From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Thu, 22 Aug 2019 12:01:52 +0000 (+1000)
Subject: [spec] normal distributions
X-Git-Tag: yr12~59
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/33c9f9dae4ce9b4b7f23aad987826856ef44bda8

[spec] normal distributions
---

diff --git a/spec/statistics.pdf b/spec/statistics.pdf
index adbdbf0..357175e 100644
Binary files a/spec/statistics.pdf and b/spec/statistics.pdf differ
diff --git a/spec/statistics.tex b/spec/statistics.tex
index f546e20..7c89068 100644
--- a/spec/statistics.tex
+++ b/spec/statistics.tex
@@ -12,6 +12,7 @@
 \usepackage{xcolor} % used only to show the phantomed stuff
 \definecolor{cas}{HTML}{e6f0fe}
 \usepackage{mathtools}
+\pgfplotsset{compat=1.16}
 
 \pagestyle{fancy}
 \fancyhead[LO,LE]{Unit 4 Specialist --- Statistics}
@@ -61,9 +62,11 @@
 
   \section{Sample mean}
 
+  Approximation of the \textbf{population mean} determined experimentally.
+
   \[ \overline{x} = \dfrac{\Sigma x}{n} \]
 
-  where \(n\) is the size of the sample (number of sample points)
+  where \(n\) is the size of the sample (number of sample points) and \(x\) is the value of a sample point
 
   \subsubsection*{\colorbox{cas}{On CAS:}}
 
@@ -79,8 +82,41 @@
 
   \[ \overline{X} = \sum_{i=1}^n \frac{x_i}{n} = \dfrac{\sum x}{n} \]
 
-  Sample mean is distributed with mean \(\mu\) and sd \(\frac{\sigma}{\sqrt{n}}\)
-  
+  Sample mean is distributed with mean \(\mu\) and sd \(\frac{\sigma}{\sqrt{n}}\) (approaches these values for increasing sample size \(n\)).
 
+  \colorbox{cas}{On CAS:} Spreadsheet \(\rightarrow\) Catalog \(\rightarrow\) \verb;randNorm(sd, mean, n); where \verb;n; is the number of samples. Show histogram with Histogram key in top left \\
+  To calculate parameters of a dataset: Calc \(\rightarrow\) One-variable
+  
+  \section{Normal distributions}
+
+  mean = mode = median
+
+  \[ Z = \frac{X - \mu}{\sigma} \]
+
+  Normal distributions must have are (total prob.) of 1 \(\implies \int^\infty_{-\infty} f(x) \> dx = 1\)
+\pgfmathdeclarefunction{gauss}{2}{%
+  \pgfmathparse{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))}%
+}
+
+\begin{tikzpicture}
+\begin{axis}[every axis plot post/.append style={
+  mark=none,domain=-3:3,samples=50,smooth}, % All plots: from -2:2, 50 samples, smooth, no marks
+  axis x line*=bottom, % no box around the plot, only x and y axis
+  axis y line*=left, % the * suppresses the arrow tips
+  enlargelimits=upper,
+  ytick={0.5},
+  yticklabels={\(\frac{1}{\sigma \sqrt{2\pi}}\)}, 
+  xtick={-2,-1,0,1,2},
+  xticklabels={\(\mu-2\sigma\), \(\mu-\sigma\), \(\mu\), \(\mu+\sigma\), \(\mu+2\sigma\)},
+  xlabel={\(x\)},
+  every axis x label/.style={at={(current axis.right of origin)},anchor=north west},
+  ylabel={\(\Pr(X=x)\)}]
+  \addplot {gauss(0,0.75)};
+\end{axis}
+\end{tikzpicture}
+
+  \section{Central limit theorem}
+
+  If \(X\) is randomly distributed with mean \(\mu\) and sd \(\sigma\), then with an adequate sample size \(n\) the distribution of the sample mean \(\overline{X}\) is approximately normal with mean \(E(\overline{X})\) and \(\operatorname{sd}(\overline{X}) = \frac{\sigma}{\sqrt{n}}\).
 
 \end{document}