Significance of Centroid of a Gaussian distribution

Question

I am learning some of the parameters used for a gaussian function (or normal distribution). What is a centroid really? Is that just referring to the point where the y is the highest?

What are some instances that centroid is used?

Thanks!

Answer 1

For the normal distribution the centroid happens to be the point $x_0$ where $y$ (the PDF) is the highest because the normal distribution is symmetrical around that point. But in general for other distributions it is the point $x_0$ where the CDF of the random variable $X$ equals $1/2$.

This $x_0$ is also called the mean of the distribution.

In other words it is a point $x_0$ where if you put a stick perpendicular to the $X$ axis and pointing upwards at $x_0$ (or say a rope pointing downwards at $x_0$), the graph of your density function (and what is under it) will be in perfect balance i.e. will not bend either to the left or to the right.

Think of the object under the graph of the PDF as some mass (a unity mass). The centroid is that point $x_0$ on the $X$ axis such that half of the mass is to the left, and half of it is to the right.

Here is a formal definition which applies not only to functions which are PDFs, but to any type of function.