633 lines
19 KiB
TeX
Executable File
633 lines
19 KiB
TeX
Executable File
\documentclass{beamer}
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\usepackage{xcolor}
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\usepackage{listings}
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\usepackage{pgf}
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%\usepackage{pgf,pgfarrows,pgfnodes,pgfautomata,pgfheaps,pgfshade}
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%\usepackage{multimedia}
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\usepackage[latin1]{inputenc}
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\usepackage{amsmath}
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\usepackage{bm}
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\usepackage[T1]{fontenc}
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\usepackage{hyperref}
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\usepackage{ulem}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>
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{
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\usetheme{Singapore}
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\setbeamercovered{opaque}
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\usecolortheme{tuebingen}
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\setbeamertemplate{navigation symbols}{}
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\usefonttheme{default}
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\useoutertheme{infolines}
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% \useoutertheme{miniframes}
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}
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\AtBeginSection[]
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{
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\begin{frame}<beamer>
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\begin{center}
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\Huge \insertsectionhead
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\end{center}
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% \frametitle{\insertsectionhead}
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% \tableofcontents[currentsection,hideothersubsections]
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\end{frame}
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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
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\setbeamertemplate{blocks}[rounded][shadow=true]
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\title[]{Scientific Computing -- Statistics}
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\author[Statistics]{Fabian Sinz\\Dept. Neuroethology,
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University T\"ubingen\\
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Bernstein Center T\"ubingen}
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\institute[Scientific Computing]{}
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\date{11/27/2013}
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%\logo{\pgfuseimage{logo}}
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\subject{Lectures}
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%%%%%%%%%% configuration for code
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\lstset{
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basicstyle=\ttfamily,
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numbers=left,
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showstringspaces=false,
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language=Matlab,
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commentstyle=\itshape\color{darkgray},
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keywordstyle=\color{blue},
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stringstyle=\color{green},
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backgroundcolor=\color{blue!10},
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breaklines=true,
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breakautoindent=true,
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columns=flexible,
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frame=single,
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captionpos=b,
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xleftmargin=1em,
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xrightmargin=1em,
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aboveskip=10pt
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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\mycite}[1]{
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\begin{flushright}
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\tiny \color{black!80} #1
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\end{flushright}
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}
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\input{../latex/environments.tex}
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\makeatother
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\begin{document}
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\begin{frame}
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\titlepage
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\end{frame}
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\begin{frame}
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\frametitle{plan}
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\setcounter{tocdepth}{1}
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\tableofcontents
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\end{frame}
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\begin{frame}
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\frametitle{information}
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\begin{itemize}
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\item Samuels, M. L., Wittmer, J. A., \& Schaffner,
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A. A. (2010). Statistics for the Life Sciences (4th ed.,
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p. 668). Prentice Hall.
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\item Zar, J. H. (1999). Biostatistical Analysis. (D. Lynch,
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Ed.)Prentice Hall New Jersey (4th ed., Vol. 4th, p. 663). Prentice
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Hall. doi:10.1037/0012764
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\item \url{http://stats.stackexchange.com}
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\end{itemize}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% errorbars (error bar paper)
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% confidence intervals (sources of error)
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% plotting (the right plot for the right data, Dan plotting paper)
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% statistical test structure (bootstrapping, resampling, permutation)
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% Don'ts: repeated testing, exclude data points
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% study design
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% PCA
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section[Prelude]{Prelude}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% ----------------------------------------------------------
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\begin{frame}
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\frametitle{my expectations to this course}
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\begin{itemize}
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\item interest and participation
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\item motivation to understand and question concepts
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\item high scientific standard
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\item intellectual honesty
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\item sincere cooperation
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\end{itemize}
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\end{frame}
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% ----------------------------------------------------------
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\begin{frame}
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\frametitle{this week will be ...}
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\only<1>{
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\framesubtitle{... no \sout{fun} piece of cake}
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\begin{center}
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\includegraphics[height=0.7\textheight]{figs/feeding.jpg}
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\end{center}
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}
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\only<2>{
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\framesubtitle{... no \sout{fun} piece of cake}
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\begin{center}
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\includegraphics[height=0.7\textheight]{figs/nacho-trainer.jpg}
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\end{center}
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}
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\only<3>{
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\framesubtitle{... no lecture (please!)}
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\begin{center}
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\includegraphics[height=0.7\textheight]{figs/soccer.jpg}
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\end{center}
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}
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\end{frame}
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% ----------------------------------------------------------
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\begin{frame}
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\frametitle{What you should learn this week}
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\begin{itemize}
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\item What makes good plots?
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\item What is descriptive/inferential statistics?
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\item What is the general structure of a statistical test?
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\item What does a p-value mean?
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\item How can I build my own tests?
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\item How large should my $n$ be?
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\item What is {\em maximum likelihood} and why is it important?
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\end{itemize}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section[descriptive statistics, errorbars, and plots]{Day 1 -- descriptive statistics, errorbars, and plots}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{types of data}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
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\frametitle{data scales}
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\framesubtitle{What data types are distinguished in statistics?}
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\Large
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{\bf Why are data types important?}
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\pause
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\begin{itemize}
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\item selection of statistics
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\item selection of plots
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\item selection of correct tests
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\end{itemize}
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\end{frame}
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%-------------------------------------------------------------
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\begin{frame}
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\frametitle{data scales}
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\framesubtitle{nominal/categorial scale}
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\begin{itemize}
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\item properties like cell type, experimental group (i.e. treatment
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1, treatment 2, control)
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\item each observation/sample is put into one category
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\item there is no reasonable order among the categories
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\item example: [rods, cones] vs. [cones, rods]
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\end{itemize}
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\end{frame}
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%-------------------------------------------------------------
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\begin{frame}
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\frametitle{data scales}
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\framesubtitle{ordinal scale}
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\begin{itemize}
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\item like nominal scale, but there is an order
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\item {\bf but:} there is no reasonable measure of {\em distance}
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between the classes
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\item examples: ranks, ratings
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\end{itemize}
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\end{frame}
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%-------------------------------------------------------------
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\begin{frame}
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\frametitle{data scales}
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\framesubtitle{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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\item examples: temperature in $^\circ$C
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\end{itemize}
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\end{frame}
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%-------------------------------------------------------------
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\begin{frame}
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\frametitle{data scales}
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\framesubtitle{absolut/ratio scale}
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\begin{itemize}
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\item like interval scale but with absolute zero
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\item example: temperature in $^\circ$K
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\end{itemize}
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\pause
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\begin{emphasize}{relationsships between scales}
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\begin{itemize}
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\item scales exhibit increasing information content from nominal
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to absolute
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\item conversion ,,downwards'' always possible
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\end{itemize}
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\end{emphasize}
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\end{frame}
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%-------------------------------------------------------------
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\begin{frame}
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\frametitle{examples from neuroscience and psychology}
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\begin{itemize}
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\item {\bf nominal:}\pause
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\begin{itemize}
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\item treatment group
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\item stimulus class
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\item cell type
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\end{itemize}
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\item {\bf ordinal:} \pause
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\begin{itemize}
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\item ratings
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\item clinical stages of a disease
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\item states of an ion channel
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\end{itemize}
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\item {\bf Absolut-/Ratioskala:}\pause
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\begin{itemize}
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\item firing rate
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\item membrane potential
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\item ion concentration
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\end{itemize}
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\end{itemize}
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\end{frame}
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%-------------------------------------------------------------
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{statistics}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%-------------------------------------------------------------
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\begin{frame}
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\frametitle{What is "a statistic"?}
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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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sample (e.g., its arithmetic mean value). It is calculated by
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applying a function (statistical algorithm) to the values of the
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items of the sample, which are known together as a set of data.
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\source{http://en.wikipedia.org/wiki/Statistic}
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\end{definition}
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\end{frame}
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%-------------------------------------------------------------
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\begin{frame}
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\frametitle{Beispiele f\"ur Teststatistiken}
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\begin{itemize}
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\item {\bf nominal:}\pause
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\begin{itemize}
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\item count
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\item relative frequency/proportion
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\end{itemize}
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\item {\bf ordinal:} \pause
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\begin{itemize}
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\item median
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\item quantile/percentile
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\item rank correlation
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\end{itemize}
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\item {\bf absolute/ratio:}\pause
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\begin{itemize}
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\item mean
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\item variance/ standard deviation
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\item Pearson correlation
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\end{itemize}
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\end{itemize}
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\end{frame}
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%-------------------------------------------------------------
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\begin{frame}
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\frametitle{exercise}
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\begin{task}{Spearman rank correlation}
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\begin{enumerate}
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\item Use {\tt randi} to generate two 100-dimensional vectors
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{\tt x,y} of random integers between $0$ and $10$.
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\item Find out how to compute the Spearman
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rank correlation $$\rho = 1- {\frac {6 \sum
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d_i^2}{n(n^2 - 1)}}$$ with Matlab. $d_i = x_i - y_i$ is the
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difference in the rank between the single data points.
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\item Compute $\rho$ between $x$ and $y$, between $x$ and
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$y^2$, between $\log(x+1)$ and $y^2$.
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\item Compute the "standard" (Pearson) correlation coefficient
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between these values.
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\item What can you observe and why does it make sense?
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\end{enumerate}
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\end{task}
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\end{frame}
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%-------------------------------------------------------------
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\begin{frame}[fragile]
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\frametitle{solution}
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\begin{solution}{Spearman rank correlation }
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\scriptsize
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\begin{lstlisting}
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>>> x = randi(10, 100, 1);
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>>> y = randi(10, 100, 1);
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>>> corr(x,y,'type','Spearman')
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ans =
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0.1220
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>>> corr(x,y.^2,'type','Spearman')
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ans =
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0.1220
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>>> corr(x,y,'type','Pearson')
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ans =
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0.1074
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>>> corr(x,y.^2,'type','Pearson')
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ans =
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0.0551
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\end{lstlisting}
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The rank correlation does not change under a monotone transformation
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of the data. Therefore, it can be used for ordinal data. The Pearson
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correlation coefficient does not have that property.
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\end{solution}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{description of data and plotting}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{nominal scale}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%-------------------------------------------------------------
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\begin{frame}[fragile]
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\frametitle{}
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\begin{center}
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\Huge What makes a good plot?
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\end{center}
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\end{frame}
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%-------------------------------------------------------------
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\begin{frame}[fragile]
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\frametitle{features of a good plot}
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A good plot
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\begin{itemize}
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\item helps the reader to clearly understand your point.\pause
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\item is not misleading and let's the reader judge the information
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on her own (different y-axis/length scales in two related plots,
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"squeezing" via log-plots). \pause
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\item contains information about the data (a comic might be
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illustrative, but does not contain information about the
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data).\pause
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\item adheres to the principle of {\em ink minimization}.
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\end{itemize}
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\end{frame}
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%-------------------------------------------------------------
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\begin{frame}[fragile]
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\frametitle{features of a good plot}
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\framesubtitle{design/organization}
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\begin{itemize}
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\item Is the display consistent with the model or hypothesis
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being tested?\pause
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\item Are there "empty dimensions" in the display that could be
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removed (A 3D pie chart for 2D categorical data, extraneous colors
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that do not encode meaningful information)?\pause
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\item Does the display provide an honest and transparent portrayal
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of the data (hiding, smoothing, modifying data points should be
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avoided or explicitly mentioned)?
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\end{itemize}
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\mycite{Allen et al. 2012, Neuron}
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\end{frame}
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%-------------------------------------------------------------
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\begin{frame}[fragile]
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\frametitle{features of a good plot}
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\framesubtitle{axes}
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\begin{itemize}
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\item Are axes scales defined as linear, log, or radial?\pause
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\item Does each axis label describe the variable and its units (use
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"a.u." for arbitrary units)?\pause
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\item Are axes limits appropriate for the data (The graphic should
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not be bounded at zero if the data can take on both positive and
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negative values.)?\pause
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\item Is the aspect ratio appropriate for the data (When x and y
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axes contrast the same variable under different conditions the
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graphic should be square.)?
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\end{itemize}
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\mycite{Allen et al. 2012, Neuron}
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\end{frame}
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%-------------------------------------------------------------
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\begin{frame}[fragile]
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\frametitle{features of a good plot}
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\framesubtitle{color mapping}
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\begin{itemize}
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\item Is a color bar provided?\pause
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\item Is the color map sensible for the data type (does the data
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extend to both $\pm$, does it live in an interval, is it
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circular)?\pause
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\item Are contrasting colors consistent with a natural interpretation?
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\item Can features be discriminated when printed in grayscale?
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\item Has red/green contrast been avoided to accommodate common
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forms of colorblindness?
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\end{itemize}
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\mycite{Allen et al. 2012, Neuron}
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\end{frame}
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%-------------------------------------------------------------
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\begin{frame}[fragile]
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\frametitle{features of a good plot}
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\framesubtitle{uncertainty}
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\begin{itemize}
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\item Does the display indicate the uncertainty of estimated parameters?\pause
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\item Is the type of error surface appropriate for the data?
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\begin{itemize}
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\item Use standard deviations to describe variability in the population.\pause
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\item Use standard errors or confidence intervals to make inferences
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about parameters estimated from a sample.\pause
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\item Parametric confidence intervals should only be used if data
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meet the assumptions of the underlying model.\pause
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\end{itemize}
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\item Are the units of uncertainty defined (is it standard error, is
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it $95\%$ confidence interval)?
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\end{itemize}
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\mycite{Allen et al. 2012, Neuron}
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\end{frame}
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%-------------------------------------------------------------
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\begin{frame}[fragile]
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\frametitle{features of a good plot}
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\framesubtitle{annotation}
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\begin{itemize}
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\item Are all symbols defined, preferably by directly labeling objects?\pause
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\item Is the directionality of a contrast between conditions obvious?\pause
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\item Is the number of samples or independent experiments indicated?\pause
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\item Are statistical procedures and criteria for significance described?\pause
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\item Are uncommon abbreviations avoided or clearly defined?\pause
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\item Are abbreviations consistent with those used in the text?
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\end{itemize}
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\mycite{Allen et al. 2012, Neuron}
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\end{frame}
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%-------------------------------------------------------------
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\begin{frame}[fragile]
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\frametitle{suboptimal example}
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\begin{center}
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\includegraphics[width=.5\linewidth]{figs/nobelbad}
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\end{center}
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\mycite{Hafting et al. 2005, nature}
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\end{frame}
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%-------------------------------------------------------------
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\begin{frame}[fragile]
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\frametitle{different axes}
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\end{frame}
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%-------------------------------------------------------------
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\begin{frame}
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\frametitle{Bad bar plot}
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\begin{center}
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\includegraphics[width=.8\linewidth]{figs/badbarplot}
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\end{center}
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\source{www.enfovis.com}
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\end{frame}
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%-------------------------------------------------------------
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\begin{frame}[fragile]
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\frametitle{plotting nominal data}
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\framesubtitle{bar plot for count and relative frequency}
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\begin{center}
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\includegraphics[width=.8\linewidth]{figs/nominaldataplot}
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\end{center}
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\end{frame}
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%-------------------------------------------------------------
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\begin{frame}[fragile]
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\frametitle{plotting nominal data}
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\framesubtitle{bar plot for count and relative frequency}
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\scriptsize
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\begin{lstlisting}
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% plot
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bar([1,2], [50, 90], 'facecolor', 'k')
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% labels axes
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ylabel('cell count')
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xlabel('cell type')
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% cosmetics
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xlim([0.5,2.5])
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ylim([0, 100])
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box('off')
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set(gca,'XTick',1:2,'XTickLabel',{'pyramidal','interneuron'},'FontSize',20)
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% settings for saving the figure
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set(gcf, 'PaperUnits', 'centimeters');
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set(gcf, 'PaperSize', [11.7 9.0]);
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set(gcf, 'PaperPosition',[0.0 0.0 11.7 9.0]);
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\end{lstlisting}
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\end{frame}
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%----------------------------------------------------------
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\begin{frame}[fragile]
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\frametitle{plotting nominal data}
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\framesubtitle{pie chart for count and relative frequency}
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\begin{center}
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\includegraphics[width=.8\linewidth]{figs/nominaldataplot2}
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\end{center}
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\end{frame}
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%-------------------------------------------------------------
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\begin{frame}[fragile]
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\frametitle{Darstellung nominaler Daten}
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\framesubtitle{exercise}
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\begin{task}{pie chart}
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Plot the same data ($n_{py}=50$, $n_{in}=90$) as a pie chart in Matlab.
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\end{task}
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\end{frame}
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%-------------------------------------------------------------
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\begin{frame}[fragile]
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\frametitle{Darstellung nominaler Daten}
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\framesubtitle{pie chart for relative frequency}
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\scriptsize
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\begin{lstlisting}
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data = [50, 90];
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h = pie(data, [1,0], {'pyramidal (n=50)', 'interneuron (n=90)'})
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hText = findobj(h,'Type','text') % text object handles
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set(h(1), 'FaceColor', [.2,.2,.2]);
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set(h(2), 'Rotation', 45);
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set(h(3), 'FaceColor', [.8,.8,.8]);
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set(h(4), 'Rotation', 45);
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title('cell count')
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set(gca,'XTick',1:2,'XTickLabel',{'pyramidal', 'interneuron'})
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box('off')
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set(gcf, 'PaperUnits', 'centimeters');
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set(gcf, 'PaperSize', [11.7 9.0]);
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set(gcf, 'PaperPosition',[0.0 0.0 11.7 9.0]);
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\end{lstlisting}
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\end{frame}
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%-------------------------------------------------------------
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\begin{frame}[fragile]
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\frametitle{plotting interval/ratio/absolute data}
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\framesubtitle{histogram}
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\begin{center}
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\includegraphics[width=.8\linewidth]{figs/histogram}
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\end{center}
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\end{frame}
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%-------------------------------------------------------------
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\begin{frame}[fragile]
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\frametitle{plotting interval/ratio/absolute data}
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\framesubtitle{bad choice of bins}
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\begin{center}
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\includegraphics[width=.4\linewidth]{figs/histogrambad}
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\includegraphics[width=.4\linewidth]{figs/histogrambad2}
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\end{center}
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\begin{summary}{Rule of thumb}
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Choose the bins $b\approx n/20$.
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\end{summary}
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\end{frame}
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%-------------------------------------------------------------
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\begin{frame}[fragile]
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|
\frametitle{plotting interval/ratio/absolute data}
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\framesubtitle{how to do in Matlab}
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\scriptsize
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|
\begin{lstlisting}
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x = randn(2000,1); % generate Gaussian data
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|
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hist(x, 50); % generate histogram
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|
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% set facecolor to gray
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h = findobj(gca, 'Type','patch');
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set(h(1), 'FaceColor',[.2,.2,.2], 'EdgeColor','w', 'linewidth',2)
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% plot a white grid over it
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h = gridxy([],get(gca,'ytick'),'color','w','linewidth',2)
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uistack(h, 'top')
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% cosmetics
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box('off');
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xlabel('Data')
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ylabel('Count')
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|
\end{lstlisting}
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|
\end{frame}
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|
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|
%-------------------------------------------------------------
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\begin{frame}[fragile]
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|
\frametitle{plotting interval/ratio/absolute data}
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|
\framesubtitle{other ways}
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|
There are other ways to plot a sample $x_1, ..., x_n$ of
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|
interval/ratio/absolute scale data. E.g.
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|
\begin{itemize}
|
|
\item box plot
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|
\item bar plot
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|
\item smoothed histogram
|
|
\item ...
|
|
\end{itemize}
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|
We will look at them while plotting mixed data in the following.
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|
\end{frame}
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|
|
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\end{document}
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