Canberra Distance
Canberra Distance Metric · Also known as: Canberra metric, normalized Manhattan distance
Canberra distance is a weighted version of the Manhattan distance that normalizes differences by the sum of absolute values. Introduced by Geoffrey Lance and William Williams in 1967 as part of their work on clustering classification methods, this metric emphasizes differences in small values and is sensitive to changes in relative proportions. It is commonly used in taxonomy, ecology, decision-making, and any application where normalized relative differences matter.
Read the full method
Sign in with a free account to read this section.
Method map
The neighbourhood of related methods — select a node to explore.
When to use it
Canberra distance is appropriate when relative differences are more meaningful than absolute differences, especially in ecological data, preference vectors, or proportional data. It is sensitive to changes in small values and robust against scale differences. Use it when comparing compositional or normalized data and when you need to penalize proportional mismatches. Avoid it when zero values are present or when absolute magnitude is the primary concern.
Strengths & limitations
- Emphasizes relative differences over absolute ones; suitable for proportional data
- Naturally sensitive to small-value differences without explicit weighting
- Intuitive interpretation in contexts where percentage or ratio changes matter
- Useful for comparing vectors with different scales or compositional data
- Undefined or problematic when both components are zero in any dimension
- Biased toward small values; can produce overly large distances when comparing near-zero numbers
- Computationally more expensive than simple Manhattan distance due to normalization
- Less suitable when absolute magnitude differences are the primary concern
Frequently asked
What is the key difference between Canberra and Manhattan distance?
Manhattan distance sums absolute differences directly. Canberra distance normalizes each difference by the sum of the absolute values of the two components, then sums these normalized ratios. Canberra emphasizes relative differences and is sensitive to small-value changes.
How do zero values affect Canberra distance?
If both components are zero in any dimension, the formula's denominator is zero, making the metric undefined. Most implementations skip that dimension or treat it as zero contribution. Handle zero values carefully in preprocessing.
When should I use Canberra over Euclidean distance?
Use Canberra when relative/proportional differences matter more than absolute spatial distance, such as with compositional data, ecological communities, or normalized preference profiles. Use Euclidean when measuring absolute spatial separation.
Does Canberra distance scale-invariant?
Not fully. While it normalizes at the component level, scaling both vectors by the same factor does not preserve the distance. However, it is more robust to scale differences than absolute metrics like Manhattan distance.
Sources
- Lance, G. N., & Williams, W. T. (1967). A general theory of classificatory sorting strategies. Computer Journal, 10(3), 271-277. DOI: 10.1093/comjnl/10.3.271 ↗
- Cantrell, C. D. (1971). A review of taxonomic methods. Taxon, 20(2), 157-175. link ↗
How to cite this page
ScholarGate. (2026, June 3). Canberra Distance Metric. ScholarGate. https://scholargate.app/en/decision-making/canberra-distance
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Bray-Curtis DissimilarityDecision-making↔ compare