17 June 2015, Created with GeoGebra

Vector algebra provides a simple explanation to the Simpson paradox.

With the help of the above applet, it is easy to find four vectors $A_{i} = (x_{ai}, y_{ai})$ and $B_{i} = (x_{bi}, y_{bi}),$ $i = 1, 2,$ such that:

1. $y_{bi}/x_{bi} \gt y_{ai}/x_{ai},$ $i = 1, 2,$ whereas

2. $(y_{b1} + y_{b2})/(x_{b1} + x_{b2}) \lt (y_{a1} + y_{a2})/(x_{a1} + x_{a2}).$

1. means that $Slope(B_{1}) \gt Slope(A_{1})$ and $Slope(B_{2}) \gt Slope(A_{2}).$ However, this does not imply the same inequality for the sums: $Slope(B_{1} + B_{2}) \gt Slope(A_{1} + A_{2})$ may not be true.