Add first chapter

2023-01-20 00:01:37 +01:00 · 2023-01-20 00:01:37 +01:00 · ab05595812
commit ab05595812
parent 52a5cbc553
8 changed files with 1052 additions and 2 deletions
--- a/Statistik/Boxplot.png
+++ b/Statistik/Boxplot.png
--- a/Statistik/Diagrams.ipynb
+++ b/Statistik/Diagrams.ipynb
--- a/Statistik/Histogram.png
+++ b/Statistik/Histogram.png
--- a/Statistik/Probabilities.png
+++ b/Statistik/Probabilities.png
--- a/Statistik/Quantiles.png
+++ b/Statistik/Quantiles.png
--- a/Statistik/Summary.md
+++ b/Statistik/Summary.md
@ -1,9 +1,22 @@
+<style>
+  img {
+    max-height: 300px;
+  }
+</style>
+
 # Stochastik und Statistik

 - [Stochastik und Statistik](#stochastik-und-statistik)
  - [Deskriptive Statistik](#deskriptive-statistik)
    - [Merkmals-Typen](#merkmals-typen)
-  - [Häufigkeiten](#häufigkeiten)
+    - [Häufigkeiten](#häufigkeiten)
+      - [Klassierte Stichproben im Histogramm](#klassierte-stichproben-im-histogramm)
+    - [Kennwerte](#kennwerte)
+      - [Quantile](#quantile)
+      - [Boxplot](#boxplot)
+      - [Lage-Kennwerte](#lage-kennwerte)
+      - [Streuungs-Kennwerte](#streuungs-kennwerte)
+      - [Lagewerte in Klassierten Daten](#lagewerte-in-klassierten-daten)
 - [Glossar](#glossar)

 ## Deskriptive Statistik
@ -66,7 +79,7 @@ Frage: Wieviel kostet ein Mars-Riegel?
 | ----------------------- | ----------- | ------------------------------------------------------------------- |
 | Beliebiger Preis in CHF | Diskret     | Beträge, die nicht durch 5 Rappen teilbar sind, sind nicht möglich. |

-## Häufigkeiten
+### Häufigkeiten
 Eine Häufigkeit ist die Anzahl Male, die ein Merkmalsträger in der Stichprobe eine bestimmte Eigenschaft erfüllt.
 Diese kann auf verschiedene Weisen dargestellt werden.

@ -84,6 +97,14 @@ Die Funktion für die Häufigkeitsfunktion (auch genannt: Dichtefunktion) hat fo
  - Für diskrete Merkmale: _PMF_ (probability mass function)
  - Für stetige Merkmale: _PDF_ (probability density function)

+Zudem gibt es folgende Verteilungsfunktionen:
+  - $H(x)$ Absolute Summenhäufigkeit: Anzahl Merkmalträger mit Merkmal $x_i$ mit $x_i < x$
+  - $F(x)$ Kummulative Verteilungsfunktion _CDF_: Relative Häufigkeit der Merkmalträger mit Merkmal $x_i$ mit $x_i < x$
+
+Beispiele der wichtigsten Häufigkeits-Funktionen:
+
+![](./Probabilities.png)
+
 <div class="letters">

  - $f_i$: Relative Häufigkeit
@ -92,5 +113,112 @@ Die Funktion für die Häufigkeitsfunktion (auch genannt: Dichtefunktion) hat fo

 </div>

+#### Klassierte Stichproben im Histogramm
+Klassierte Stellen werden durch Grösse der Klasse geteilt, um diese zu berücksichtigen.
+
+Daraus gewonnene Daten können in einem Histogramm dargestellt werden.
+
+Beispiel:
+| Klasse                       | $[100,200[$                | $[200,500[$                 | $[500,800[$                 | $[800,1000[$               | $[1000,2000[$                | Total |
+| ---------------------------- | -------------------------- | --------------------------- | --------------------------- | -------------------------- | ---------------------------- | ----- |
+| Absolute Häufigkeit          | $35$                       | $182$                       | $317$                       | $84$                       | $132$                        | $750$ |
+| Relative Häufigkeit          | $\frac{35}{750}$           | $\frac{182}{750}$           | $\frac{317}{750}$           | $\frac{84}{750}$           | $\frac{132}{750}$            | $1$   |
+| Klassen-Grösse               | $100$                      | $300$                       | $300$                       | $200$                      | $1000$                       |
+| Säulenhöhe für Absolut       | $\frac{35}{100}$           | $\frac{182}{300}$           | $\frac{317}{300}$           | $\frac{84}{200}$           | $\frac{132}{1000}$           |       |
+| Säulenhöhe für Relativ (PDF) | $\frac{35}{750 \cdot 100}$ | $\frac{182}{750 \cdot 300}$ | $\frac{317}{750 \cdot 300}$ | $\frac{84}{750 \cdot 200}$ | $\frac{132}{750 \cdot 1000}$ |       |
+
+Histogramm der genannten Daten:
+
+![](./Histogram.png)
+
+### Kennwerte
+#### Quantile
+Ein $q$-Quantil definiert den Wert des $\lceil n \cdot q \rceil$-te Element.
+
+Folgende bekannte $q$-Quantile gibt es:
+  - $0.25$-Quantil: Das 1. Quartil
+  - $0.50$-Quantil: Das 2. Quartil auch "Median" oder "Zentralwert"
+  - $0.75$-Quantil: Das 3. Quartil
+
+Beispiel einer Statistik mit eingezeichneten Quartilen:
+
+![](./Quantiles.png)
+
+#### Boxplot
+  - Boxplots zeigen folgende Informationen
+    - Das 1. Quartil $Q_1$
+    - Den Median $Q_2$
+    - Das 3. Quartil $Q_3$
+    - Den Minimalwert (min. $1.5 \cdot (Q_3 - Q_1)$)
+    - Den Maximalwert (max. $1.5 \cdot (Q_3 - Q_1)$)
+
+![](Boxplot.png)
+
+#### Lage-Kennwerte
+  - Arithmetisches Mittel $\overline{x}$: Mittelwert der Stichprobenwerte
+  - Median $x_\text{med}$: Wert des 2. Quartil $Q_2$
+  - Modus $x_\text{mod}$: Der häufigste Wert in der Stichprobe
+
+#### Streuungs-Kennwerte
+
+<div class="formula">
+
+***Varianz $\tilde{s}^2$:***
+
+$$\tilde{s}^2 = \left(\frac{1}{n} \cdot \sum_{i = 1}^m{a_i^2 \cdot h_i}\right) - \tilde{x}^2$$
+
+alternative Schreibweisen:
+
+$$\begin{aligned}
+ \tilde{s}^2 &= \frac{1}{n} \cdot \sum_{i = 1}^n(x_i - \overline{x})^2 = \frac{1}{n} \cdot \sum_{i = 1}^m{h_i \cdot (a_i - \overline{x})^2} \\
+ &= \left(\frac{1}{n} \cdot \sum_{i = 1}^n x_i^2\right) - \overline{x}^2 = \left(\sum_{i = 1}^m{a_i^2 \cdot f_i}\right) - \overline{x}^2
+\end{aligned}
+$$
+
+***Standardabweichung $\tilde{s}$:***
+
+$$\tilde{s} = \sqrt{\tilde{s}^2}$$
+
+***Korrigierte Varianz $s^2$:***
+
+$$s^2 = \frac{n}{n - 1} \tilde{s}^2$$
+
+***Korrigierte Standardabweichung $s$:***
+
+$$s = \sqrt{s^2} = \sqrt{\frac{n}{n - 1}} \cdot \tilde{s}$$
+
+***Interquartilsabstand $IQR$:***
+
+$$IQR = Q_3 - Q_1$$
+
+</div>
+
+<div class="letters">
+
+  - $a_i$: $i$-te Merkmals-Ausprägung
+  - $m$: Anzahl unterschiedlicher Merkmals-Ausprägungen (oder Klassen)
+
+</div>
+
+#### Lagewerte in Klassierten Daten
+Einige Werte berechnen sich speziell in klassierten Daten.
+
+***Quantile:***
+
+Der Wert $R_q$ eines $q$-Quantils berechnet sich, indem folgendes durchgeführt wird:
+
+ 1. Erste Klasse $K$ mit einem $CDF > q$ finden
+ 2. $K_0$ auf untere und $K_1$ auf obere Grenze der Klasse $K$ setzen
+ 3. Folgendes berechnen:
+    $$R_q = K_0 + \frac{(K_1 - K_0) \cdot (q - CDF(K_0))}{CDF(K_1) - CDF(K_0)}$$
+ 4. $R_q$ entspricht nun dem $q$-Quartil
+
+***Modus:***
+
+ 1. Klasse $K$ mit grösster relativer Häufigkeit bestimmen
+ 2. $K_0$ auf untere und $K_1$ auf obere Grenze der Klasse $K$ setzen
+ 3. Folgendes berechnen:
+    $$x_\text{mod} = K_0 + \frac{K_1 - K_0}{2}$$
+
 # Glossar
  - Univariate Daten: Daten, welche nur ein Merkmal haben
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