Mean distance of the points from a fixed target point - r

Question

I have some points in my data and I am trying to find the mean (arithmetic mean) distance to a target point.
I am taking two routes here:
One, using the 'Distance Between Two Points' formula to compute the distance between each point to the target and then getting a mean distance of those distance values.
Other, find the mean point of all points and then find the distance between this mean point and the target point.
I am not sure which approach is correct (both give different answers) if I want to get the average distance of all points to a target point?
My target point is in green and mean point is in red.
Following is my R code:
# three data points
a <- c(1.6, 2.3, 3.4)
b <- c(3.1, 4.1, 0.5)
# target point
t_x <- 1.1
t_y <- 0.1
df <- data.frame("x" = a, "y" = b)
# mean of the distances
df$distance <- sqrt(((df$x - t_x)^2) + ((df$y - t_y)^2))
print(mean(df$distance))
# distance from the mean point to the target
mean_x <- mean(df$x)
mean_y <- mean(df$y)
print(sqrt((mean_x - t_x)^2 + (mean_y - t_y)^2))
# plotting all
ggplot(df, aes(x = x, y = y)) +
geom_point() +
coord_cartesian(xlim = c(-5,5), ylim = c(-5,5)) +
geom_point(aes(x=mean_x, y=mean_y), color = "red") +
geom_point(aes(x=t_x, y=t_y), color = "green")

Answer 1

Just consider the case with one target point and two other points
Target: (0,0)
Point1: (-1,0)
Point2: (1, 0)
If you take the average of Points 1 and 2, you get (0,0) so the average distance to the target is 0. But the distance from each of the points to the target is 1 so the mean distance in 1. In general these two calcuations are quite different
It really just comes down to how you define the problem. In this example, do you expect the answer to be 0 or 1.

Answer 2

Here's an intuitive explanation of why your first method is the correct one. Imagine your target is at (0, 0):
t_x <- 0
t_y <- 0
Now suppose we draw some points around it that are all the same distance from it - in fact, they all lie on the unit circle and by definition are a distance of 1 from the target:
library(ggplot2)
t_x <- 0
t_y <- 0
rads <- seq(0, 2 * pi, length.out = 17)[-17]
df <- data.frame(x = cos(rads), y = sin(rads), xend = 0, yend = 0)
ggplot(df, aes(x, y)) +
geom_point(col = "red") +
geom_segment(aes(xend = xend, yend = yend), linetype = 2) +
coord_equal() +
geom_point(x = t_x, y = t_y, size = 5, colour = "red")
Now, not surprisingly, since all points are distance 1 from the target, the average distance by Pythagoras will also be 1:
# mean of the distances
df$distance <- sqrt(((df$x - t_x)^2) + ((df$y - t_y)^2))
print(mean(df$distance))
#> [1] 1
But now consider what happens if we take the average of all our x values - they cancel to 0. So do the y values, so the average of all points is (0, 0). When you measure the distance from (0, 0) to (0, 0), the answer, of course, is 0:
mean_x <- mean(df$x)
mean_y <- mean(df$y)
print(sqrt((mean_x - t_x)^2 + (mean_y - t_y)^2))
#> [1] 0
Created on 2020-08-22 by the reprex package (v0.3.0)