Definition:Centroid of Set of Points/Definition 2

Definition
Let $S = \set {A_1, A_2, \ldots, A_n}$ be a set of $n$ points in Euclidean space. Let the Cartesian coordinates of the elements of $S$ be $\tuple {x_j, y_j, z_j}$ for each $j \in \set {1, 2, \ldots, n}$.

Let $G$ be the point whose Cartesian coordinates are given by:


 * $G = \tuple {\dfrac 1 n \ds \sum_{j \mathop = 1}^n x_j, \dfrac 1 n \ds \sum_{j \mathop = 1}^n y_j, \dfrac 1 n \ds \sum_{j \mathop = 1}^n z_j}$

That is, the arithmetic mean of the Cartesian coordinates of the elements of $S$

Then $G$ is known as the centroid of $S$.

Also see

 * Equivalence of Definitions of Centroid of Set of Points