Given n stock time series X₁, X₂, ... X_n, I want to measure how much they have dispersed over time. For example, are they moving "more together" this year comparing to last year?

If there are only 2 time series, then I can just calculate the correlation of X₁ and X₂.

For a large number of n, I would like to know if there are proper statistical measurements for this.

There are many ways to quantify how assets disperse over time.

The most common approach is called "Cross-sectional Return Dispersion". At each monthly cross-section, it calculates the universal dispersion as the standard deviation of monthly returns:

Fei et al. (2019) modified the formula above with market cap weights as follows:

To make the estimate less noisy, you could also use portfolio or factor returns instead of stock returns. The authors also introduce a heterogeneous autoregressive (HAR) version of CSSD

You can calculate the "Average Pairwise Correlation"

where w_i are the portfolio or market cap weights. It excludes the diagonal elements of the correlation matrix.

Note that it is not the same as dispersion but still gives an idea how assets move together

You can also run a PCA and compare if the first principal component takes up more variance