I measured the distance between the center of the north anchor and north of a node and got 7.20068 pt which is 0.1 inches. This is for a especial case as below:

Here is my code:
\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary {positioning}
\usetikzlibrary { calc}
\begin{document}
\begin{tikzpicture}
[simple/.style={rectangle,draw, inner sep=0pt,minimum size=10 pt}]
\node[simple] (the node) {};
\node (the anchor) [red,above] at (the node.north) {$a < b$};
\draw[latex-latex] let \p1 = ($(the anchor.center)-(the node.north)$), \n1={veclen(\x1,\y1)} in (the
node.north) -- node[above] {\n1} (the anchor.center);
\end{tikzpicture}
\end{document}
The main question is: assuming the fontsize is constant let us say 10 pt, how the length of the string inside anchor affects this distance? If instead of $a > b$ we make a longer string such as $aaaaaaaaaaaaa$ the distance described above changes. The question is how it varies.
