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@@ -36,3 +36,5 @@ saved_model/**/* filter=lfs diff=lfs merge=lfs -text
 Manually[[:space:]]Disecting[[:space:]]arXiv2512.15720/main.pdf filter=lfs diff=lfs merge=lfs -text
 simple_trend_filter[[:space:]]dominant_poc_displacement/main.pdf filter=lfs diff=lfs merge=lfs -text
 Using[[:space:]]My[[:space:]]Flesh[[:space:]]to[[:space:]]Make[[:space:]]TeX-written[[:space:]]Notes/main.pdf filter=lfs diff=lfs merge=lfs -text
+variance[[:space:]]and[[:space:]]standard[[:space:]]deviation/image.png filter=lfs diff=lfs merge=lfs -text
+variance[[:space:]]and[[:space:]]standard[[:space:]]deviation/main.pdf filter=lfs diff=lfs merge=lfs -text

variance and standard deviation/main.pdf

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+version https://git-lfs.github.com/spec/v1
+oid sha256:9abfb4be117539e4407b26d7a1f3a340914d071ccf6e29c2d274694abec4758a
+size 195804

variance and standard deviation/main.tex

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+
+\documentclass[8pt]{article}
+\usepackage[margin=0.5in]{geometry}
+
+% Core packages
+\usepackage{amsmath,amssymb}
+\usepackage{tikz-cd}
+\usepackage{multicol}
+\usepackage{pgfplots}
+\pgfplotsset{compat=1.18}
+\newcommand{\mathcolorbox}[2]{\colorbox{#1}{$\displaystyle #2$}}
+
+% Paragraphs
+\setlength{\parindent}{0pt}
+\setlength{\parskip}{1\baselineskip}
+
+\title{Variance and Standard Deviation}
+\author{algorembrant}
+\date{\today}
+
+\begin{document}
+\maketitle
+
+\includegraphics[width=1\textwidth]{image.png}
+
+\begin{align*}
+    \intertext{What shown above is the population version of $\sigma^2$ and $\sigma$ let $X$ element of the dataset, and $t$ as timestep (but forward), the following solutions is how you get the varience and standard deviation and what perspective it meant to us.}
+\end{align*}
+\newpage
+\newpage
+\begin{align*}
+    \sigma^2 &= \frac{1}{T}\sum_{t=1}^T(X_t - \mu)^2 \quad \text{or} \quad \sigma^2 = \frac{1}{T}\sum_{t=1}^T(X_t - \frac{1}{T}\sum_{t=1}^T X_t)^2\\
+    \sigma^2 &= \frac{1}{T}\sum_{t=1}^T(\mathcolorbox{yellow!50}{X_t} - \mu)^2, \quad X\in \{109,183,153,132,102,123,96,58,98,125,143,110,164,143,189\} \\
+    \sigma^2 &= \frac{1}{T}\sum_{t=1}^T(X_t - \mathcolorbox{yellow!50}{\mu})^2, \quad \text{the mean of the dataset, represented by one value} \\
+    \sigma^2 &= \frac{1}{T}\sum_{t=1}^T(\mathcolorbox{yellow!50}{X_t - \mu})^2, \quad \text{the deviation, or distance, from $X_t$ to $\mu$} \\
+    \sigma^2 &= \frac{1}{T} \sum_{t=1}^T \mathcolorbox{yellow!50}{(X_t - \mu)^2}, \quad \text{squaring the deviation will turn all values into positive}\\
+    \sigma^2 &= \frac{1}{T} \mathcolorbox{yellow!50}{\sum_{t=1}^T(X_t - \mu)^2}, \quad \text{is the sum of all squared deviation} \\
+    \sigma^2 &= \mathcolorbox{yellow!50}{\frac{1}{T}\sum_{t=1}^T(X_t - \mu)^2}, \quad \text{is the \textcolor{red}{variance}} \\
+    \sigma &= \mathcolorbox{yellow!50}{\sqrt{\frac{1}{T}\sum_{t=1}^T(X_t - \mu)^2}}, \quad \text{is the \textcolor{red}{standard deviation}} \\
+    \text{or} \\
+    \sigma^2 &= \frac{1}{T}\sum_{t=1}^T(X_t - \frac{1}{T}\sum_{t=1}^T X_t)^2, \quad \intertext{is the average of the squared differences from the mean; it tells us how scattered the data points are around the mean, we cant visualize this on the same plot due to the unit mismatched (squared)}
+    \sigma &= \sqrt{\frac{1}{T}\sum_{t=1}^T(X_t - \frac{1}{T}\sum_{t=1}^T X_t)^2}, \quad \intertext{meanhile, this is more idial due to the squaroot operation, making the values simillar (same unit) to the original datapoints}
+    \intertext{both variance and standard deviation meansures how compact or scattered the datapoints from the mean. What shown above was population $\sigma^2$ and $\sigma$ and to calculate for sample version, we simply subtract $T$ by 1}
+\end{align*}
+\begin{align*}
+    \sigma^2_{sample} = \frac{1}{\textcolor{red}{T-1}}\sum_{t=1}^T(X_t - \frac{1}{T}\sum_{t=1}^T X_t)^2, \quad \sigma_{sample} = \sqrt{\frac{1}{\textcolor{red}{T-1}}\sum_{t=1}^T(X_t - \frac{1}{T}\sum_{t=1}^T X_t)^2},
+\end{align*}
+\begin{align*}
+    \sigma^2_{population} = \frac{1}{\textcolor{red}{T}}\sum_{t=1}^T(X_t - \frac{1}{T}\sum_{t=1}^T X_t)^2, \quad \sigma_{population} = \sqrt{\frac{1}{\textcolor{red}{T}}\sum_{t=1}^T(X_t - \frac{1}{T}\sum_{t=1}^T X_t)^2},
+\end{align*}
+
+\end{document}
+