# 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 known as

A **centroid** is also referred to as a **center of mean position**.

Some sources refer to it as a **mean point**.

Approaches to this subject from the direction of **physics** and **mechanics** can be seen referring to it as a **center of gravity**.

However, it needs to be noted that the latter is merely a special case of a **centroid**.

Beware that some sources use the term **center of gravity** even when approaching the topic from a pure mathematical perspective, which can cause confusion.

## Also see

- Results about
**centroids**can be found**here**.

## Sources

- 1933: D.M.Y. Sommerville:
*Analytical Conics*(3rd ed.) ... (previous) ... (next): Chapter $\text I$. Coordinates: $12$. Mean points and centres of gravity