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I have a set of points whose positions are represented by 3D vectors. Each point has the associated weight in range from 0 to 1. The sum of all weights doesn't equal to 1.

How should the weighted mean point be calculated from the whole set of points?

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I don't understand "I have a set of points set by 3D vectors representing their positions." Do you mean something like "I have a set of points whose positions are represented by 3D vectors"? – joriki Oct 31 '12 at 12:42
Yes, you're right. – BartoNaz Oct 31 '12 at 12:42
up vote 1 down vote accepted

The process is called normalization, and you simly divide each weight by the sum of all the weights: $$w_i \rightarrow \frac{w_i}{\sum w_i}$$ You can easily verify that the sum of the new weights is now $1$.

The mean vector is given by the sum: $${\bf{v}} = \sum w_i \bf{v}_i$$

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Well, that is obvious. But that just makes the sum of weights equal to 1. But how to use these weights to calculate the weighted mean vector? – BartoNaz Oct 31 '12 at 13:23
@BartoNaz - Edited my answer. Is this what you meant? – nbubis Oct 31 '12 at 13:27
Probably yes. I think this should be correct. Thank you. – BartoNaz Oct 31 '12 at 13:53

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