A choropleth is read by area

Two hundred and sixty-six essays in this collection measure what a map does to the ground. Not one of them measures what it does to the number a reader carries away — and that number is the reason most maps are drawn at all.

A thematic map is not a picture of the ground. It is an argument about a quantity, delivered by colouring regions, and the reader’s whole apparatus for reading it is the eye’s estimate of how much of the page each colour occupies. That estimate is a weighted mean with the page areas as weights. The truth is the same weighted mean with the ground areas as weights. The two disagree, and by exactly how much is not a matter of taste.

The rule everybody states #

The advice is universal and it is correct. Use an equal-area projection for a thematic map. It appears in every textbook, every style guide and every well-meaning thread about Mercator against Peters, and it is offered as a rule of thumb — a sensible default, a professional courtesy, the sort of thing a careful cartographer does.

It is not a rule of thumb. It is the sufficient condition of a theorem, one of several, and the theorem says something the rule does not: exactly how wrong a reading is when the condition fails, and exactly when it is right anyway.

Both halves matter. A rule with no magnitude attached cannot be traded against anything — it cannot tell a designer whether the equal-area map’s shapes are worth the cost, because it does not price the alternative. And a rule with no exceptions attached is wrong about the cases where it does not bind, which turn out to be common.

The identity #

Let each region i carry a value vᵢ, occupy a ground area Aᵢ, and be drawn on the page with area Aᵢsᵢ, where sᵢ is the projection’s areal scale factor there. The areal factor is not a new quantity: it is the determinant of the Jacobian over the area element on the sphere, and it is what Tissot’s indicatrix has been handing over at every point since the first essay in this collection.

The ground’s answer is the area-weighted mean

$$\bar{v}_{\text{ground}} = \frac{\sum_i A_i v_i}{\sum_i A_i},$$

and the page shows

$$\bar{v}_{\text{page}} = \frac{\sum_i A_i s_i v_i}{\sum_i A_i s_i}.$$

Subtract them and the second is a ratio of two ground-weighted sums, so both are means under one weighting and the difference collapses:

$$\bar{v}_{\text{page}} - \bar{v}_{\text{ground}} = \frac{\operatorname{Cov}(v, s)}{\bar{s}},$$

with the covariance and the mean both taken under the ground weighting. Three lines of algebra, no approximation, and the map’s entire contribution to the misreading is one covariance.

What was computed, and how #

Nothing here comes from a dataset, for the reason this collection’s coastline decision records at length: a polygon has a simplification level in it, and a number computed off one is partly a measurement of somebody’s generalisation.

The regions are latitude–longitude cells, 24 by 16 between ±80°, three hundred and eighty-four of them. A cell’s ground area is exact —

$$A = \Delta\lambda\,(\sin\varphi_2 - \sin\varphi_1)$$

on the unit sphere, the same closed form an area on the grid is not an area rests on. Its page area is the shoelace of its projected boundary, densified to sixteen points a side, and that is the only quadrature in the calculation.

The values are stated functions of position rather than statistics of anything. The headline field is a single northern concentration, v = 10 + 90 exp(−d²/25°²) with d the angular distance from 20°E 55°N, which is the shape almost every real thematic map has: a quantity that is large in one part of the world and small elsewhere. A mid-latitude band, v = 20 + 70 exp(−((|φ| − 45°)/18°)²), is the second, and three controls complete the set.

The identity is exercised rather than quoted. Every call computes the bias both ways — as a difference of weighted means and as a covariance over a mean — and requires them to agree. Across five projections and two fields the largest disagreement anywhere is 4 × 10⁻¹⁶ of the reading itself, which is the arithmetic and not the identity.

The quadrature is the only soft number, and it has an order #

A page area is the one quantity here that is not closed-form, so it is the one place an artefact could hide. The separation is the same refinement argument the indicatrix is a limit makes for a different quantity.

Ten equal-area projections are read against three fields, thirty numbers, and all thirty are supposed to be zero. Seven of the ten come back at 10⁻¹⁵ — Gall–Peters, the Lambert cylindrical, Behrmann, the sinusoidal, Albers and the two conic forms — because their cells project to straight-sided polygons and the shoelace of a straight-sided polygon is exact. The three whose parallels bend do not: the Lambert azimuthal reads 2.35 × 10⁻⁴ high at sixteen points a side.

Refine the ring by three and that falls to 2.61 × 10⁻⁵; refine again and it is 2.90 × 10⁻⁶. The fitted orders are 1.9994 and 1.9999. It is second-order in the densification, which is what a polygonal approximation to a curve has to be, and therefore a fact about the drawing routine rather than about the projection. Mercator’s 24.85 per cent does not move at all.

Four ways the bias vanishes #

Here is where the rule of thumb stops describing what is going on.

The map is equal-area. Then s is constant, the covariance of anything with a constant is zero, and the bias vanishes for every field there could ever be. This is the case the rule of thumb is about, and it is the only one of the four that is a property of the map alone.

The field is constant. Then v has no variation to covary with, and the bias vanishes for every projection there could ever be. Trivial, and worth stating because it identifies what the bias is actually about: not the map’s distortion, but the interaction between the map’s distortion and the data’s variation.

The field varies with longitude alone and the map’s areal factor with latitude alone. Every cylindrical projection has an areal factor that is a function of latitude only. A field v = 50 + 40 cos 2λ therefore has covariance exactly zero with it over a full covering, term by term, and reads off Mercator to 10⁻¹⁵. A thematic map of such a quantity on Mercator is perfect.

The field is odd about the equator and the map is even. This one is the least expected and the most instructive. Take v = 50 + 40 sin φ, a quantity rising steadily from the south pole to the north. Mercator’s areal factor is even about the equator; the field’s departure from its mean is odd; the product is odd; the sum over a symmetric covering is zero. Mercator reads that field exactly right, to 10⁻¹⁵, while inflating the far north by a factor of fifteen.

That last one is not a curiosity. It says that a projection which manifestly ruins the picture can leave the statistic untouched, and that a reader who has learned “Mercator distorts, therefore Mercator misleads” has learned a conflation rather than a fact.

Distortion does not predict the error #

The four zeros already imply that the bias cannot be a function of the map alone. The library shows how far that goes.

Read the two columns together on the mid-latitude band rather than the concentration, and the relation falls apart.

Miller’s cylindrical spreads area by nearly eight to one and moves this reading by two parts in a thousand. Robinson spreads by two to one and moves it by eleven times as much. A designer choosing between them on the strength of an areal audit would choose exactly backwards for this field.

The mechanism behind Miller’s near-miss is visible in the sign column. Miller’s areal factor grows monotonically away from the equator; the band field rises to a maximum at 45° and falls away again. The product of the two departures is positive in the middle latitudes and negative at both ends, and on this field the two nearly cancel. Change the field and the cancellation goes: the same Miller map moves the northern concentration by 17.98 per cent.

Distortion does bound the error, even though it does not predict it #

The table above says a projection’s areal spread is not a predictor of the reading error. That is the finding and it is easily over-read into a stronger claim that is false: that the map contributes nothing knowable on its own. It contributes a ceiling, and the ceiling is exactly computable from the map alone.

A covariance is bounded by the product of the two standard deviations, so writing the bias as a fraction of the ground’s own answer gives

$$\frac{\bar v_{\text{page}} - \bar v_{\text{ground}}}{\bar v_{\text{ground}}} = \rho\;\cdot\;\frac{\sigma_v}{\bar v}\;\cdot\;\frac{\sigma_s}{\bar s},$$

three factors, each of which knows about only one thing. The middle one is the field’s coefficient of variation and belongs to the data. The last is the map’s coefficient of variation of areal factor, and it belongs to the projection — one number per projection, computable before any data exists. The first is the correlation between the two, and it is the only factor that requires both.

Measured over the same covering the rest of this essay uses, the map’s factor is 1.262 for Mercator, 0.813 for Miller, 0.463 for the plate carrée, 0.204 for Robinson and 0.200 for the Winkel tripel — an ordering that is a monotone rearrangement of the areal-spread column and carries the same information more usefully, because it is a standard deviation rather than a range and is therefore not decided by two extreme cells.

The northern concentration has a coefficient of variation of 0.946, so the largest reading error Mercator could produce on any field with that much variation is 119 per cent. What it actually produces is 24.85, which fixes the correlation between the field and the areal factor at 0.208. On the mid-latitude band the same decomposition gives Miller a ceiling of 42.9 per cent and an actual error of 0.19, a correlation of −0.005 — the near-cancellation the essay attributes to the shapes of the two functions, now a number rather than a description.

So the two quantities that a designer can know separately give a bound, and the third — which needs the data and the sheet together — decides where inside it the answer lands. That is a more useful statement than either the rule of thumb or its refutation. A projection with a small areal coefficient of variation cannot produce a large reading error whatever the data is; a projection with a large one might produce none at all, and there is no way to know without looking at the field. Robinson’s ceiling on the band field is 10.8 per cent and it delivers 2.11; Mercator’s is 66.7 and it delivers 5.17. The projection with six times the ceiling delivers two and a half times the error, which is precisely why the ranking by spread came out backwards.

Where the weight actually goes #

The weights are worth looking at on their own, because they are the half of the calculation that has nothing to do with the data and therefore prices every field at once.

The band between 70° and 80° holds 1.8 per cent of the Earth’s surface and 12.3 per cent of a Mercator sheet. That is the arithmetic behind every complaint about Greenland, and it is the reason the complaint is usually framed as being about shape when it is about voting weight: the far north gets nearly seven times the say in any impression formed by area.

The equatorial band runs the other way, holding 17.4 per cent of the ground and 7.1 per cent of the page.

The bound has one further use, which is negative and worth stating. It rules out the design move a reader might reach for on seeing the table — picking a projection by minimising areal spread and then treating the reading as sound. Minimising the ceiling does not minimise the error, and on the two fields measured here it does not even order the projections the same way. What the ceiling licenses is the opposite inference: a projection whose coefficient of variation is small enough that its ceiling sits under the tolerance is safe for every field, and that is a claim about the sheet alone, which is the only kind of claim a projection can honestly make.

Where the model stops #

The eye is not a planimeter. This ladder computes the page-area-weighted mean and calls it what a reader takes away, and a reader is not an integrator. The perception literature is clear that judged area is closer to a power of true area with an exponent somewhere under one, so the real impression is a compressed version of the page-area mean rather than the page-area mean itself. What the compression does is shrink the bias, not remove it or change its sign: a monotone transform of the weights leaves the covariance’s sign alone, and the covariance is the whole mechanism.

A choropleth is usually read for its pattern rather than its mean. The mean is the simplest statistic that has the property, and it is the one with a closed form. The classification a reader actually sees is a different statistic, and it is priced at the class breaks were computed on the page at the top of this ladder.

The regions here are cells rather than countries. Real thematic units are administrative and their sizes correlate with everything — with population, with terrain, with history. That correlation is a second source of bias and it is the modifiable areal unit problem, which the answer depends on the cells it was counted in prices in full. It is a different mechanism: it survives on an equal-area map, and this one does not.

The generalisation #

The pattern here is one this collection has now met from several directions, and it has a name inside the site: an operation performed after the map is an operation performed on the map.

A centroid computed in a plane is a statement about the plane. A mean of noisy positions moves because the averaging happens on the page. A simplification tolerance is a promise about the picture rather than the ground. A strain rate differenced from grid coordinates is partly a measurement of the grid.

Averaging a thematic value by eye is the same operation with the reader as the machine. What makes this instance different from the others is that the reader cannot be told to do it in a better chart. A pipeline can be repaired; a glance cannot. The only place the repair fits is the projection, which is why the rule of thumb exists, and why knowing what the rule is a special case of is worth four essays.

Who found it, and when #

The advice is old and the arithmetic is not. Equal-area projections were being recommended for distribution maps by the middle of the nineteenth century, and the argument was made qualitatively: a map that enlarges some regions will make the phenomena in them look more important. Judgements of area on maps have been measured experimentally since the 1950s, and the psychophysical exponent has been re-measured many times since.

What is unusual about stating it as a covariance is that it makes the exceptions computable, and the exceptions are what a rule of thumb cannot carry. A cartographer told to use an equal-area projection has been told a sufficient condition. A cartographer shown the covariance can look at a specific field, on a specific sheet, and find out whether the condition binds — and on a field that varies with longitude, or one that is antisymmetric about the equator, it does not bind at all.

Where the ladder goes next #

The mean is the easiest reading to price and the least interesting. Three quantities a thematic map delivers are not means: the size of a proportional symbol against the region under it, the visible density of a dot map, and the class boundaries a classifier computes. Each inherits the areal factor by a different route, and the next rung takes the first — where the error turns out to be the reciprocal of the areal factor, exactly, and the correction is one multiplication that no atlas performs.