Is it possible to find the mean of more than two points on a number line?

Question

I know you can find the mean of two points on a number line, just as you can find the mean of two numbers. However, I would like to know, is it possible to find the mean of THREE or more points on a number line? And if it is possible, how? And if this is possible, can you find the mean of three or more points on a line that is on a graph? Any answers would be appreciated.

Answer 1

The centroid of a triangle with vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ is the point whose cartesian components are the means of the individual $x$-coordinates and $y$-coordinates:
$$ (x_C, y_C) = \frac{1}{3} (x_1 + x_2 + x_3, y_1 + y_2 + y_3) $$ The centroid of a triangle can also be easily constructed by bisecting each side of the triangle and then connecting the three midpoints to the vertices opposite them. So if you want to find the arithmetic means of these three points, you just need to find the centroid.

If you only need to find the average of a single set of three numbers (rather than two sets, which is effectively what the above technique does), then you can construct a triangle whose $x$-coordinates are the set of numbers you want to average. (You also need to make sure that the three points don't all lie on a straight line, i.e., that you actually have a triangle.) Then just follow the above construction, and project the result back down onto the number line to get your answer. Here's how it would look if you were averaging 1, 3, and 8:

As desired, the average of the three numbers is 4.

This method would, in principle, generalize to sets of more than three numbers. However, to average $n$ numbers in this way would require drawing things in an $(n-1)$-dimensional space (I think). This is tricky enough for $n = 4$ (requiring 3D space); good luck with $n \geq 5$ (4-D space or higher.)

Answer 2

An attempt to solve the problem for any number of points on the number line while always being in 2-D space:

1. Label the points $P_1,P_2,...,P_n$ progressively from left to right;
2. Let $P_{i,j,...}$ represent the mean of points $P_i, P_j,...$
3. Find $P_{1,2}$
4. Find $P_{1,2,3}$ by dividing the line segment joining $P_{1,2},P_3$ in the ratio $1:2$ (Note $P_{1,2,3}$ will be twice as close to $P_{1,2}$ as it is to $P_3$. This can be done as explained here
5. In general, find $P_{1,2,...,n-1,n}$ by dividing the line segment joining $P_{1,2,...,n-1}$ and $P_n$ in the ratio $1:n-1$