Gearys CEdit
Gearys C, commonly written as Geary's C, is a statistic used in spatial statistics to measure how much nearby locations resemble each other in the values of a variable of interest. The measure was introduced by the statistician R. C. Geary in the mid-twentieth century as part of a family of tools designed to detect spatial autocorrelation — the degree to which similar values cluster in space. Gearys C sits alongside Moran's I as a standard instrument in the toolbox of spatial analysis and is widely used in geography and related fields such as urban planning, criminology, epidemiology, and environmental science. It is especially valued for its sensitivity to local differences, which makes it a useful complement to global measures of spatial association.
Gearys C is defined with a weighting structure that encodes which locations are considered neighbors and how strongly they influence one another. The core idea is to compare pairs of neighboring units i and j by looking at how different their values x_i and x_j are, then to relate that sum of squared differences to the overall variation in the data. In practice, researchers specify a spatial weights matrix W, which captures contiguity or distance-based relationships (for example rook contiguity or queen contiguity in grid-like maps, or distance-based schemes for irregular regions). The statistic is then computed as a function of the weighted squared differences relative to the overall variance of the data.
Gearys C takes values roughly between 0 and 2, with 1 serving as a baseline corresponding to spatial randomness. Values below 1 indicate positive spatial autocorrelation: similar values tend to occur near each other (clustering of like values). Values above 1 indicate negative spatial autocorrelation: neighboring units tend to be dissimilar. Unlike some other spatial measures, Gearys C is particularly responsive to local dissimilarities, which makes it a useful diagnostic when one is concerned about neighborhood-scale patterns as well as broader trends. For continuous data, the interpretation follows this general rule; for binary or categorical data, adaptations of the method are used, often with the same guiding intuition about local versus global structure.
History
Gearys C was developed as part of the early development of formal spatial statistics in the 1950s and 1960s. It is often presented in relation to the better-known Moran's I; while Moran's I emphasizes global similarity or dispersion across the study area, Gearys C places more emphasis on the magnitude of local differences between neighboring units. The introduction of Gearys C contributed to a broader understanding that spatial patterns could be detected and quantified even when the overall mean and variance of a dataset suggested little structure. Today Gearys C remains a standard reference point in discussions of spatial autocorrelation and is supported by a wide range of software tools, including packages in the ecosystem around R (programming language) and dedicated programs like GeoDa for exploratory spatial data analysis.
Calculation and interpretation
Gearys C requires three ingredients: the data values x_i for each spatial unit i, the mean x̄ of those values, and a spatial weights matrix W with elements w_ij that describe the neighbor relation between units i and j. A common, published form of the statistic is
C = ((n − 1) / (2W)) × [ Σ_i Σ_j w_ij (x_i − x_j)^2 ] / [ Σ_i (x_i − x̄)^2 ],
where n is the number of spatial units, W is the sum of all weights (W = Σ_i Σ_j w_ij), and the double sum runs over all pairs of units. The denominator standardizes by the overall variance, making the ratio comparable across data sets. In practice, many users adopt row-standardized weights so that each unit’s neighbors sum to one, which can affect the scale and interpretation of C. Significance testing is commonly performed with permutation tests, in which the observed value is compared to a distribution created by randomly reallocating the observed values across units.
The weights matrix is central to the interpretation of Gearys C. Different choices of neighborhoods — for example, neighbors defined by sharing a boundary (contiguity-based), or by being within a fixed distance — can yield different C values for the same data. This sensitivity to the chosen neighborhood structure is a standard caveat of all spatial statistics and a frequent topic of methodological debate within the field. Users should report the weights scheme plainly and consider sensitivity analyses to show how robust conclusions are to reasonable alternative definitions of neighborhood.
Gearys C also has local variants, sometimes referred to as Local Geary’s C, which produce a C value for each spatial unit to illuminate where local pockets of similarity or dissimilarity lie on the map. Local measures complement global statistics like the overall C by revealing fine-grained patterns that might otherwise be hidden in a single summary number. Researchers often compare Gearys C with the more global Moran's I to obtain a fuller sense of both local and global spatial structure.
Applications of Gearys C span many domains. In urban geography, it helps analysts understand how social, economic, or environmental attributes cluster within cities. In criminology, it can shed light on the geographic concentration of crime or disorder and inform targeted policing or prevention efforts. In epidemiology and environmental science, Gearys C contributes to the study of disease incidence, pollution, or habitat fragmentation by highlighting where neighboring areas show similar or divergent conditions. Tools for computing Gearys C are built into several standard software environments, and researchers frequently present results alongside maps produced with Geographic Information Systems to communicate spatial patterns effectively.
Controversies and debates
As with many spatial measures, the interpretation of Gearys C depends on choices about the neighborhood structure encoded in the weights matrix, and this has drawn methodological critique. Critics emphasize that conclusions can be sensitive to the specification of neighbors, the scale of analysis (the modifiable areal unit problem, or MAUP), and the treatment of outliers. Proponents counter that, when the weights are chosen transparently and subjected to robustness checks, Gearys C provides valuable insight into how local interactions shape observed patterns and should be considered alongside other indicators rather than used as the sole basis for policy judgments. In practice, economists and policymakers tend to use a suite of tools, compare several neighborhood definitions, and anchor statistical conclusions with theory and domain knowledge.
From a practical governance perspective, some debates revolve around how to translate spatial autocorrelation findings into policy. Critics on the more expansive side of public policy sometimes push for broad interventions based on patterns detected by spatial statistics. Advocates of limited government and market-minded policy argue that statistical patterns are informative but should not automatically drive resource redistribution or regulation; instead, they should guide targeted investments in infrastructure, education, or services where they are most likely to improve efficiency and outcomes without imposing unnecessary costs or distortions. In these discussions, it is common to stress that statistical findings describe structure in space, not moral judgments about communities. When criticisms are framed as “woken” or as attempts to assign blame based on spatial patterns, the counterargument is that robust analytics can help tailor interventions to real needs and avoid one-size-fits-all programs that distort incentives.