Q:

Controlling tikz arrow shape end points

I can't seem to find a simple example that shows how to control where the end points are of tikz arrows.  I see many complicated examples, and I'm sure simple ones exist, but I can't find them.  If some kind soul could point me in the right direction (bad pun), I would appreciate it.  In the following example, I want the tips of the double arrows to touch the lines.
\documentclass[11pt,letterpaper]{article}
\usepackage{pgfplots}
\usetikzlibrary{shapes.arrows, fadings}

\pgfmathdeclarefunction{std_norm}{1}{%
  \pgfmathparse{1/(sqrt(2*pi))*exp(-((#1)^2)/(2))}%
}

\begin{document}

\begin{center}
  \begin{tikzpicture}[scale=0.6]
    \begin{axis}[ 
        no markers, 
        domain=-4:4, % Only display z values between -4 and 4.
        samples=100, 
        xlabel=\textbf{z},
        every axis x label/.style={at={(axis description cs:1.0, 0.0)}, anchor=west},
        height=5cm, width=12cm,
        xtick=\empty, ytick=\empty,
        enlargelimits=false, 
        clip=false, 
        axis on top=true,
        hide y axis, 
        axis x line*=middle,
        axis line style ={thick,latex-latex}
      ]

        \addplot [very thick,draw=cyan!50!black] {std_norm(x)};

        \draw [thick] (axis cs:-1.75,0) -- (axis cs:-1.75,{std_norm(-1.75)});
        \draw [thick] (axis cs:1.75,0) -- (axis cs:1.75,{std_norm(1.75)});

        \draw [thick] (axis cs:-1.75,0.53) -- (axis cs:-1.75,{std_norm(-1.75)+0.03});
        \draw [thick] (axis cs:1.75,0.53) -- (axis cs:1.75,{std_norm(1.75)+0.03});

        \node [draw=gray, 
          left color=white,
          right color=green!30,
          double arrow,
          minimum height=4cm,
          minimum width=1cm,
          double arrow head extend=0.05cm
        ] at (axis cs:0.0,0.5) {};

        \draw node at (axis cs:0, 0.5) {Confidence Interval};

    \end{axis}
  \end{tikzpicture}
\end{center}

\end{document}

A:

One of several options is to use the calc library. This allows us to measure distances between nodes/coordinates with the piec
let \p1=($(R)-(L)$) in

In general the distance will be \n1={veclen(\x1,\y1} but here the line is horizontal so it is just \x1.
\documentclass[11pt,letterpaper]{article}
\usepackage{pgfplots}
\usetikzlibrary{shapes.arrows, fadings,calc}

\pgfmathdeclarefunction{std_norm}{1}{%
  \pgfmathparse{1/(sqrt(2*pi))*exp(-((#1)^2)/(2))}%
}

\begin{document}

\begin{center}
  \begin{tikzpicture}[scale=0.6]
    \begin{axis}[ 
        no markers, 
        domain=-4:4, % Only display z values between -4 and 4.
        samples=100, 
        xlabel=\textbf{z},
        every axis x label/.style={at={(axis description cs:1.0, 0.0)}, anchor=west},
        height=5cm, width=12cm,
        xtick=\empty, ytick=\empty,
        enlargelimits=false, 
        clip=false, 
        axis on top=true,
        hide y axis, 
        axis x line*=middle,
        axis line style ={thick,latex-latex}
      ]

        \addplot [very thick,draw=cyan!50!black] {std_norm(x)};

        \draw [thick] (axis cs:-1.75,0) -- (axis cs:-1.75,{std_norm(-1.75)})
        coordinate[pos=1] (L);
        \draw [thick] (axis cs:1.75,0) -- (axis cs:1.75,{std_norm(1.75)})
        coordinate[pos=1] (R);

        \draw [thick] (axis cs:-1.75,0.53) -- (axis cs:-1.75,{std_norm(-1.75)+0.03});
        \draw [thick] (axis cs:1.75,0.53) -- (axis cs:1.75,{std_norm(1.75)+0.03});

        \path let \p1=($(R)-(L)$) in (axis cs:0, 0.5)
          node [draw=gray, 
          left color=white,
          right color=green!30,
          double arrow,
          minimum height=\x1,
          minimum width=1cm,
          double arrow head extend=0.05cm,inner xsep=0pt
        ] {Confidence Interval};

    \end{axis}
  \end{tikzpicture}
\end{center}

\end{document}

Other options worth trying out is to use fit.

