A question, and a comment, raised in my mind from recent postings.

A recent post talked about axis orientations "averages" and that
observed values varied 5 to 10 degrees from this average. What, exactly,
does that mean?

In 3D space the orientation of a non-orthogonal axis needs to be
described by two angle values -- elevation and rotation, or theta and
rho. That means that the axis orientation is a multivariate descriptor.
What is meant by "average" in this application?

If, by average, you mean "the arithmetic mean" that concept is invalid
for multivariate data. While you could calculate the mean value of theta
and the mean value of rho, those two mean values taken together are not
necessarily the appropriate central (mean) value of orientation (mean in
this case being the minimized squared distance from all other data
points, which is what the arithmetic mean does).

You could calculate the median values for theta and rho as your meaning
of "average". Taken together, the median theta-rho couplet would
describe a multivariate point that has 50% of the data distribution
around it (the meaning of median). However, you could not use the same
method to calculate percentiles (or quartiles) that would have any
descriptive validity.

Finally, people frequently make statements like "5 degrees from the
average." Since, the orientation is three dimensional, does this mean to
imply a 5 degree sphere around the centroid point? Or, does it imply a 5
degree variance on only one of the descriptor angles? From my reading of
the literature, I would infer that most authors mean the latter,
although their words more appropriately describe the former.

So, when you say "average" for multivariate biomechanical data, what do
you mean?

Thomas M. Greiner, Ph.D.
Assistant Professor of Anatomy
Dept. of Health Professions
University of Wisconsin - La Crosse