Since distance is a variable of a single dimension, it can not represent the spatial distribution of points in two dimensions.
If the points represent different observations of the same measurement, the Arithmetic Mean should be used. This is due to its properties:
- The mean is the only single number for which the residuals (deviations from the estimate) sum to zero.
- The arithmetic mean of the numbers does this best, in the sense of minimizing the sum of squared deviations from the typical value.
The Root Mean Square is used in periodic functions or continuous waveforms. Also due to its properties:
- The RMS over all time of a periodic function is equal to the RMS of one period of the function.
- The RMS value of a continuous function or signal can be approximated by taking the RMS of a sequence of equally spaced samples.
Some spatial information of the points can be inferred by analyzing the Moments of the distribution of their distances.
But if you want to analyze the spatial location of the points you need to made Multivariate Statistics, either of their cartesian (x and y coordinates) or of their polar (distances to the center and angle with respect to some axis) coordinates.