When the rotated test is included the differences between the four tests on each sample can be described by six equations, with four error terms as in Equations.... Being an over determined ill-conditioned system of linear equations, they can be solved simultaneously as a minimisation problem using python xxx. Equations --- were solved for means giving estaimates on the existing bias of each test (relative to the others). The system was also solved for varences, the resulting calculated errors of the varences provide estimates (after conversion to standard devations) of the 95% confidence intervals on an abatory measurement. ---needs more explaning---
\(RS\ -\ OS\ =\ e_{rs\ }+\ e_{os}\)
\(RS\ -\ QS1\ =\ e_{rs}\ +\ e_{qs}\)
\(RS\ -\ QS2\ =\ e_{rs\ }+e_{qs}\ +\ e_{rot}\)
\(OS\ -\ QS1\ =\ e_{os}\ +\ e_{qs}\)
\(OS\ -\ QS2\ =\ e_{os}\ +\ e_{qs}\ +\ e_{rot}\)
\(QS1\ -\ QS2\ =\ 2e_{qs}\ +\ e_{rot}\)
Where RS is the result from the rapid splitting test, OS is the result of the origonal splitting test,  QS1 is the result from the quartering test along the same plane as the rapid splitting test and QS2 is the quartering test along the plane perpendicular to QS1. e_rs and e_os are the measurement error associated with the rapid and original splitting tests, e_qs is the measurement error associated with the quartering test (note the measurement error is the same regardless of splitting plane) and e_rot is the difference resulting from the which plane is cut.