---
id:
aliases: []
title: Decrease in Sigma
tags:
  - authorship/original
  - destiny/uncertain
  - status/incomplete
  - topic/math/statistics
  - type/encyclopedia-entry
---
# Decrease in Sigma

```tikz
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}

\begin{document}
\begin{tikzpicture}
    \begin{axis}[
      width=13cm,
      height=7cm,
      axis lines=middle,
      xlabel={$x$},
      ylabel={$\varphi(x;\mu,\sigma)$},
      xmin=-6, xmax=6,
      ymin=0, ymax=0.85,
      samples=400,
      domain=-6:6,
      legend style={draw=none, fill=none, at={(0.98,0.98)}, anchor=north east},
      legend cell align=left,
      ytick=\empty,
    ]
    
    % Normal PDF: (1/(sigma*sqrt(2*pi))) * exp(-(x-mu)^2/(2*sigma^2))
    
    \addplot[thick]
      { (1/(1.8*sqrt(2*pi))) * exp(-((x-0.8)^2)/(2*1.8^2)) };
    \addlegendentry{$\mu=0,\ \sigma=1.8$}
    
    \addplot[thick, dashed]
      { (1/(0.8*sqrt(2*pi))) * exp(-((x-0.8)^2)/(2*0.8^2)) };
    \addlegendentry{$\mu=0,\ \sigma=0.8$}
    
    \end{axis}
\end{tikzpicture}
\end{document}
```

^pdf

```tikz
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}

\pgfmathdeclarefunction{erfapprox}{1}{%
  \pgfmathparse{%
    % save sign and work with |x|
    ( (#1<0) ? -1 : 1 )
    * ( 1 - (1 + 0.278393*abs(#1) + 0.230389*abs(#1)^2 + 0.000972*abs(#1)^3 + 0.078108*abs(#1)^4)^(-4) )
  }%
}

% Normal CDF using erf approximation:
% F(x;mu,sigma) = 0.5*(1 + erf((x-mu)/(sigma*sqrt(2))))
\pgfmathdeclarefunction{normcdf}{3}{%
  \pgfmathparse{ 0.5*(1 + erfapprox((#1-#2)/(#3*sqrt(2)))) }%
}

\begin{document}
\begin{tikzpicture}
    \begin{axis}[
      width=13cm,
      height=7cm,
      axis lines=middle,
      xlabel={$x$},
      ylabel={$F(x;\mu,\sigma)$},
      xmin=-6, xmax=6,
      ymin=0, ymax=1.05,
      samples=400,
      domain=-6:6,
      legend style={draw=none, fill=none, at={(0.02,0.98)}, anchor=north west},
      legend cell align=left,
      ytick={0,0.5,1},
    ]
    
    % Normal CDF: 0.5*(1 + erf((x-mu)/(sigma*sqrt(2))))
    \addplot[thick]
      { normcdf(x,0,1.8) };
    \addlegendentry{$\mu=0,\ \sigma=1.8$}
    
    \addplot[thick, dashed]
      { normcdf(x,0.8) };
    \addlegendentry{$\mu=0,\ \sigma=0.8$}
    
    \end{axis}
\end{tikzpicture}
\end{document}
```