I have multiple samples on $n$-dimensional random vectors (from a time-series). I'll like to have a unidimensional measure of its variability.
A natural one (discussed here) is to extend the variance for n-dimensional vectors $$ \mathbb{E}\left(\sum_{i=1}^n (X_i-\mathbb{E}(X_i))^2\right) = \sum_{i=1}^n \mathbb{V}(X_i)$$ however this measure can have scale-problems, for example, if the magnitude of the values in one dimension is considerably larger than the other dimensions.
Which measures are appropriate for this case?