# Name of the point whose coordinates are the mean of the coordinates of a list of points.

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?

It is the barycenter of the sets of points.

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).

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 $\frac{1}{|I|}$.

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