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The z-score of an observation is the number of standard deviations above or below the mean of the distribution of observations from which it came.
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Z-Normalized Euclidean Distance Derivation
Here's a straightforward way. Let $\mathbf u$ and $\mathbf v$ be $m$-vectors.
Let $\mu_1=\frac 1m\sum_{k=1}^m u_k$ and $\mu_2=\frac 1m\sum_{k=1}^m v_k$ denote their means.
Let $\sigma_1^2=\frac 1 …