Canberra Distance Calculator

Canberra distance is a weighted (relative) version of the Manhattan / L1 distance, introduced by Lance and Williams in 1966 for numerical taxonomy. For two equal-length vectors x = [x₁…xₙ] and y = [y₁…yₙ], it divides each absolute coordinate difference by the sum of the two magnitudes before adding them up. It is the metric SciPy calls canberra.

d(x, y) = Σ |xᵢ − yᵢ| / (|xᵢ| + |yᵢ|)

The tool computes this in three steps:

1. Per-coordinate term. For each dimension take the absolute difference |xᵢ − yᵢ| and divide it by the magnitude sum |xᵢ| + |yᵢ|. Because the numerator can never exceed the denominator, each term lands in [0, 1].
2. Zero convention. If a coordinate is zero in both vectors the term is 0/0, which is undefined. Following scipy.spatial.distance.canberra the tool sets that term to 0, so a both-zero pair contributes nothing rather than producing NaN. Each such pair is flagged in the working table.
3. Sum and normalize. Add the terms to get the Canberra distance. Since every term is at most 1, the total lies in [0, n]. The tool also reports the normalized value d ÷ n, which stays in [0, 1] and lets you compare vectors of different lengths.

The division by |xᵢ| + |yᵢ| is what sets Canberra apart from plain Manhattan distance: it turns each coordinate into a relative difference, giving disproportionate weight to gaps near zero. That is why it suits non-negative data spanning many orders of magnitude — a jump from 1 to 2 registers as a large term (0.333), while 1000 to 1001 barely moves the needle (0.0005). Unlike Mahalanobis distance it uses no covariance, and unlike cosine similarity it is not angle-based. To keep the result trustworthy, the calculator recomputes the distance a second way — through an equivalent sign-based form where opposite-sign pairs contribute 1, same-sign pairs contribute ||xᵢ| − |yᵢ|| / (|xᵢ| + |yᵢ|), and both-zero pairs contribute 0 — and confirms the two routes agree, exactly as they equal scipy.spatial.distance.canberra. All arithmetic is double-precision and rounded only for display.

• SciPy — scipy.spatial.distance.canberra (the canonical implementation and 0/0 = 0 convention this tool matches)
• Lance & Williams (1966), The Computer Journal 9(1): 60–64 — origin of the Canberra metric
• Wikipedia — Canberra distance (definition, range, and zero-handling convention)

The formula on this page was last cross-checked against these sources on 2026-07-16. Canberra distance is a stable mathematical definition, so this tool needs no rate or schedule updates — only the worked examples are periodically re-reconciled.