Cartographic Projection
every map is wrong, and by exactly how much
Concept

Choropleth — where it appears

A map that colours whole regions by a value each one carries. Its visual weight is the page area each colour covers, so the impression it delivers is an area-weighted mean, and the area doing the weighting is the projected one rather than the ground's.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The same field, the same statistic, two pages. A single northern concentration — v = 10 + 90 exp(−d²/(25°)²), d the distance from 20°E 55°N — drawn twice, with the shading rule and the data identical. The ground's own area-weighted mean is 14.009. A reader weighting by the area actually on the page takes 17.490 off Mercator, which is +24.85%, and 14.009 off Gall–Peters, which is +0.00%. Neither map has misdrawn a single cell.

A choropleth is read by area

Every cartography course states the rule — use an equal-area projection for a thematic map — and states it as advice. It is a theorem, and it has a residual: the error a page puts into a reading is exactly the covariance of the value with the areal factor, which is 24.85 per cent for a northern concentration read off Mercator and 0.00 per cent for the same field read off a map that spreads area by 7.7 to one.

distortion · Thematic One field, one classifier, two sets of breaks. A field that varies with latitude put into 5 classes by area-weighted quantiles, on Mercator. The upper panel weights each region by its ground area and the lower by the area it occupies on this page, which is what a classifier handed projected geometry does. 60 of 150 regions land in a different class, marked in the lower panel. The data has not changed and neither has the number of classes.

The class breaks were computed on the page

The three rungs below price what a reader does with a finished map. A classifier is software, it runs on the geometry it has, and the geometry it has is projected: a five-class quantile classification of one stated field puts half of the three hundred and eighty-four regions in a different colour on Mercator, and 87.5 per cent of them at nine classes.

distortion · Thematic

Named alongside it

The objects these essays reach for when they reach for this one.

Area weightingAreal factorBiasEqual-areaMercatorThematic mappingAggregationClassificationCovarianceDensityGall–PetersNatural breaks

All concepts