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

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