Written and maintained by CASRAI Editorial Board
Last updated
The modifiable areal unit problem (MAUP) is the way statistical results change when the same underlying data are summarised over different areal units. The units (census tracts, zip codes, districts, grid cells) are drawn by people and could be drawn differently, so a mean, a correlation or a regression slope computed on one set of boundaries is not guaranteed to match the one computed on another. The problem has two parts: the scale effect (the same area divided into more or fewer units) and the zoning effect (the same number of units drawn with different boundaries).
This page is for researchers who analyse area-level data and need to recognise the problem, see it in numbers, and report it properly. It sits alongside the guides on spatial data analysis and the ecological fallacy.
What is the modifiable areal unit problem?
Wong (2004) describes the MAUP as a problem that arises because boundaries such as administrative areas, political districts and census units are artificially demarcated and can be redrawn, so different boundary definitions produce different data sets and likely inconsistent results. Gehlke and Biehl (1934) had already observed aspects of it in the effect of grouping on correlation size; the name was not coined formally until Openshaw and Taylor (1979).
Scale effect
The scale effect appears when smaller units are merged into fewer, larger ones (or larger units are subdivided). The data change spatial resolution, and statistics change with them. When neighbouring units with different values are merged, their values are averaged, so variation is smoothed away: the mean of a variable is preserved by aggregation, but its variance falls.
Zoning effect
The zoning effect appears when the number of units stays roughly the same but the boundaries are drawn differently. Wong gives two examples: Washington, D.C. split into census tracts versus zip code areas, and the United States split into the nine Census divisions versus the ten EPA administrative regions. Wong also notes that the direction of change between partitioning schemes cannot be predicted.
Formula: is there one?
There is no single formula for the MAUP, because it is a property of how data are partitioned, not a statistic. What can be written down is the recomputation step. When the units i in zone z are merged, the zone value is usually the mean, x̄z = (1/nz) Σ xi, where nz is the number of units in zone z. The correlation is then recomputed on the zone values:
r = Σ (x̄z − x̿)(ȳz − y̿) / √[ Σ (x̄z − x̿)² · Σ (ȳz − y̿)² ]
Here x̄z and ȳz are the zone means of the two variables, x̿ and y̿ are the means of those zone means, and the sums run over zones. Because the zone means depend on which units were grouped, r depends on the zoning. See Pearson r step by step for the unaggregated version.
Worked example: one data set, three correlations
Illustrative toy data, constructed and computed for this page; not a real study. Eight small areal units sit on a 2 × 4 grid, numbered across the top row (1–4) then the bottom row (5–8). Each has two variables:
| Unit | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| x | 2 | 4 | 3 | 6 | 5 | 8 | 7 | 9 |
| y | 7 | 3 | 6 | 2 | 8 | 4 | 9 | 5 |
At unit level. x̄ = 44/8 = 5.5 and ȳ = 44/8 = 5.5. The sum of cross-products of deviations is −6, Σ(x − x̄)² = 42 and Σ(y − ȳ)² = 42, so r = −6 / √(42 × 42) = −6/42 = −0.143.
Zoning A (horizontal pairs): merge units (1,2), (3,4), (5,6), (7,8). Zone means: x = 3, 4.5, 6.5, 8; y = 5, 4, 6, 7. Both grand means are 5.5. Deviations of x: −2.5, −1, 1, 2.5; of y: −0.5, −1.5, 0.5, 1.5. Cross-products sum to 1.25 + 1.5 + 0.5 + 3.75 = 7.0; Σ(x − x̿)² = 14.5; Σ(y − y̿)² = 5. So r = 7 / √(14.5 × 5) = 7 / √72.5 = 7 / 8.515 = +0.822.
Zoning B (vertical pairs): merge units (1,5), (2,6), (3,7), (4,8). Zone means: x = 3.5, 6, 5, 7.5; y = 7.5, 3.5, 7.5, 3.5. Grand means are again 5.5. Deviations of x: −2, 0.5, −0.5, 2; of y: 2, −2, 2, −2. Cross-products sum to −4 − 1 − 1 − 4 = −10; Σ(x − x̿)² = 8.5; Σ(y − y̿)² = 16. So r = −10 / √(8.5 × 16) = −10 / √136 = −10 / 11.662 = −0.857.
Both zonings have four zones and the same eight underlying units, and both preserve the means, yet one correlation is strongly positive and the other strongly negative. Going from eight units to four zones is the scale effect; choosing horizontal rather than vertical pairs is the zoning effect. Openshaw and Taylor (1979) showed, using simulated zonings, that correlations between two variables could be made to span the full range from −1 to +1 (as summarised in Penn State’s GEOG 586 course materials); this toy example shows the same behaviour in miniature.
A real example: Washington, D.C.
Wong (2004, Table 93.1) reports averages for Washington, D.C. computed from census units at different levels. D.C. is divided into 188 census tracts, 433 block groups (2000 Census) and 27 zip code areas.
| Level | Proportion non-white | Proportion elderly (65+) |
|---|---|---|
| Entire D.C. (reference) | 0.6922 | 0.1222 |
| Census tracts | 0.7241 | 0.1200 |
| Block groups | 0.7181 | 0.1309 |
| Zip code areas | 0.5613 | 0.1151 |
The same source reports a standard deviation for the proportion non-white of 0.3270 at tract level and 0.3436 at block-group level, illustrating how the coarser tract data were smoothed and lost variation, and a correlation between white and elderly population counts of 0.3247 at tract level and 0.2800 at block-group level. Aggregation tends to raise correlations (Fotheringham and Wong 1991), but that is a tendency, not a rule.
How the MAUP affects correlation and regression
Wong (2004) identifies impacts on descriptive statistics, standard procedures such as multiple and logistic regression, and spatial models such as gravity-type interaction models. Because almost every multivariable method starts from correlations, results that differ across scales imply that other statistical results differ too. The MAUP is also described as one source of the ecological fallacy, in which relationships estimated for areas are read as relationships for individuals.
Reading guidance
| What you see | What it suggests |
|---|---|
| Variance falls when you aggregate; mean unchanged | Scale effect: averaging smooths neighbouring differences (Wong 2004; Penn State GEOG 586) |
| Different results from two zoning systems with similar unit counts | Zoning effect (Wong 2004) |
| Correlation or slope changes sign or size across zonings | The result is sensitive to the units chosen; report the range across data sets or zonings (Fotheringham 1989, cited by Wong 2004) |
Assumptions and common mistakes
- Treating the published boundaries as natural. Census tracts, zip codes and districts exist for administrative reasons; they are one of many possible partitions.
- Reporting one scale only. A single correlation or regression from one unit system hides how fragile it may be.
- Assuming aggregation always raises correlation. It is a tendency; the D.C. count correlations above are higher at the tract than the block-group level, and Wong states the direction of change between partitioning schemes cannot be predicted.
- Expecting a fix. Wong (2004) states that no general solutions exist. Approaches discussed include treating the zoning problem as interpolation (Fisher and Langford 1995), scale-insensitive techniques with limited success (Tobler 1989; Wong 2001), and reporting results across multiple data sets or zonings, which GIS makes feasible.
- Drawing individual-level conclusions from area data. See the ecological fallacy guide.
How to report it (APA style)
Because no standard test exists, report the sensitivity analysis you ran. A format consistent with APA 7th-edition reporting of correlations (no leading zero for r):
The area-level association between x and y was r = −.14 at the unit level (N = 8), r = .82 under horizontal zoning (N = 4) and r = −.86 under vertical zoning (N = 4), indicating that the estimate depends on the areal units used (modifiable areal unit problem; Openshaw & Taylor, 1979).
State the unit systems, the number of units in each, and the data year. See how to report a correlation coefficient for the general pattern.
How to calculate it in Excel, R or Python
There is no built-in MAUP function; the check is to aggregate under alternative zonings and recompute the statistic.
Excel
Put units in rows with columns for x, y and a zone ID per zoning. Build a zone table with =AVERAGEIFS(B:B, D:D, F2) for each zone (B = x, D = zone ID, F2 = the zone), the same for y, then =CORREL(G2:G5, H2:H5) on the zone means. Repeat for each zoning column.
R
df <- data.frame(x = c(2,4,3,6,5,8,7,9), y = c(7,3,6,2,8,4,9,5), zone_h = c(1,1,2,2,3,3,4,4), zone_v = c(1,2,3,4,1,2,3,4))cor(df$x, df$y)for (z in c("zone_h", "zone_v")) { g <- aggregate(df[c("x", "y")], by = list(zone = df[[z]]), FUN = mean) print(cor(g$x, g$y))}Python (pandas)
import pandas as pddf = pd.DataFrame({"x": [2,4,3,6,5,8,7,9], "y": [7,3,6,2,8,4,9,5], "zone_h": [1,1,2,2,3,3,4,4], "zone_v": [1,2,3,4,1,2,3,4]})print(round(df["x"].corr(df["y"]), 3)) # -0.143for z in ["zone_h", "zone_v"]: g = df.groupby(z)[["x", "y"]].mean() print(z, round(g["x"].corr(g["y"]), 3)) # 0.822, -0.857With real boundaries, the GeoPandas spatial join and dissolve operations can assign units to alternative zonings before the same aggregation.
FAQ
What is the difference between the scale effect and the zoning effect?
The scale effect is a change in results when the same area is divided into a different number of units; the zoning effect is a change when the number of units stays similar but the boundaries differ.
Is the MAUP the same as the ecological fallacy?
No. The ecological fallacy is inferring individual-level relationships from area-level data; Wong (2004) describes the MAUP as one source of it, not a synonym.
Is there a fix for the MAUP?
Wong (2004) states no general solutions exist. The practical response is to analyse at several scales or zonings and report the range of results.
Why is it called “modifiable”?
Because the areal boundaries are artificially demarcated and can be redrawn, which changes the resulting data.
Does aggregation always increase correlation?
No. It tends to, according to Fotheringham and Wong (1991) as cited by Wong (2004), but the direction cannot be predicted between partitioning schemes.
Sources
- Wong, D. W. S. (2004). The modifiable areal unit problem (MAUP). In D. G. Janelle et al. (Eds.), WorldMinds: Geographical perspectives on 100 problems (pp. 571–575). Kluwer Academic Publishers. Read 2026-11-08.
- Fotheringham, A. S., & Wong, D. W. S. (1991). The modifiable areal unit problem in multivariate statistical analysis. Environment and Planning A, 23, 1025–1044.
- Openshaw, S., & Taylor, P. J. (1979). A million or so correlation coefficients: Three experiments on the modifiable areal unit problem.
- Gehlke, C. E., & Biehl, K. (1934). Journal of the American Statistical Association, 29 (Supplement), 169–170.
- Penn State, GEOG 586, “The Pitfalls of Spatial Data, II”: courses.ems.psu.edu/geog586/node/644. Read 2026-11-08.
