I am thinking of the vector as a point in multidimensional space. The Mean would be the location of a vector point with the minimum squared distances from all of the other vector points in the sample. Similarly, the Median would be the location of the vector point with the minimum absolute distance from all the other vector points.
Conventional thinking would have me calculate the Mean vector as the vector formed from the arithmetic mean of all the vector elements. However, there is a problem with this method. If we are working with a set of unit vectors the result of this method would not be a unit vector. So conventional thinking would have me normalize the result into a unit vector. But how would that method apply to other, non-unit, vectors? Should we divide by the arithmetic mean of the vector magnitudes? When calculating the Median, should we divide by the median of the vector magnitudes?
Do these methods produce a result that is mathematically correct? If not, what is the correct method?
An approach might be (stolen from mechanics) to think of vectors as locations of point masses in space. Its center of gravity (cog) is calculated then by
r_c = 1/M * \sum{m_i*r_i} with M = \sum{m_i}
where r_i are the vectors in a space of any dimension.
Setting m_i in your application to unit mass '1' gives ...
r_c = 1/N * \sum{r_i} with N = total number of vectors
The same problem appears with sets of scalars. Say, if you have a set of just two scalars of modulus 1: {-1, 1}, then the natural mean is 0. If you intend to find a "mean value" of modulus one, I don't see any natural choice for this.
By the way, analogous questions is a popular subject in combinatorial geometry. One may minimize several different quantities related to a given finite set in a normed space. See, for example https://link.springer.com/article/10.1023%2FA%3A1004934900506
Another keyword related to this kind of problems is Steiner point of a set.
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