For the most curious students

QQ-plots and Box-Whisker plots usually become part of the statistical toolbox for the students attending my course of ‘Experimental methods in agriculture’. Most of them learn that the QQ-plot can be used to check for the basic assumption of gaussian residuals in linear models and that the Box-Whisker plot can be used to describe the experimental groups, when their size is big enough and we do not want to assume a gaussian distribution. Furthermore, most students learn to use the plot() method on an ‘lm’ object and the boxplot() function in the base ‘graphic’ package and concentrate on the interpretation of the R output. To me, in practical terms, this is enough; however, there is at least a couple of students per year who think that this is not enough and they want to know more. What is the math behind those useful plots? Can we draw them by hand?