\ bigl (} 23 {,} 478 \, 1 {\ begin {smallmatrix} +0 {,} 003 \, 3 \\ - 0 {,} 003 \, 1 \ end {smallmatrix}} {\ bigr)} \; \ mathrm {m}}$
• The spelling can be found in .${\ displaystyle l = 23 {,} 478 \, 2 (0 {,} 003 \, 2) \; \ mathrm {m}}$
• Especially in connection with the standard uncertainty, the short notation is common (sometimes also called "brackets", in English concise notation ). The numerical value of the standard uncertainty in units of the place value of the last specified digit is shown here in brackets .${\ displaystyle l = 23 {,} 478 \, 2 (3 \, 2) \; \ mathrm {m}}$

## Questioning the error calculation

The "classic" Gaussian error calculation only deals with random deviations. However, Gauss had already pointed out the existence and importance of so-called unknown systematic measurement errors. These arise from disturbance variables that are constant over time and whose magnitude and sign are unknown; they are usually of an order of magnitude comparable to the random deviations . Unknown systematic measurement deviations must be limited with the help of intervals.

Today's mainstream metrology interprets the process of estimating the measurement uncertainty as a “technical regulation” that is to be practiced uniformly. In the area of legal metrology and the calibration service in Germany , it is recommended to define measurement uncertainties according to DIN . This guideline for specifying the uncertainty during measurement corresponds to the European prestandard ENV 13005, which adopts the recommendation of the ISO ; it is also known under the acronym " GUM ".

DIN V ENV 13005 has been withdrawn. The rule maker recommends the application of the "Technical Rule" ISO / IEC Guide 98-3: 2008-09 Measurement Uncertainty - Part 3: Guidelines for specifying the measurement uncertainty .

## Exact values

“Exact value” is a term from metrology. In this context, exact values ​​have no measurement uncertainty and no systematic deviation.

Some fundamental constants of nature are exactly by definition, others are not or no longer (see definition of the SI base units ). For example, the magnetic field constant is now provided with an uncertainty. In the case of the quantities precisely defined with a certain number of digits, it is not the numerical value that is uncertain, but the implementation of the unit defined by the quantity and the numerical value.

Other exact values ​​are mathematically defined irrational numbers , like the circle number as the ratio of the circumference and diameter of circles (in Euclidean geometry). ${\ displaystyle \ pi}$

Some even numbers in calculations are exact values, such as the arbitrarily defined conversion factors 12 between troy pounds and troy ounces and 90 between the size of right angles and degrees .

Exact rational numbers can be written as fractions in formulas, for example and not to avoid the false assumption that there could be an implicit uncertainty in the last decimal place. ${\ displaystyle 1/2}$${\ displaystyle 0 {,} 5}$

## literature

• DIN 1319 "Basics of measurement technology"
Part 1: Basic Terms (Edition: 1995-01)
Part 2: Terms for measuring equipment (Edition: 2005-10)
Part 3: Evaluation of measurements of a single measurand, measurement uncertainty (Edition: 1996-05)
Part 4: Evaluation of measurements; Measurement uncertainty (Edition: 1999-02)
• DIN, German Institute for Standardization e. V. (Hrsg.): Guidelines for specifying the measurement uncertainty when measuring. 1st edition. Beuth Verlag GmbH, Berlin 1995, ISBN 3-410-13405-0
• DIN V ENV 13005: 1999-06, edition 1999-06 "Guidelines for specifying the uncertainty when measuring" German version ENV 13005: 1999, Beuth Verlag GmbH, Berlin
• DIN ISO 5725 "Accuracy (correctness and precision) of measuring methods and measurement results"
Part 1: General principles and terms (ISO 5725-1: 1994) (Edition: 1997-11)
Part 2: Basic method for determining the repeatability and comparability of a standardized measuring method (ISO 5725-2: 1994 including Technical Corrigendum 1: 2002) (Edition: 2002-12)
Part 3: Precision dimensions of a standardized measurement method under intermediate conditions (ISO 5725-3: 1994 including Technical Corrigendum 1: 2001) (Edition: 2003-02)
Part 4: Basic methods for determining the correctness of a standardized measuring method (ISO 5725-4: 1994) (Edition: 2003-01)
Part 5: Alternative methods for determining the precision of a standardized measurement method (ISO 5725-5: 1998) (Edition: 2006-04)
Part 6: Application of accuracy values ​​in practice [ISO 5725-6: 1994 including Technical Corrigendum 1: 2001] (Edition 2002-08)
• Guide to the Expression of Uncertainty in Measurement, ISO, International Organization for Standardization
• ISO 21748 "Guidance for the use of repeatability, reproducibility and trueness estimates in measurement uncertainty estimation" (Edition: 2010-10)
• Way, Klaus; Wöger, Wolfgang: Measurement uncertainty and measurement data evaluation. Weinheim: Wiley-VCH 1999. ISBN 3-527-29610-7

## Individual evidence

1. a b DIN 1319-1: 1995 Fundamentals of measurement technology - Part 1: Basic terms.
2. JCGM 200: 2012 International vocabulary of metrology - Basic and general concepts and associated terms (VIM) , Definition 2.26.
3. Michael Krystek: Calculation of the measurement uncertainty. Basics and instructions for practical use . Beuth, 2012, p. 279 ( limited preview in Google Book search).
4. Susanne Heinicke: One becomes wise from mistakes. A genetic-didactic reconstruction of the measurement error . Berlin 2012, ISBN 978-3-8325-2987-1 , pp. 208–211 ( limited preview in Google Book search).
5. ^ A b Franz Adunka: Measurement uncertainties. Theory and practice . 2007, ISBN 978-3-8027-2205-9 , pp. 93–95 ( limited preview in Google Book search).
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9. a b Susanne Heinicke: From Errors One Becomes Smart: A Genetic-Didactic Reconstruction of the Measurement Error . Logos Verlag Berlin GmbH, 2012, ISBN 978-3-8325-2987-1 , p. 208 ( limited preview in Google Book search).
10. a b c ISO 21748: 2017 Guidance for the use of repeatability, reproducibility and trueness estimates in measurement uncertainty evaluation .
11. Bruno Wampfler, Samuel Affolter, Axel Ritter, Manfred Schmid: Measurement uncertainty in plastics analysis - determination with round robin test data . Carl Hanser Verlag, Munich 2017, ISBN 978-3-446-45286-2 , pp. 13-18 .
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13. EN ISO 80000-1: 2013, sizes and units - Part 1: General.
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15. ^ Standard Uncertainty and Relative Standard Uncertainty , The NIST Reference on Constants, Units and Uncertainty, accessed March 16, 2018.
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