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Let $ X = \{ (x_i,y_i) \, | \, i \in I\}$ be a set of points (where $I$ is a finite index set).

Does the point $x_0 = \frac{1}{|I|} \sum_{i \in I} (x_i,y_i) $ have any name?

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It is the barycenter of the sets of points.

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In geometry it's called centroid, geometric center or barycenter.

Please note that depending on context, the latter term may be ambiguous because in physics barycenter has a slightly different meaning: the center of mass (which may or may not be the same as the centroid, depending on the distribution of mass).

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It can be called the first moment of the set of points. This notion of 'moments' generalizes a lot. For example, suppose that the points have an associated "weight" $w_i$. Then,

$$p = \sum_i w_i \cdot (x_i, y_i)$$

is the first moment with respect to $w_i$. In your case, $w_i$ is constantly $\frac{1}{|I|}$.

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  • $\begingroup$ I think you mean "first moment" or "first raw moment", rather than "first central moment". The first central moment is zero. $\endgroup$ – ruakh Aug 3 '14 at 4:15

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