Measuring Uncertainty

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Measuring uncertainty

Definition of measurement uncertainty

The measurement uncertainty of the estimated value of a physical quantity limits a value range within which the true value of the measured parameter lies with a probability to be specified (95 % is usually specified). The result of a measurement is only defined by the estimated value itself and the corresponding measurement uncertainty. The measurement uncertainty is specified without a sign and the measurement uncertainties are themselves estimated values. The aim of estimating measurement uncertainties is to define intervals that include or ‘localise’ the true values of the measured variables, i.e. the measurement uncertainty sets the limits within which a result is regarded as accurate or precise and true.

As a rule, the measurement uncertainty defines a value range that is symmetrical to the estimated value of the measurand, whereby the estimated value has been freed from known systematic errors. Random measurement errors are normally the subject of the error calculation and are not part of the determination of the measurement uncertainty. Known, i.e. eliminable, systematic measurement errors are also not involved in the estimation of the measurement uncertainty. The measurement result is represented by an expression of the form

estimated value [unit of measurement] ± measurement uncertainty [unit of measurement or per cent]

is given.

The measurement uncertainty U can be multiplied by a coverage factor k > 1 to increase the confidence interval. This product is then referred to as the expanded measurement uncertainty U for a specific coverage factor:

$U=k\cdot u\!$

In laboratory technology, expanded uncertainties are usually specified as k = 2. The range or interval defined in this way has approximately the width of a 95 % probability of occurrence.

The uncertainty can always be estimated by determining all factors that contribute to the measurement uncertainty. The contribution of individual factors is usually estimated using standard deviations, either from repeated measurements (for random factors) or from other sources of information (for systematic factors). The total uncertainty is calculated using the variances of the individual uncertainty factors and presented as a standard deviation. The total uncertainty multiplied by a scatter factor of 2 gives (approximately) a 95 % confidence interval [1–5].

Accredited calibration laboratories or testing laboratories that carry out internal calibrations themselves must have procedures for determining the measurement uncertainty and must also apply these. The laboratory must determine the components of measurement uncertainty and use this knowledge to produce a reasonable estimate of the measurement uncertainty for the various test methods. This also applies to the validation of new procedures or test methods and the use of newer test technology in laboratory operations. It is necessary to state the measurement uncertainty in the test report if it is significant for the test result, if the customer requires it or if the measurement uncertainty may jeopardise compliance with limits or tolerances [6].

Procedure for estimating measurement uncertainty in the test laboratory using the example of mechanical plastics testing

Test method

The testing methods used in a testing laboratory for mechanical testing of plastics are usually based on the following testing techniques:

• Electromechanical universal testing machine
• Hardness testing machines
• Pendulum impact testers or drop testing systems

The universal testing machine is used to carry out all accredited test methods involving tensile, compression and bending stresses as well as the tear test and quasi-static fracture mechanics methods.

The hardness testing machine is only used for the accredited test method ISO 2039-1 to determine the ball indentation hardness, which is used as an example to explain the estimation of the measurement uncertainty.

Variables influencing measurement uncertainty in the hardness testing machine

The measurement of ball indentation hardness on plastics is based on the measurement of the penetration depth h of a 5 mm steel ball into the material to be analysed at a selected constant load. Using this method with measurement of the total deformation, the elastic, viscoelastic and plastic deformation components are thus recorded in the hardness, whereby a constant retardation time of 30 s is specified.

The measured total deformation h allows the surface area of the resulting spherical indentation to be calculated using the following equation.

Surface area $A_{m}=\pi d\cdot h$, where $d={\sqrt {Dh-h^{2}}}$ is

with

h		penetration depth
D		spherical diameter
d		diameter of the spherical cap at the surface Surface

In general, the hardness H is calculated as follows:

$H={\frac {F}{A_{m}}}(N\ mm^{-2})$

During the test, the preload and main load cause the load frame to bend upwards, which is directly taken into account in the evaluation programme of the testing machine, for example, with the reduced penetration depth h_r and the reduced force F_r. In accordance with the DIN EN ISO 2039-1 standard, the hardness is then calculated using the following numerical value equation:

$HB={\frac {1}{5\pi }}{\frac {F_{r}}{h_{r}}}$

There are no hardness reference plates or standards for measuring the hardness of plastics, which means that indirect calibration or uncertainty determination is not possible and only the force, displacement or time are accessible.

Influences variables

The preload F₀ = 9.8 N and the main loads of 49, 132, 358 and 961 N may have a limit deviation of ± 1 %. The measuring range of the indentation depth must be 0.4 mm with a measurement uncertainty of ± 0.005 mm. The time measurement unit may have an error limit of ± 0.1 s.

The test specimens used must have a flat surface and a thickness of 4 mm.

The test and calibration results of the Saxony-Anhalt State Materials Testing Office (LMPA) generally show limit deviations of less than ± 1 % for the force.

A check of the indentation depth accuracy and the load time is not carried out with this measuring system due to the technical problems, which is why the measurement uncertainty is determined in accordance with the requirements of the standard. There is no reading uncertainty in this test.

Measuring uncertainty

Exeample: Ball indentation hardness

$y=0.0637{\frac {(x_{1}+\Delta x_{1})}{x_{2}+\Delta x_{2}}}$

with

x1		pre- and mainload Fr
x2		reduced penetration depth hr
0.0637		= 1/5 π

Values:

y	(N/mm2)	Ball indentation hardness
x1	(N)	constant pre- and mainload
Δx1	(N)	standard and manufacturer specification (Wert 1 ± 0.01) (rectangular distribution)
x2	(mm)	penetration depth
Δx2	(mm)	standard requirement (Wert 1.0 ± 0.005) (rectangular distribution)

The combined measurement uncertainty Δy_komb is as follows

$\Delta y_{komb}={\sqrt {\left({\frac {U(x_{1})}{x_{1}}}\right)^{2}+\left({\frac {U(x_{2})}{x_{2}}}\right)^{2}}}$

with two uncertainty components U(x₁) and U(x₂).

Example PA6: (standard conditioned) – Ball indentation hardness HB = 55 N/mm²

x₁ = pre- and mainload = 9.8 N + 132 N = 141.8 N

U(x₁) = 141.8 ⋅ 0.01 = 1.418 N ≡ Half-width of the upper and lower limits

Assumption of rectangular distribution:

$U(x_{1})={\frac {1.418}{\sqrt {3}}}=0.8187\ N$

x₂ = 0.25 mm

Note:

The penetration depth must lie in the interval from 0.15 to 0.35 mm. If h is less than 0.15, the load is increased; if h is greater than 0.35, the load is reduced. The device does not display the value of the penetration depth actually achieved.

For this reason, a value of 0.25 mm is assumed for h, which lies exactly in the centre of the penetration depth interval.

⇒

U(x₂) = 0.25 ⋅ 0.005 = 0.0013 mm

Assumption of rectangular distribution:

$U(x_{2})={\frac {0.0013}{\sqrt {3}}}=0.00075\ mm$

The combined standard uncertainty results in:

$Uy_{komb}={\sqrt {\left({\frac {0.8187}{141.8}}\right)^{2}+\left({\frac {0.0013}{0.25}}\right)^{2}}}=0,01554$

with U_erw = 2 U_komb → U_erw = 0.0311

The combined measurement uncertainty results in:

$\Delta y_{komb}=0.0637{\frac {141.8}{0.25}}\cdot 0.01554=0.56\ Nmm^{-2}$

and the expanded measurement uncertainty is then:

Δy_erw = 1.12 N/mm² or 2.04 % (of HB = 55 N/mm²)

The result is then: HB = 55 N/mm2 ± 1.12 N/mm2.

In the book by Gottfried W. Ehrenstein entitled ‘Massenanalyse – Einfache Bestimmungsmethoden, Feuchtigkeitsbestimmung, Dichtebestimmung, Gehalt an Füll- und Verstärkungsstoffen, Thermogravimetrische Analyse, Bestimmung flüchtiger Bestandteile, Thermogravimetrie, Messunsicherheit’, the topic of measurement uncertainty is dealt with from the perspective of plastics analysis. Explanations of terms, principles for determining measurement uncertainty on the basis of round robin tests and the application of uncertainty data in practice are presented in the textbook ‘Measurement uncertainty in plastics analysis’ [8]. Examples include thermogravimetric analysis (TGA), elemental analysis of polymer materials (Cl, Br, S, N, VA in E/VA, water), the detection of heavy metals in plastics (Pb, Hg, Zn, Cr, Cd and Al) and the detection of stabilisers or residual solvents.

See also

• Measure
• Measured value
• Measuring accuracy
• Measuring device monitoring
• Measurement deviation

References

[1]	Brinkmann, B.: Internationales Wörterbuch der Metrologie ‒ Grundlegende und allgemeine Begriffe und zugeordnete Benennungen. Beuth Verlag, Berlin, Wien, Zürich, 4. Auflage (2012) (ISBN 978-3-410-22472-3)
[2]	DIN V ENV 13005 (1999-06): Leitfaden zur Angabe der Unsicherheit beim Messen (Deutsche Version des GUM) (zurückgezogen)
[3]	DIN 1319-3 (1996-05): Grundlagen der Messtechnik ‒ Teil 3: Auswertung von Messungen einer einzelnen Messgröße, Messunsicherheit
[4]	DIN 1319-4 (1999-02): Grundlagen der Messtechnik ‒ Teil 4: Auswertung von Messungen, Messunsicherheit
[5]	Weise, K., Wöger, W.: Messunsicherheit und Messdatenauswertung. Wiley-VCH, Weinheim (1999) (ISBN 978-3-527-29610-7)
[6]	ISO 21748 (2017-04): Leitfaden zur Verwendung der Schätzwerte der Wiederholpräzision, der Vergleichspräzision und der Richtigkeit beim Schätzen der Messunsicherheit (DIN ISO 21748 zurückgezogen)
[7]	ISO 2039-1 (2001-12): Plastics – Determination of Hardness – Part 1: Ball Indentation Method
[8]	Wampfler, B., Affolter, S., Ritter, A., Schmid, M.: Messunsicherheit in der Kunststoffanalytik – Ermittlung mit Ringversuchsdaten. Carl Hanser Munich (2017) (ISBN 978-3-446-45286-2)