Mahalanobis Distance: Why Units and Correlation Matter

State the technical definition before substituting numbers

Chaudhuri’s 2020 review gives D²=ΔᵀΣ⁻¹Δ, where Δ is the difference between means and Σ is their common nonsingular covariance matrix. This squared distance accounts for the covariance structure, instead of simply adding squared coordinate differences.

For the invented example, supply covariance matrix S=[[4,2],[2,9]] and a difference vector d=(2,3). S is symmetric, its first diagonal entry is positive and its determinant is 32, so this two-dimensional covariance model is positive definite. These supplied values have no connection with an observed population.

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Calculate before converting the units

The inverse is S⁻¹=(1/32)[[9,−2],[−2,4]]. Multiplying out gives dᵀS⁻¹d=(9×2²−4×2×3+4×3²)/32=48/32=1.5. This is the squared score. Its square root is approximately 1.225; reporting 1.5 as the unsquared distance would change the quantity.

For comparison, raw squared Euclidean distance is 2²+3²=13. If the off-diagonal entries were replaced by zero while keeping the supplied variances, the weighted squared score would be 2²/4+3²/9=2. That is a changed model, not an alternative calculation using the original covariance.

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Convert the vector and the covariance together

Rescale only the first measurement by 100. The new difference is (200,3), and the covariance becomes [[40000,200],[200,9]]. The first variance changes by 100², while each cross-covariance changes by 100. Applying the same quadratic calculation gives 1.5 again. Raw squared Euclidean distance instead becomes 40,009.

The invariance holds because both the measurements and their covariance were transformed consistently. Converting the vector while leaving the original covariance matrix unchanged would mix units. It would not be a valid test of this model. Replacing or estimating the covariance differently is another change that should be disclosed.

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Exercise: equal raw lengths, different directions

Keep the original S and compare d₊=(2,3) with d₋=(2,−3). Both have raw squared length 13. Calculate the quadratic score for the second direction, then explain what the comparison establishes.

Reference answer: the second numerator is 36+24+36=96, giving a squared score of 3. The first score was 1.5. The sign change affects the cross term even though the raw lengths match. Under this supplied covariance, direction matters. The calculation does not prove that a real observation is abnormal, assign a causal explanation or validate an estimated covariance matrix.

As a final check, distinguish the assumptions from the outputs: the matrix and vectors were supplied; the inverse and scores were derived. A real application would need suitable measurements and a justified covariance estimate. No human classification, historical result or probability threshold is inferred from this exercise.

Sources for this section: [1]