Common mistake 1 with Bivariate map

Using it on two variables that are nearly the same. The map looks rich and says nothing; the diagonal check in the result is there to catch exactly this.

Common mistake 2 with Bivariate map

Comparing colours across two different bivariate maps. The classes are cut on each map’s own data, so the same colour means different numbers.

Common mistake 3 with Bivariate map

Four classes each way because more looks more thorough. Sixteen colours on a map is a puzzle, not a chart.

Common mistake 4 with Bivariate map

Reading a bivariate pattern as a relationship. Two variables being high in the same places is a spatial correlation, and spatial correlation has its own tests - and its own reasons to be spurious.

Common mistake 5 with Bivariate map

Forgetting the key is a square. A reader who assumes it is a ramp will rank the colours in one dimension and get the off-diagonal cases exactly backwards.

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Bivariate map

Shades every place by TWO variables at once, using a square colour key instead of a line. The places that are high on one and low on the other get their own colour - and those are exactly the places a pair of separate maps hides, because nobody can hold two maps in their head and difference them.

What it needs

Place column
First variable (across the key)
Second variable (up the key)

The intuition

Put two maps side by side and ask which places are high on the first and low on the second. Nobody can do it. The eye cannot hold one choropleth in memory accurately enough to difference it against another, and the failure is silent - people believe they have compared the maps. A bivariate map does the differencing for you by giving every combination of classes its own colour, so the question becomes a matter of finding a colour rather than of remembering a pattern. The cost is that the key is now two-dimensional, and a reader who mistakes it for a ramp will get the interesting cases precisely backwards.

When to use it

When the question is about the RELATIONSHIP between two things across places - where schooling is high but investment is not, which districts are poor and also badly connected, where growth and inequality moved together. Two separate maps cannot answer it, because nobody can hold one map in their head while looking at another.

Where the estimator comes from

Cut each variable into k classes independently, giving class indices i(x) and j(y) in \{0,\dots,k-1\}. Fix four corner colours c_{00},c_{10},c_{01},c_{11} - low-low, high-x, high-y, high-high - and fill the grid by bilinear interpolation in CIE Lab, c(i,j)=(1-u)(1-v)c_{00}+u(1-v)c_{10}+(1-u)v c_{01}+uv\,c_{11},\qquad u=\tfrac{i}{k-1},\ v=\tfrac{j}{k-1}. Interpolating in Lab rather than in sRGB matters here more than it does for a one-dimensional ramp: the middle of the square is where two hues meet, and a straight line between them through RGB passes through a muddy, unpredictably dark region that makes the centre of the key look like a class of its own.

The model

c(i,j)=\text{bilinear}\big(c_{00},c_{10},c_{01},c_{11}\big),\qquad i=\text{class}(x),\ j=\text{class}(y)

What it assumes

Both variables are measured for the same places and are comparable across them - the classes are cut on the whole sample, so a place in the top class is top relative to the others on the map.
Three classes each way. Nine colours is already at the limit of what a reader can hold in mind while scanning; four gives sixteen and is offered mainly to show why three is the advice.
The two variables are not the same thing measured twice. If they correlate at 0.9, every place lands on the diagonal and the map is a one-variable map wearing two colours.
Quantile classes by default, so each class holds roughly the same number of places. With equal-interval classes a skewed variable can leave whole rows of the key empty.
The place names match the boundaries. Anything unmatched is named rather than dropped quietly.

Statistical properties

The design is Stevens’ (2015) popularisation of a technique the US Census used in the 1970s. Three by three is the near-universal choice; the constraint is human memory, not the arithmetic.
Classes are cut on each variable separately, so the key is always fully spanned even when the data occupy only part of the square. The counts per cell are what reveal that, which is why they are printed.
When the two variables correlate strongly, every place falls near the diagonal and the map degenerates into a single ramp drawn in two hues. This is the commonest failure and it LOOKS like success, so the result reports the share on the diagonal and says when it has happened.
The off-diagonal corners carry the information a pair of separate maps cannot: on the growth sample, 28 of 110 countries are high on schooling and low on investment or the reverse, against a correlation of -0.07.
Colours are not comparable between two bivariate maps, because each map cuts its own classes. The same is true of any quantile choropleth, but the square key makes it easier to forget.
A bivariate map shows spatial co-occurrence, not a relationship. Two variables high in the same places is exactly the pattern spatial autocorrelation produces from nothing at all.

How to read the output

Read a colour by moving ALONG the key for the first variable and UP it for the second. The four corners are the four extremes.
The off-diagonal corners are the point of the whole exercise: high on one variable and low on the other.
The count table under the map says how many places are in each square of the key. Use it before reading anything into the picture.
The correlation is reported. Near ±1 means the map has little to add over a single shaded map; near zero means the off-diagonal squares are genuinely populated.
A pale place is low on both, and a dark place is high on both - so the diagonal reads as a single ramp and the interesting cases are the ones that break it.

A worked reading

Schooling against investment for 110 countries. The correlation is -0.07 - effectively independent - so the key is populated nearly uniformly: between 10 and 14 countries in each of the nine squares. Twenty-eight sit in the opposite corners, high on one and low on the other, and those are precisely the cases the growth-regression literature argues over. Run the same map with schooling against log initial GDP instead and the diagonal fills up, the warning fires, and the honest conclusion is that one map would have done.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Everything on the diagonal	The two variables are nearly the same	Report one shaded map; the warning says so
Whole rows of the key empty	Equal-interval classes on a skewed variable	Switch to quantiles
Key looks muddy in the middle	Two hues meeting	Try another palette - the corners differ in how they blend
Very few places matched	Both variables must be present in the same row	The map drops rows missing either
Pattern looks strong but the correlation is near zero	Spatial clustering, not covariation	Test it with a spatial autocorrelation statistic
Four classes and unreadable	Sixteen colours	Three each way is the practical limit

Common mistakes

Using it on two variables that are nearly the same. The map looks rich and says nothing; the diagonal check in the result is there to catch exactly this.
Comparing colours across two different bivariate maps. The classes are cut on each map’s own data, so the same colour means different numbers.
Four classes each way because more looks more thorough. Sixteen colours on a map is a puzzle, not a chart.
Reading a bivariate pattern as a relationship. Two variables being high in the same places is a spatial correlation, and spatial correlation has its own tests - and its own reasons to be spurious.
Forgetting the key is a square. A reader who assumes it is a ramp will rank the colours in one dimension and get the off-diagonal cases exactly backwards.

Where to read this properly

Stevens (2015), "Bivariate Choropleth Maps: A How-to Guide", joshuastevens.net.
Olson (1975), "Spectrally Encoded Two-Variable Maps", Annals of the AAG 71(2).
Brewer (1994), "Color Use Guidelines for Mapping and Visualization", in Visualization in Modern Cartography.
Elmer (2013), "Symbol Considerations for Bivariate Thematic Maps", Proceedings of ICC.
Slocum, McMaster, Kessler and Howard (2022), Thematic Cartography and Geovisualization, 4th edition.

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Bivariate map

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

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