Show that one can compute the variance of a set of...

Show that one can compute the variance of a set of measurements without reference to the mean by the following formula
$$
\sigma_x^2=\frac{\nu}{2 m} \sum_{i=1}^m \sum_{j=1}^m\left(x_i-x_j\right)^2=\frac{\nu}{m} \sum_{i<j}\left(x_i-x_j\right)^2 .
$$

Key Concepts

Double Summation and Combinatorial Symmetry

The double summation over all pairs, and the recognition that each unique pair is counted twice, illustrate the role of combinatorial symmetry in statistical derivations. This symmetry simplifies the calculation by allowing a reformulation of the variance formula in terms of summing over only the distinct pairs, which emphasizes the inherent structure and redundancy in the data relationships.

Pairwise Differences

The pairwise differences approach to variance calculates the spread of data by considering the squared differences between every pair of measurements rather than differences from the mean. This method leverages the relationships among the data points and, with the proper normalization, yields the same variance result as the mean-based calculation.

Mean

The mean is the arithmetic average of a set of measurements, often used as the central or typical value in a dataset. While the standard formula for variance directly references the mean, alternative formulations of variance, such as those involving pairwise differences, allow computation of spread without explicitly calculating the mean.

Variance

Variance is a fundamental statistical measure of dispersion that quantifies the spread of a set of data points around their central value. Traditionally, variance is computed as the average of the squared differences between each measurement and the mean, providing insight into the degree to which individual data points differ from this typical value.

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