Some corrections
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@ -21,7 +21,7 @@
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%%%%% section style %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage[sf,bf,it,big,clearempty]{titlesec}
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\setcounter{secnumdepth}{-1}
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\setcounter{secnumdepth}{1}
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%%%%% units %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -144,7 +144,7 @@
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%%%%% equation references %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\eqref}[1]{(\ref{#1})}
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%\newcommand{\eqref}[1]{(\ref{#1})}
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\newcommand{\eqn}{Eq.}
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\newcommand{\Eqn}{Eq.}
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\newcommand{\eqns}{Eqs.}
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@ -229,7 +229,7 @@
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\chapter{\tr{Descriptive statistics}{Deskriptive Statistik}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Statistics of real-valued data}
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%\section{Statistics of real-valued data}
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\begin{itemize}
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\item Location, central tendency
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@ -259,7 +259,7 @@
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\end{itemize}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Median, Quartile, Percentile}
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\section{\tr{Median, quartile, etc.}{Median, Quartil, etc.}}
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\begin{figure}[t]
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\includegraphics[width=1\textwidth]{median}
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@ -307,7 +307,7 @@
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{Schreibe eine Funktion, die das erste, zweite und dritte Quartil als Vektor zur\"uckgibt.}
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\end{exercise}
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\subsection{Histogram}
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\section{\tr{Histogram}{Histogramm}}
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Histogramme z\"ahlen die H\"aufigkeit $n_i$ des Auftretens von
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$N=\sum_{i=1}^M n_i$ Messwerten in $M$ Messbereichsklassen $i$ (Bins).
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@ -349,7 +349,7 @@ des Auftretens der Gr\"o{\ss}e $x_i$ in der $i$-ten Klasse an
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\[ P_i = \frac{n_i}{N} = \frac{n_i}{\sum_{i=1}^M n_i} \; . \]
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\subsection{Probability density function}
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\section{\tr{Probability density function}{Wahrscheinlichkeitsdichte}}
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Meistens haben wir es jedoch mit reellen Messgr\"o{\ss}en zu tun.
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@ -371,11 +371,14 @@ Meistens haben wir es jedoch mit reellen Messgr\"o{\ss}en zu tun.
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unterschiedliche Klassenbreiten vergleichbar.}}
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\end{figure}
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Histogramme von reellen Messwerten m\"ussen auf das Integral 1 normiert werden, so dass
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das Integral (nicht die Summe) \"uber das Histogramm eins ergibt. Das Integral
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ist die Fl\"ache des Histograms. Diese setzt sich zusammen aus der Fl\"ache der einzelnen
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Histogrammbalken. Diese haben die H\"ohe $n_i$ und die Breite $\Delta x$. Die Gesamtfl\"ache
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$A$ des Histogramms ist also
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Histogramme von reellen Messwerten m\"ussen auf das Integral 1
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normiert werden, so dass das Integral (nicht die Summe) \"uber das
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Histogramm eins ergibt --- denn die Wahrscheinlichkeit, dass
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irgendeiner der Messwerte auftritt mu{\ss} Eins sein. Das Integral ist
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die Fl\"ache des Histogramms. Diese setzt sich zusammen aus der
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Fl\"ache der einzelnen Histogrammbalken. Diese haben die H\"ohe $n_i$
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und die Breite $\Delta x$. Die Gesamtfl\"ache $A$ des Histogramms ist
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also
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\[ A = \sum_{i=1}^N ( n_i \cdot \Delta x ) = \Delta x \sum_{i=1}^N n_i \]
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und das normierte Histogramm hat die H\"ohe
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\[ p(x_i) = \frac{n_i}{\Delta x \sum_{i=1}^N n_i} \]
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@ -421,16 +424,17 @@ spricht von einer Wahrscheinlichkeitsdichte.
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\end{figure}
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\subsection{Korrelation}
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\section{\tr{Correlations}{Korrelationen}}
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\begin{figure}[t]
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\includegraphics[width=1\textwidth]{correlation}
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\caption{\label{correlationfig} Korrelationen zwischen zwei Datens\"atzen $x$ und $y$.}
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\end{figure}
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Bisher haben wir Eigenschaften einer einzelnen Me{\ss}gr\"o{\ss}e angeschaut.
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Bei mehreren Me{\ss}gr\"o{\ss}en, kann nach Abh\"angigkeiten gefragt werden.
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Der Korrelationskoeffizient
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Bisher haben wir Eigenschaften einer einzelnen Me{\ss}gr\"o{\ss}e
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angeschaut. Bei mehreren Me{\ss}gr\"o{\ss}en, kann nach
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Abh\"angigkeiten zwischen den beiden Gr\"o{\ss}en gefragt werden. Der
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Korrelationskoeffizient
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\[ r_{x,y} = \frac{Cov(x,y)}{\sigma_x \sigma_y} = \frac{\langle
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(x-\langle x \rangle)(y-\langle y \rangle) \rangle}{\sqrt{\langle
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(x-\langle x \rangle)^2} \rangle \sqrt{\langle (y-\langle y
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@ -452,9 +456,9 @@ Korrelationskoeffizienten nahe 0 (\figrefb{correlationfig}).
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Data types}
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\section{Data types}
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\subsubsection{Nominal scale}
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\subsection{Nominal scale}
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\begin{itemize}
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\item Binary
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\begin{itemize}
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@ -475,7 +479,7 @@ Korrelationskoeffizienten nahe 0 (\figrefb{correlationfig}).
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\item Statistics: mode, i.e. the most common item
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\end{itemize}
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\subsubsection{Ordinal scale}
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\subsection{Ordinal scale}
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\begin{itemize}
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\item Like nominal scale, but with an order
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\item Examples: ranks, ratings
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@ -489,7 +493,7 @@ Korrelationskoeffizienten nahe 0 (\figrefb{correlationfig}).
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\item Statistics: mode, median
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\end{itemize}
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\subsubsection{Interval scale}
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\subsection{Interval scale}
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\begin{itemize}
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\item Quantitative/metric values
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\item Reasonable measure of distance between values, but no absolute zero
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@ -505,7 +509,7 @@ Korrelationskoeffizienten nahe 0 (\figrefb{correlationfig}).
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\end{itemize}
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\end{itemize}
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\subsubsection{Absolute/ratio scale}
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\subsection{Absolute/ratio scale}
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\begin{itemize}
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\item Like interval scale, but with absolute origin/zero
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\item Examples:
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@ -524,7 +528,7 @@ Korrelationskoeffizienten nahe 0 (\figrefb{correlationfig}).
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\end{itemize}
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\end{itemize}
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\subsubsection{Data types}
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\subsection{Data types}
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\begin{itemize}
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\item Data type selects
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\begin{itemize}
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@ -539,7 +543,7 @@ Korrelationskoeffizienten nahe 0 (\figrefb{correlationfig}).
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categories ``small/medium/large'' (ordinal scale)
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\end{itemize}
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\subsubsection{Examples from neuroscience}
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\subsection{Examples from neuroscience}
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\begin{itemize}
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\item {\bf absolute:}
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\begin{itemize}
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@ -615,7 +619,7 @@ aus der Stichprobe. Das hat mehrere Vorteile:
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\section{Bootstrap des Standardfehlers}
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Beim Bootstrap erzeugen wir durch resampling neue Stichproben und
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Beim Bootstrap erzeugen wir durch Resampling neue Stichproben und
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benutzen diese um die Stichprobenverteilung einer Statistik zu
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berechnen. Die Bootstrap Stichproben haben jeweils den gleichen Umfang
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wie die urspr\"unglich gemessene Stichprobe und werden durch Ziehen
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@ -639,7 +643,7 @@ Stichprobe vorkommen.
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\end{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Statistics}
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\section{Statistics}
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What is "a statistic"? % dt. Sch\"atzfunktion
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\begin{definition}[statistic]
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A statistic (singular) is a single measure of some attribute of a
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