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Statistik und wahrscheinlichkeit 1_cd_phst

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Statistik und wahrscheinlichkeit 1_cd_phst

  1. 1. Statistik und Wahrscheinlichkeit 1<br />LV-Leiter: Arno Raunegger<br />
  2. 2. Stochastik<br />Statistik<br />Wahrscheinlich-keitstheorie<br />Deskriptive (beschreibende) Statistik<br />Induktive (schließende) Statistik<br />
  3. 3. Begriffsklärung 1<br />Deskriptive Statistik: Ausgangspunkt sind Daten, die in<br />ihrer Rohform oft als Urlisten vorliegen. Meistens sind<br />diese Daten sehr umfangreich und im Einzelnen gar<br />nicht mehr überblickbar.<br />Um die Daten systematisch zu ordnen,<br />zu charakterisieren und übersichtlich<br />darzustellen wendet man die Methoden<br />der beschreibenden Statistik an.<br />
  4. 4. Begriffsklärung 2<br />Häufig wird in statistischen Erhebungen eine bestimmte Grundgesamtheit (eine z.B. Anzahl von Personen mittels Fragebogen) hinsichtlich bestimmter Variablen (Merkmale) befragt. Beispiele dazu sind:<br />
  5. 5. Ungeordnete/Geordnete Datenreihe<br />Bsp.: Gewicht von Schülern in einer Schulklasse<br />Ungeordnete Datenreihe (Urliste):<br />38, 29, 33, 31, 35, 30, 28, 33<br />Geordnete Datenreihe:<br />28, 29, 30, 31, 33, 33, 35, 38<br />Variable (Merkmal)<br />Variablenwerte (Merkmalsaus-prägungen)<br />
  6. 6. Absolute und relative Häufigkeiten<br />Absolute Häufigkeit ha:<br />28, 29, 30, 31, 33, 33, 35, 38<br />
  7. 7. Absolute und relative Häufigkeiten<br />Relative Häufigkeit:<br />Relative Häufigkeit in Prozent:<br />
  8. 8. Absolute und relative Häufigkeiten<br />Relative Häufigkeit:<br />
  9. 9. Anwendungsbeispiele<br />a) Geschlecht Seminargruppe:<br /> => Absolute, relative und prozentuelle Häufigkeit<br /> => Säulen- und Kreisdiagramm<br />b) Zweitfachverteilung Seminargruppe:<br /> => Absolute, relative und prozentuelle Häufigkeit<br /> => Säulen- und Kreisdiagramm<br />
  10. 10. Spannweite, Minimum und Maximum<br />Geordnete Datenreihe:<br />28, 29, 30, 31, 33, 33, 35, 38<br />Spannweite: <br />R = xMax - xMin<br />R =<br />xMin<br />xMax<br />
  11. 11. Arithmetischer Mittelwert<br />Geordnete Datenreihe:<br />28, 29, 30, 31, 33, 33, 35, 38<br />
  12. 12. Modalwert (Modus)<br /><ul><li>Der Modalwert (Modus) ist jener Wert, welcher am häufigsten in einer Datenreihe vorkommt.</li></ul>28, 29, 30, 31, 33, 33, 35, 38<br />Es ist auch möglich, dass eine Datenreihe mehrere Modalwerte enthält.<br />m = 32<br />
  13. 13. Median (Zentralwert)<br /><ul><li>Der Median (Zentralwert) teilt eine sortierte Datenreihe in genau zwei Hälften.</li></ul>28, 29, 30, 31, 33, 33, 35, 38<br />Bei einer geraden Anzahl von Elementen in einer geordneten Datenreihe, ergibt sich der Median aus dem arithmetischen Mittelwert der beiden zentralen Elemente.<br />z = 32<br />
  14. 14. Median (Zentralwert)<br />Ungerade Anzahl von Datenelementen:<br />28, 29, 30, 31, 33, 33, 35, 38, 40<br />Bei einer ungeraden Anzahl von Elementen in einer geordneten Datenreihe, ergibt sich der Median genau aus dem mittleren Element.<br />xMed=33<br />
  15. 15. Robustheit von Zentralmaßen<br />Gehaltsstruktur in einer Firma:<br />1000, 1000, 1200, 1200, 1200, 1200, 1500, 4000<br />„Ausreißer“ bereinigen:<br />1000, 1000, 1200, 1200, 1200, 1200, 1500, 4000<br />x = <br />z = <br />m = <br />x = <br />z = <br />m = <br />
  16. 16. Quartile<br />Mit den Quartilen q0, q1, q2, q3, q4 wird eine geordnete Datenreihe „geviertelt“.<br />xMin<br />xMax<br />geordnete Datenreihe<br />q0<br />q1<br />q2<br />q3<br />q4<br />
  17. 17. Quartile<br />Es gilt definitionsgemäß: <br />q2 = z<br />q3 und q1 sind die Mediane der oberen und unteren Hälfte.<br />q3=zo<br />q4=xMax<br />q0=xMin<br />q1=zu<br />q2=z<br />
  18. 18. Quartile<br />Gerade Anzahl von Datenelementen:<br />28, 29, 30, 31, 33, 33, 35, 38<br />q2=z  q2=<br />q1=zu q1=<br />q3=zo q3=<br />q2=z<br />q3<br />q1<br />
  19. 19. Quartile<br />Ungerade Anzahl von Datenelementen:<br />28, 29, 30, 31, 33, 33, 35, 38, 40<br />q2=z  q2=<br />q1=zu q1=<br />q3=zo q3=<br />q2=z<br />q3<br />q1<br />
  20. 20. Grafische Darstellung von Quartilen<br />Die grafische Darstellung von Quartilen erfolgt in so genannten Kastenschau-bildern (Boxplots)<br />Analog kann man geordnete Listen auch z.B. in 10 Teile (Dezile) oder 100 Teile (Centile) teilen.<br />Gewicht / kg<br />40<br />Klasse 1a<br />Klasse 1b<br />36,5<br />33<br />29,5<br />Schulklasse<br />28<br />
  21. 21. Standardabweichung<br />Die Standardabweichung s ist ein Maß für die Streuung der Merkmalswerte xi um den arithmetischen Mittelwert .<br />
  22. 22. Standardabweichung<br />Geordnete Datenreihe:<br />28, 29, 30, 31, 33, 33, 35, 38<br />Arithm. Mittelwert<br />
  23. 23. Varianz<br />Die Standardabweichung s ergibt sich aus der Quadratwurzel der Varianz, daher gilt für die Varianz:<br />Häufig ist in der Literatur auch folgende Formel zu finden:<br />Stichproben-umfang „Ausreißer bereinigt“<br />

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