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<title>Intervallskala</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Intervallskala</span></h1>

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<p>Die <b>Intervallskala</b> (eine von drei <b>Kardinalskalen</b>) ist ein <a href="Skalenniveau" title="Skalenniveau">Skalenniveau</a> in der <a href="Statistik" title="Statistik">Statistik</a>. Sie zählt zum <i><a href="Kennzahl" title="Kennzahl">metrischen</a> Messniveau</i>, da sich die Ausprägungen dieses Skalenniveaus <a href="Quantit%C3%A4t" title="Quantität">quantitativ</a> mittels Zahlen darstellen lassen. Insbesondere bedeutet das auch, dass <a href="Rangordnung" title="Rangordnung">Rangunterschiede</a> und Abstand zwischen Werten gemessen werden können; das heißt, quantitative Merkmale gehen in ihren Anforderungen über <a href="Ordinalskala" title="Ordinalskala">ordinale</a> oder gar <a href="Nominalskala" title="Nominalskala">nominale</a> Eigenschaften hinaus.

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<div class="mw-heading mw-heading2"><h2 id="Beschreibung">Beschreibung</h2></div>

<p>Bei intervallskalierten Merkmalen lassen sich zusätzlich zu den Eigenschaften der <a href="Ordinalskala" title="Ordinalskala">Ordinalskala</a> die Abstände zwischen den verschiedenen Merkmalsausprägungen exakt bestimmen.

</p><p>Allerdings existiert kein natürlicher <a href="Nullpunkt" title="Nullpunkt">Nullpunkt</a> für die Intervallskala. Willkürlich definierte Nullpunkte, wie z. B. bei der <a href="Grad_Celsius" title="Grad Celsius">Grad-Celsius</a>-Temperaturskala, zählen nicht zu den natürlichen Nullpunkten, während der Nullpunkt der <a href="Kelvin" title="Kelvin">Kelvin</a>-Temperaturskala, der dem <a href="Absoluter_Nullpunkt" title="Absoluter Nullpunkt">absoluten Nullpunkt</a> entspricht, ein natürlicher Nullpunkt ist.

</p><p>Der Unterschied lässt sich daran ablesen, dass 20 °C nicht doppelt so viel bedeuten wie 10 °C (z. B. doppelt so viel Hitze). Bei Kelvin hingegen stehen die Zahlwerte tatsächlich im Verhältnis: 20 Kelvin bedeuten auch doppelt so viel Energie wie 10 Kelvin. Die Celsius-Temperaturskala ist also intervallskaliert, Kelvin-Temperaturangaben sogar auf <a href="Verh%C3%A4ltnisskala" title="Verhältnisskala">Verhältnisskalenniveau</a>, dem nächsthöheren und zugleich höchsten <a href="Skalenniveau" title="Skalenniveau">Skalenniveau</a>. Beide Skalen gehören dabei zu den metrischen Skalenniveaus (auch <i>Kardinalskala</i>).

</p><p>Sind zwei Datenpaare (a,b) und (c,d) äquivalent (siehe unten), dann ist bei Intervallskalen der <a href="Quotient" title="Quotient">Quotient</a> aus <a href="Subtraktion" title="Subtraktion">Differenzen</a> (a−b)/(c−d) immer gleich.

</p><p>Zulässige Aussagen bei Intervallskalen lassen sich an folgendem Beispiel illustrieren. Dabei werden zwei Intervallskalen in einem zweiten Schritt in ein Verhältnis gesetzt (<a href="Verh%C3%A4ltnisskala" title="Verhältnisskala">Verhältnisskala</a>). Dies entspricht einer weiteren Datenverarbeitung der Intervallskala: Wir kennen die Temperaturen von Tag A, Tag B und Tag C. Jetzt bilden wir das Verhältnis der Differenzen: (A−B)/(A−C). Angenommen, das Verhältnis ist 2. Dann wäre eine zulässige Aussage: „Der Temperaturunterschied zwischen Tag A und B ist doppelt so groß wie der Temperaturunterschied zwischen Tag A und C.“

</p><p>Jede Intervallskala ist so geartet, dass die Rangfolge der Differenz zwischen Zahlen gleich der Rangfolge der Merkmalsunterschiede zwischen den entsprechenden Objekten ist.

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<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>

<p>Beispiele für intervallskalierte Merkmale mit einer mathematischen Paarbildung <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 S\times S}">

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<annotation encoding="application/x-tex">{\displaystyle S\times S}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b0f88969df0adf75522ab2d85717167d1069d28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.839ex; height:2.176ex;" alt="{\displaystyle S\times S}" loading="lazy"></span> aus der Skala <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 S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"></span> sind:

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<ul><li>Temperatur auf der Celsius-Skala mit <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 ({\text{Temperatur}}_{12\colon 00{\text{Uhr}}},{\text{Temperatur}}_{6\colon 00{\text{Uhr}}})\in S\times S}">

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<mtext>Temperatur</mtext>

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<mtext>Temperatur</mtext>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\text{Temperatur}}_{12\colon 00{\text{Uhr}}},{\text{Temperatur}}_{6\colon 00{\text{Uhr}}})\in S\times S}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ba6a023c3439162a2b6e8a3ef19a8135fe54dfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.546ex; height:2.843ex;" alt="{\displaystyle ({\text{Temperatur}}_{12\colon 00{\text{Uhr}}},{\text{Temperatur}}_{6\colon 00{\text{Uhr}}})\in S\times S}" loading="lazy"></span></li>

<li>Jahreszahlen mit <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 ({\text{Sterbejahr}},{\text{Geburtsjahr}})\in S\times S}">

<mtext>Sterbejahr</mtext>

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<mtext>Geburtsjahr</mtext>

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<annotation encoding="application/x-tex">{\displaystyle ({\text{Sterbejahr}},{\text{Geburtsjahr}})\in S\times S}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5953b81be933060d64b19aef3f3ae7f00da1f765.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.32ex; height:2.843ex;" alt="{\displaystyle ({\text{Sterbejahr}},{\text{Geburtsjahr}})\in S\times S}" loading="lazy"></span></li>

<li><a href="Zeitpunkt" title="Zeitpunkt">Zeitpunkte</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 ({\text{Zielzeit}},{\text{Startzeit}})\in S\times S}">

<mtext>Zielzeit</mtext>

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<mtext>Startzeit</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\text{Zielzeit}},{\text{Startzeit}})\in S\times S}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c406701dbf7bdfc088d839bc19518d3a8fc8ff26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.678ex; height:2.843ex;" alt="{\displaystyle ({\text{Zielzeit}},{\text{Startzeit}})\in S\times S}" loading="lazy"></span></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Mögliche_Operationen"><span id="M.C3.B6gliche_Operationen"></span>Mögliche Operationen</h2></div>

<p>Zusätzlich zu Größenvergleichen sind Differenzen und Summen aus intervallskalierten Merkmalen sinnvoll, da hier die Abstände zwischen den einzelnen Merkmalsausprägungen exakt definiert sind. Damit lassen sich hier auch <a href="Mittelwert" title="Mittelwert">Durchschnittswerte</a> berechnen.

Aufgrund des fehlenden Nullpunkts stellt die <a href="Multiplikation" title="Multiplikation">Multiplikation</a> keine sinnvolle Operation für intervallskalierte Merkmale dar.

</p><p>Ein Beispiel:

</p><p>War es gestern 10 Grad <a href="Celsius" class="mw-redirect" title="Celsius">Celsius</a> warm, und heute sind es zwanzig Grad, dann kann man zwar behaupten: „Es ist zehn Grad Celsius wärmer“, aber nicht: „Es ist doppelt so warm wie gestern“. Dies wird besonders deutlich, wenn man Celsius in <a href="Kelvin" title="Kelvin">Kelvin</a> oder <a href="Grad_Fahrenheit" title="Grad Fahrenheit">Grad Fahrenheit</a> umrechnet.

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<div class="mw-heading mw-heading2"><h2 id="Erlaubte_Transformationen">Erlaubte Transformationen</h2></div>

<p>Zulässig sind positiv-<a href="Affine_Abbildung" title="Affine Abbildung">lineare Transformationen</a> der Art <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 y=\alpha x+\beta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=\alpha x+\beta }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/67882b22bda7d6f894842d8935e3549030832134.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.244ex; height:2.509ex;" alt="{\displaystyle y=\alpha x+\beta }" loading="lazy"></span>

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<div class="mw-heading mw-heading2"><h2 id="Mathematische_Deutung">Mathematische Deutung</h2></div>

<p>Aus mathematischer Sicht ist eine <b>Intervallskala</b> <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 S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"></span> eine <a href="Menge_(Mathematik)" title="Menge (Mathematik)">Menge</a>, für die Folgendes gilt:

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<ol><li>Es existiert eine <a href="%C3%84quivalenzrelation" title="Äquivalenzrelation">Äquivalenzrelation</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 E\subseteq P\times P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E\subseteq P\times P}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c9e3c97b7e0cc649521e516ea857b1664669e72a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.205ex; height:2.343ex;" alt="{\displaystyle E\subseteq P\times P}" loading="lazy"></span> mit <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 P:=S\times S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P:=S\times S}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dcfe35c4f51f99327bf605eaf5317e9a08dae1f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.33ex; height:2.176ex;" alt="{\displaystyle P:=S\times S}" loading="lazy"></span> (Menge der Paare aus <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 S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"></span>). <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 E=\left\{\left(m,n\right)\vert m=(m_{1},m_{2})\in P\wedge n=(n_{1},n_{2})\in P\wedge m_{1}-m_{2}=n_{1}-n_{2}\right\}}">

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

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</mrow>

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</mrow>

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</mrow>

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</mrow>

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</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E=\left\{\left(m,n\right)\vert m=(m_{1},m_{2})\in P\wedge n=(n_{1},n_{2})\in P\wedge m_{1}-m_{2}=n_{1}-n_{2}\right\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd0d48464872208ca80abf827a727fd4738cc377.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:75.121ex; height:2.843ex;" alt="{\displaystyle E=\left\{\left(m,n\right)\vert m=(m_{1},m_{2})\in P\wedge n=(n_{1},n_{2})\in P\wedge m_{1}-m_{2}=n_{1}-n_{2}\right\}}" loading="lazy"></span> (Nominalskalen-Eigenschaft). Bezogen auf das Beispiel <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 ({\text{Zielzeit}},{\text{Startzeit}})\in P=S\times S}">

<mtext>Zielzeit</mtext>

</mrow>

<mtext>Startzeit</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\text{Zielzeit}},{\text{Startzeit}})\in P=S\times S}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/207ed45dc0f7b36758e612a9260c874ede981ac8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.521ex; height:2.843ex;" alt="{\displaystyle ({\text{Zielzeit}},{\text{Startzeit}})\in P=S\times S}" loading="lazy"></span> werden alle Paare <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 ({\text{Zielzeit}},{\text{Startzeit}})}">

<mtext>Zielzeit</mtext>

</mrow>

<mtext>Startzeit</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\text{Zielzeit}},{\text{Startzeit}})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4c0fc4c1048868b2a507cd9041b524dc73aa5490.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.998ex; height:2.843ex;" alt="{\displaystyle ({\text{Zielzeit}},{\text{Startzeit}})}" loading="lazy"></span> zu einer Äquivalenzklasse zusammengefasst, die die gleiche Zeitdauer benötigt haben, also z. B. <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 m=(m_{1},m_{2})=(20,7)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m=(m_{1},m_{2})=(20,7)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eebc111e3f1c61c8365fe8985cb9b31c6c11a7ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.6ex; height:2.843ex;" alt="{\displaystyle m=(m_{1},m_{2})=(20,7)}" loading="lazy"></span> und <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 n=(n_{1},n_{2})=(30,17)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=(n_{1},n_{2})=(30,17)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/336dfabc28341fb08d6a436aaecfd52255fd0aef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.826ex; height:2.843ex;" alt="{\displaystyle n=(n_{1},n_{2})=(30,17)}" loading="lazy"></span> sind in einer Äquivalenzklasse (formal: <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 (m,n)\in E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (m,n)\in E}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e6b84a7582f26727aba1fb206be0b84bf616dfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.895ex; height:2.843ex;" alt="{\displaystyle (m,n)\in E}" loading="lazy"></span>), weil beide Datenpaare <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 m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span> und <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 n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> die gleiche Zeitdauer zwischen Start und Ziel benötigt haben. Siehe auch Differenzfunktion.</li>

<li>Es existiert eine <a href="Lineare_Ordnungsrelation" class="mw-redirect" title="Lineare Ordnungsrelation">lineare Ordnungsrelation</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 O\subseteq P\times P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O\subseteq P\times P}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24e956a1d25c777d6f44d7f00013377b8d61e1de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.203ex; height:2.343ex;" alt="{\displaystyle O\subseteq P\times P}" loading="lazy"></span> mit <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 P:=S\times S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P:=S\times S}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dcfe35c4f51f99327bf605eaf5317e9a08dae1f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.33ex; height:2.176ex;" alt="{\displaystyle P:=S\times S}" loading="lazy"></span> (Ordinalskalen-Eigenschaft). <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 O=\left\{\left(m,n\right)\vert m=(m_{1},m_{2})\in P=S\times S\wedge n=(n_{1},n_{2})\in P=S\times S\wedge m_{1}-m_{2}\leq n_{1}-n_{2}\right\}}">

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O=\left\{\left(m,n\right)\vert m=(m_{1},m_{2})\in P=S\times S\wedge n=(n_{1},n_{2})\in P=S\times S\wedge m_{1}-m_{2}\leq n_{1}-n_{2}\right\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40cd50ce20d6e06558caf60edb29c8240e5c7e54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:92.993ex; height:2.843ex;" alt="{\displaystyle O=\left\{\left(m,n\right)\vert m=(m_{1},m_{2})\in P=S\times S\wedge n=(n_{1},n_{2})\in P=S\times S\wedge m_{1}-m_{2}\leq n_{1}-n_{2}\right\}}" 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 \leq }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \leq }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/440568a09c3bfdf0e1278bfa79eb137c04e94035.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \leq }" loading="lazy"></span>-Beziehung kann z. B. auch durch eine andere Ordnungsrelation auf der Differenz in <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 D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span> ersetzt werden (z. B. <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 \geq }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \geq }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bcef7c0e95bb77a35fd1a874ca91f425215f3c26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \geq }" loading="lazy"></span>), wenn die mathematischen Eigenschaften der Ordnungsrelation erhalten bleiben. Bezogen auf das Beispiel <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 ({\text{Zielzeit}},{\text{Startzeit}})\in P=S\times S}">

<mtext>Zielzeit</mtext>

</mrow>

<mtext>Startzeit</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\text{Zielzeit}},{\text{Startzeit}})\in P=S\times S}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/207ed45dc0f7b36758e612a9260c874ede981ac8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.521ex; height:2.843ex;" alt="{\displaystyle ({\text{Zielzeit}},{\text{Startzeit}})\in P=S\times S}" loading="lazy"></span> werden alle Paare <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 ({\text{Zielzeit}},{\text{Startzeit}})}">

<mtext>Zielzeit</mtext>

</mrow>

<mtext>Startzeit</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\text{Zielzeit}},{\text{Startzeit}})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4c0fc4c1048868b2a507cd9041b524dc73aa5490.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.998ex; height:2.843ex;" alt="{\displaystyle ({\text{Zielzeit}},{\text{Startzeit}})}" loading="lazy"></span> bezogen auf die Zeitdifferenz geordnet, also z. B..mit <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 m=(m_{1},m_{2})=(40,37)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m=(m_{1},m_{2})=(40,37)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/39224531a0485f2dc975a27721d315b031ed67c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.763ex; height:2.843ex;" alt="{\displaystyle m=(m_{1},m_{2})=(40,37)}" loading="lazy"></span> und <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 n=(n_{1},n_{2})=(30,7)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=(n_{1},n_{2})=(30,7)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a1cfe1de8e28088dfc3d5097cd3c12aea7f76a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.663ex; height:2.843ex;" alt="{\displaystyle n=(n_{1},n_{2})=(30,7)}" loading="lazy"></span> wäre <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 (m,n)\in O}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (m,n)\in O}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4887a9a44530dbc5d93e012cbc5e40268dd8eab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.892ex; height:2.843ex;" alt="{\displaystyle (m,n)\in O}" loading="lazy"></span> (<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 m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span> ist kleiner als <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 n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>), weil <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 m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span> weniger Zeit zwischen Start und Ziel benötigt hat als <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 n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>. Es wird eine Ordnungsrelation auf der Menge der Zahlenpaare in <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 P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> über die Differenz der Komponenten von <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 m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span> bzw. <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 n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> definiert (siehe nachfolgende Definition der Differenzfunktion auf <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 S\times S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S\times S}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b0f88969df0adf75522ab2d85717167d1069d28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.839ex; height:2.176ex;" alt="{\displaystyle S\times S}" loading="lazy"></span>).</li>

<li>Intervallskalen-Eigenschaft:

<ol><li>Es existiert eine Funktion <b>(Differenzfunktion)</b> <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 \Box -\Box :S\times S\longrightarrow D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Box -\Box :S\times S\longrightarrow D}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0bdebc2ef09ace2279c924b2cb78a13a5a9e6a27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.253ex; height:2.343ex;" alt="{\displaystyle \Box -\Box :S\times S\longrightarrow D}" loading="lazy"></span> (Man kann <a href="Mengenlehre#Differenz_und_Komplement" title="Mengenlehre">Differenzen</a> bilden, z. B. <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 ({\text{Zielzeit}},{\text{Startzeit}})\in S\times S}">

<mtext>Zielzeit</mtext>

</mrow>

<mtext>Startzeit</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\text{Zielzeit}},{\text{Startzeit}})\in S\times S}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c406701dbf7bdfc088d839bc19518d3a8fc8ff26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.678ex; height:2.843ex;" alt="{\displaystyle ({\text{Zielzeit}},{\text{Startzeit}})\in S\times S}" loading="lazy"></span> wird der Zeitdauer <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 d={\text{Zielzeit}}-{\text{Startzeit}}\in D}">

<mtext>Zielzeit</mtext>

</mrow>

<mtext>Startzeit</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d={\text{Zielzeit}}-{\text{Startzeit}}\in D}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1de5ef69c7f486c92b0bace402c8afa49fca39ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:28.074ex; height:2.343ex;" alt="{\displaystyle d={\text{Zielzeit}}-{\text{Startzeit}}\in D}" loading="lazy"></span> zugeordnet).</li>

<li>Es existiert eine <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktion</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 \Box +\Box :S\times D\longrightarrow S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Box +\Box :S\times D\longrightarrow S}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/843b67b21274ddf13e0be7af550b7ed3f71be53a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.253ex; height:2.343ex;" alt="{\displaystyle \Box +\Box :S\times D\longrightarrow S}" loading="lazy"></span> (Man kann die Differenzen wieder auf Ausprägungen von <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 S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"></span> addieren), für die außerdem gilt:

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \forall \left(m\in S\right):\left(m+0=m\right)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5fb4ab164b7865a023b1eb03b043d42f3e732040.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.798ex; height:2.843ex;" alt="{\displaystyle \forall \left(m\in S\right):\left(m+0=m\right)}" loading="lazy"></span> (Das Addieren von Null bringt keine Änderung)</li>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \forall \left(m_{0}\in S\right):\forall \left(m_{1}\in S\right):\left(m_{0}+\left(m_{1}-m_{0}\right)=m_{1}\right)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c4bdef2bdbc2018ff87bd6114a72062b86679576.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.498ex; height:2.843ex;" alt="{\displaystyle \forall \left(m_{0}\in S\right):\forall \left(m_{1}\in S\right):\left(m_{0}+\left(m_{1}-m_{0}\right)=m_{1}\right)}" loading="lazy"></span> (Differenzbildung ist konsistent mit Addierung).</li>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \forall \left(d_{0}\in D\right):\forall \left(d_{1}\in D\right):\forall \left(m\in S\right):\left(\left(m+d_{0}\right)+d_{1}=m+\left(d_{0}+d_{1}\right)\right)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e892bae0f15d9fdc23eb0bb4c4a5ad26d4a7c31b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:69.736ex; height:2.843ex;" alt="{\displaystyle \forall \left(d_{0}\in D\right):\forall \left(d_{1}\in D\right):\forall \left(m\in S\right):\left(\left(m+d_{0}\right)+d_{1}=m+\left(d_{0}+d_{1}\right)\right)}" loading="lazy"></span> (eine Art einseitiges <a href="Assoziativgesetz" title="Assoziativgesetz">Assoziativgesetz</a>)</li></ol></li>

<li>Die Menge der Differenzen <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 D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span> ist den <a href="Reelle_Zahlen" class="mw-redirect" title="Reelle Zahlen">reellen Zahlen</a> in folgender Hinsicht ähnlich:

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(D,\Box +\Box \right)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5dec95a068fbea67988c7f386327d5a02ab374c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.224ex; height:2.843ex;" alt="{\displaystyle \left(D,\Box +\Box \right)}" loading="lazy"></span> ist ein <a href="Untermonoid" class="mw-redirect" title="Untermonoid">Untermonoid</a> von <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 \left(\mathbb {R} ,\Box +\Box \right)}">

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(\mathbb {R} ,\Box +\Box \right)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b91c2faad2605c4b7e482e6f7940b3ff311be9f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.978ex; height:2.843ex;" alt="{\displaystyle \left(\mathbb {R} ,\Box +\Box \right)}" loading="lazy"></span> (<a href="Reelle_Zahlen" class="mw-redirect" title="Reelle Zahlen">reelle Zahlen</a> mit der <a href="Addition" title="Addition">Addition</a>).</li></ol></li></ol></li></ol>

<p>Jedes Element <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 m\in S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m\in S}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ae56e325b3a04ac26a9b18523cf6803ebdd042a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.38ex; height:2.176ex;" alt="{\displaystyle m\in S}" loading="lazy"></span> heißt <a href="Auspr%C3%A4gung" class="mw-redirect" title="Ausprägung">Ausprägung</a> von <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 S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"></span>.

</p><p>Jede <i>Intervallskala</i> <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 (S,-)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (S,-)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/873bf1a0b4d7ca1475463234b6bbee4fb3825799.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.151ex; height:2.843ex;" alt="{\displaystyle (S,-)}" loading="lazy"></span> ist eine <a href="Ordinalskala" title="Ordinalskala">Ordinalskala</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 S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"></span> mit einer Differenzfunktion <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 -}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04bd52ce670743d3b61bec928a7ec9f47309eb36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}" loading="lazy"></span> auf <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 S\times S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S\times S}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b0f88969df0adf75522ab2d85717167d1069d28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.839ex; height:2.176ex;" alt="{\displaystyle S\times S}" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Ordinales_Merkmal" class="mw-redirect" title="Ordinales Merkmal">ordinales Merkmal</a></li>

<li><a href="Nominales_Merkmal" class="mw-redirect" title="Nominales Merkmal">nominales Merkmal</a></li></ul>

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<div class="klappleiste-kopf"><a href="Skalenniveau" title="Skalenniveau">Skalenniveaus</a></div>

<p><a href="Nominalskala" title="Nominalskala">Nominalskala</a> |

<a href="Ordinalskala" title="Ordinalskala">Ordinalskala</a> |

<a class="mw-selflink selflink">Intervallskala</a> |

<a href="Verh%C3%A4ltnisskala" title="Verhältnisskala">Verhältnisskala</a> |

<a href="Absolutskala" title="Absolutskala">Absolutskala</a>

</p>

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