Where \(X\) and \(Z\) are the design matrices, \(\beta\) is the fixed effect of testing type, \(\mu\) is the random effect of sample, and \(\epsilon\) is the random error.  
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 ---to --- Being an over determined ill-conditioned system of linear equations, they can be solved simultaneously as a minimisation problem using python xxx --ref--. Equations --- were solved for means giving estimates on the existing bias of each test (relative to the others). The system was also solved for variances, the resulting calculated errors of the variances provide estimates (after conversion to standard deviations) of the 95% confidence intervals on an abatory measurement. 
\(RS\ -\ OS\ =\ e_{rs\ }+\ e_{os}\)
\(RS\ -\ QS1\ =\ e_{rs}\ +\ e_{qs}\)
\(RS\ -\ QS2\ =\ e_{rs\ }+e_{qs}\ +\ e_{rot}\)
\(OS\ -\ QS1\ =\ e_{os}\ +\ e_{qs}\)
\(OS\ -\ QS2\ =\ e_{os}\ +\ e_{qs}\ +\ e_{rot}\)
\(QS1\ -\ QS2\ =\ 2e_{qs}\ +\ e_{rot}\)
Where RS is the result from the rapid splitting test, OS is the result of the original 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.
Again if no measurement error exists in any of the the three testing procedures, Equations 1, 2 and 4 and hence \(e_{rs}\)\(e_{os}\)and \(e_{qs}\) would be 0. Equations 3, 5 and 6 (the equations containing \(e_{rot}\)) would only also be zero if all of the stems contain a completely homogeneous strain field - in reality this is not the case, and therefore, if \(e_{rs}\)\(e_{os}\) and \(e_{qs}\) are zero, any differences between QS2 and other tests would be the result of the change in strain by cutting the sample at a different angle. If this assumption were to hold, the difference between the results from the first cut and the results from the second would provide a distribution with a mean of zero and a non-zero variance (assuming the samples are arbitrarily aligned). The distribution would be the expected difference between the cuts and it would become possible to estimate 95% confidence bounds on the value of another hypothetical cut on the same sample. 
Unfortunately none of these tests have zero measurement error, however, because there are three tests which should all produce the same result (as they are measuring the same thing) we can estimate the error associated with each test.  Note that this is not necessarily the error from the "true" value, but error in the sense of what confidence we can hold that the result of one sample from one test can estimate the hypothetical mean of the distribution formed by the same test run on the same sample in the same orientation an infinite number of times. The QS2 test encompasses two types of error, \(e_{rot}\), discussed above, and  \(e_{qs}\), the error associated with the quartering test with respect to the RS and OS tests. Fortunately because this test is used twice on each sample, the two errors are separable.

Results and Discussion

Population wide means and standard deviations for growth strain measured by the three different tests are shown in Table ----.