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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Fehlermodell</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Ein <b>Fehlermodell</b> klassifiziert <a href="Messfehler" class="mw-redirect" title="Messfehler">Messfehler</a>, hier die der <a href="Metrologie" title="Metrologie">Metrologie</a>. Die Eigenschaften der Messfehler, die in den <a href="Messprozess" class="mw-redirect" title="Messprozess">Messprozess</a> einfließen, sind durch die physikalische Arbeitsweise der <a href="Messapparatur" class="mw-redirect" title="Messapparatur">Messapparatur</a> festgelegt.
</p><p>Bezeichne
</p>
<ul><li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{l}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{l}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4c951fcb4e4a883d93799fe8de37c70f01b8113.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.009ex;" alt="{\displaystyle x_{l}}" loading="lazy"></span> die <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>l</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle l.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d770377c901876265c4675eb010cd64bac0bb9aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.34ex; height:2.176ex;" alt="{\displaystyle l.}" loading="lazy"></span> von n Wiederholungsmessungen</li>
<li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"></span> den <a href="Wahrer_Wert" title="Wahrer Wert">wahren Wert</a> der <a href="Messgr%C3%B6%C3%9Fe" title="Messgröße">Messgröße</a></li>
<li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> den unbekannten systematischen Messfehler, der allen n Wiederholungsmessungen gemeinsam ist</li>
<li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{l}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{l}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ff106ded1adf91c43ece9c04aed1f66cbb16944.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.806ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{l}}" loading="lazy"></span> den zufälligen Messfehler.</li></ul>
<p>Dann liegt der Revision der Gaußschen <a href="Fehlerrechnung" title="Fehlerrechnung">Fehlerrechnung</a> eine Fehlergleichung folgender Form zugrunde:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{l}=x_{0}+f+\varepsilon _{l};\quad -f_{s}\leq f=const.\leq +f_{s}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>+</mo>
<mi>f</mi>
<mo>+</mo>
<msub>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>l</mi>
</mrow>
</msub>
<mo>;</mo>
<mspace width="1em"></mspace>
<mo>−<!-- − --></mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>s</mi>
</mrow>
</msub>
<mo>≤<!-- ≤ --></mo>
<mi>f</mi>
<mo>=</mo>
<mi>c</mi>
<mi>o</mi>
<mi>n</mi>
<mi>s</mi>
<mi>t</mi>
<mo>.</mo>
<mo>≤<!-- ≤ --></mo>
<mo>+</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>s</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{l}=x_{0}+f+\varepsilon _{l};\quad -f_{s}\leq f=const.\leq +f_{s}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a81f7d792e7358828d79edd399712a118ddac28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:43.98ex; height:2.509ex;" alt="{\displaystyle x_{l}=x_{0}+f+\varepsilon _{l};\quad -f_{s}\leq f=const.\leq +f_{s}}" loading="lazy"></span></dd></dl>
<p>Die <a href="Messwert" title="Messwert">Messwerte</a> einer stationär arbeitenden Messapparatur <a href="Fehlerrechnung#Normalverteilung" title="Fehlerrechnung">streuen</a> also <i>nicht</i> um den wahren Wert <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"></span> der Messgröße, sondern um den seitens des unbekannten systematischen Messfehlers <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> verschobenen Wert <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}+f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>+</mo>
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{0}+f}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b6e12defd598af50bbba40a8de956f25a09c7000.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.503ex; height:2.509ex;" alt="{\displaystyle x_{0}+f}" loading="lazy"></span>. Letzteren bezeichnet die <a href="Statistik" title="Statistik">Statistik</a> als <i>belasteten</i> oder <i>nicht <a href="Erwartungstreu" class="mw-redirect" title="Erwartungstreu">erwartungstreuen</a></i> <a href="Erwartungswert" title="Erwartungswert">Erwartungswert</a> – nicht erwartungstreu soll heißen: der wahre Wert <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"></span> wurde verfehlt.
</p><p>Weder <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"></span> noch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}+f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>+</mo>
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{0}+f}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b6e12defd598af50bbba40a8de956f25a09c7000.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.503ex; height:2.509ex;" alt="{\displaystyle x_{0}+f}" loading="lazy"></span> sind dem Experimentator bekannt.
</p>
<div class="mw-heading mw-heading2"><h2 id="Interpretation_unbekannter_systematischer_Fehler">Interpretation unbekannter systematischer Fehler</h2></div>
<p>Gegenwärtig werden unbekannte systematische Fehler unterschiedlich interpretiert:
</p>
<ul><li>Der <a href="GUM_(Norm)" title="GUM (Norm)">ISO-Guide</a> weist ihnen formal eine <a href="Zufallsvariable" title="Zufallsvariable">Zufallsvariable</a> zu und dieser, per <a href="Postulat" title="Postulat">Postulat</a>, eine Rechteckdichte, definiert über dem <a href="Intervall_(Mathematik)" title="Intervall (Mathematik)">Intervall</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -f_{s}\ldots f_{s}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>−<!-- − --></mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>s</mi>
</mrow>
</msub>
<mo>…<!-- … --></mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>s</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle -f_{s}\ldots f_{s}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb48c26f703f22079d7b2b89eb778e4fd62dbe2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.591ex; height:2.509ex;" alt="{\displaystyle -f_{s}\ldots f_{s}}" loading="lazy"></span>. <a href="Implizit" class="mw-redirect" title="Implizit">Implizit</a> wird unterstellt, es sei zulässig, die Realisierungen jener Zufallsvariable im Wesentlichen auf ihre einfache <a href="Standardabweichung" class="mw-redirect" title="Standardabweichung">Standardabweichung</a> zu begrenzen, d. h. auf ein Intervall der Länge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm f_{s}/{\sqrt {3}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>±<!-- ± --></mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>s</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \pm f_{s}/{\sqrt {3}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db5a80af808b8c16cc5eb69f48512c700380a531.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.211ex; height:3.009ex;" alt="{\displaystyle \pm f_{s}/{\sqrt {3}}}" loading="lazy"></span>. Wie zu zeigen ist, bringt diese Interpretation allerdings nichtbehebbare mathematische und physikalische Probleme mit sich; siehe <a href="Messunsicherheit" title="Messunsicherheit">Messunsicherheit</a>.</li>
<li>Die zum Guide alternative Revision der Gaußschen Fehlerrechnung betrachtet den unbekannten systematischen Messfehler – physikalisch realistischer – als zeitkonstante, durch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -f_{s}\ldots f_{s}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>−<!-- − --></mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>s</mi>
</mrow>
</msub>
<mo>…<!-- … --></mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>s</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle -f_{s}\ldots f_{s}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb48c26f703f22079d7b2b89eb778e4fd62dbe2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.591ex; height:2.509ex;" alt="{\displaystyle -f_{s}\ldots f_{s}}" loading="lazy"></span> eingegrenzte Größe, die im Sinne einer <a href="Worst-case" class="mw-redirect" title="Worst-case">worst-case</a>-Abschätzung zum Tragen zu bringen ist, d. h. in Gestalt der Intervallgrenzen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm f_{s}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>±<!-- ± --></mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>s</mi>
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<annotation encoding="application/x-tex">{\displaystyle \pm f_{s}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d215267aa4853f3864abc1b8351c2c51cefb5338.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.951ex; height:2.509ex;" alt="{\displaystyle \pm f_{s}}" loading="lazy"></span>. Diese Vorgehensweise bedingt eine neue Art der Fehlerrechnung, die sich nicht durch Fortschreiben der Gaußschen Fehlerrechnung darstellen lässt. Andererseits führt der neue Formalismus zu sicheren Messunsicherheiten, insbesondere zeigt er sich frei von <a href="Inkonsistenz" title="Inkonsistenz">Inkonsistenzen</a>.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li>Fehler (Statistik)</li>
<li><a href="Indexfehler" title="Indexfehler">Indexfehler</a></li>
<li><a href="Systematische_Abweichung" title="Systematische Abweichung">Drift</a></li>
<li>Verzerrung (Empirie)</li>
<li><a href="Designeffekt" class="mw-redirect" title="Designeffekt">Designeffekt</a></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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