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A choropleth should shade rates, not counts: count over population times 100,000, or people per square mile, read through the legend with an eye for area bias.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
By the end of this lesson you will be able to turn counts into rates for a choropleth, compare places by rate, and spot the common ways choropleths mislead.
You can judge a map's purpose, scale and limits, and compute percent changes. A choropleth is the most common thematic map in newspapers and reports, and the easiest to misread. This lesson shows what its shading should measure and how to compute it.
| Term | What it means |
|---|---|
| Choropleth | A map that shades areas, such as counties, by a value in classes. |
| Count | A total number of people or events in an area. |
| Rate | A count divided by a base, such as population, and scaled. |
| Normalize | To divide a count by population or area so places can be compared. |
| Population density | People per unit of area, such as per square mile. |
| Enumeration unit | The areas a choropleth shades: states, counties, tracts. |
A choropleth shades each area by a value. The value should be a rate, because areas differ in size and population.
Darker shading means a higher value of what the legend states, nothing more.
Another way: picture
Picture two counties with thirty cases of a disease each. One has three million people and the other thirty thousand. On a count map they look the same; on a rate map the small county is a hundred times darker, because each resident there is far more likely to be affected.
Another way: steps
Here are three invented counties.
| County | Cases | People | Rate per 100,000 |
|---|---|---|---|
| A | 30 | 100,000 | 30 |
| B | 90 | 600,000 | 15 |
| C | 8 | 20,000 | 40 |
On a count map, B is darkest. On a rate map, C is darkest and B lightest. Only the rate map answers the question most readers care about: where is a person most at risk?
Dividing a count by population gives a tiny number, such as $0.0003$ cases per person. Multiplying by a round base makes it readable: $30$ per $100000$. Health agencies use per $100000$ for rare events; per $1000$ is common for births and deaths.
Whatever the base, it must be the same for every area on the map and printed in the legend.
Some choropleths normalize by area instead: population density is people per square mile. A county of $240000$ people over $800$ square miles has $300$ people per square mile.
The United States averaged about $94$ people per square mile in 2020, but densities range from under one in parts of Alaska to more than $70000$ in Manhattan.
On a county map of the United States, the largest counties are in the thinly settled West. They cover much of the map with few people, so they dominate the eye whatever their values. After the 2016 election, county maps of the winning party looked overwhelmingly one color, though the national vote was close.
Cartograms, which resize areas by population, and dot maps are two ways to correct for area bias.
In counties with few people, rates swing wildly by chance: two cases among $5000$ people is $40$ per $100000$, and one more case makes it $60$. Such counties often show the highest and the lowest rates on a map, not because of real differences but because of small numbers.
Analysts combine several years or neighboring counties, or mark rates based on few cases as unreliable.
The legend tells you what the shading means: the variable, its unit, the base of a rate, the classes and the date. A map of rates per $100000$ read as percents would overstate everything a thousandfold.
Read the legend before the map, every time.
A choropleth shades whatever areas the data come in: states, counties, census tracts. The choice changes the picture: a state average hides differences between its counties, and a county average hides differences between neighborhoods.
Choose the smallest units the data support reliably, and say which units the map uses.
Checking an answer. A rate must fall when the population rises and the count stays the same. If a map's darkest areas are simply its biggest cities, suspect counts.
Dividing by population is allowed, and needed, because it turns a total that depends on size into a per-person measure that does not. Multiplying by a fixed base changes only the number's size, not the ranking of places.
Comparing rates by a ratio is allowed because both rates are on the same base.
The Centers for Disease Control and Prevention publishes county maps of disease rates, death rates and vaccination rates. Health departments use them to decide where to send clinics, nurses and outreach.
A count map would send resources to big cities alone; a rate map shows the rural counties where risk per person is highest, even when their totals are small.
A choropleth can divide by people or by land, and the two tell different stories. Per person, a rate shows how common something is among residents: cases per 100,000, income per household. Per square mile, a density shows how crowded something is on the ground: people, roads, wells.
Alaska has fewer than two people per square mile, so a density map paints it almost empty, while a rate map can show its residents with some of the highest or lowest rates in the country. Neither is wrong. Choose the denominator that matches the question, name it in the legend, and never map a raw count on areas of different sizes.
The most common slip is mapping counts instead of rates. Another is reading darker shading as more people.
A third is reading a rate's base wrongly, per $100000$ as a percent. A fourth is trusting extreme rates in counties with very few people.
A choropleth is only one way to map a value. The next lesson shows proportional symbols and dot maps, which suit counts better than shading does, and the one after shows how the classes of a choropleth can change its story.
The Centers for Disease Control and Prevention and state health departments publish county choropleths of death rates, disease rates and vaccination coverage. Each county is shaded by its rate per 100,000 people, often adjusted for the ages of its residents so that a county with many older people is not unfairly shaded high.
Such maps reveal patterns that counts hide. Some of the highest rates of heart disease deaths in the country fall in rural counties of the South, a region a count map would show as pale beside the big cities. Health agencies use the rate maps to decide where to send mobile clinics and outreach.
Mapmakers also flag rates based on very few deaths as unreliable, or combine several years, because a small county's rate can jump with a single case.
After the 2016 presidential election, a county map of the United States shaded by the winning party circulated widely. It appeared overwhelmingly red, suggesting a landslide, though the national popular vote was nearly even.
The map showed a winner for each county, not how many votes it cast, and the largest counties by area are in the thinly settled rural West and Plains. Los Angeles County, with about ten million people, took up less of the map than several ranch counties with a few thousand residents each.
Cartographers answered with other maps: shading by vote share instead of winner, dot maps with a dot for so many votes, and cartograms that resize each county by its voters. Each told a truer story about people. The episode became a standard lesson in how area bias shapes what a choropleth seems to say.
It is natural to read a dark county as a crowded one or as the place with the most cases. On a well-made choropleth, darker means only a higher value of the variable in the legend, usually a rate, and a small rural county can be the darkest on the map.
When a map shades counts rather than rates, its darkest areas are simply where most people live, and it tells you little else. Check the legend, and normalize counts before mapping them.
A county of $150000$ people had $45$ cases. Divide.
$\dfrac{45}{150000} = 0.0003$
Cases per person.
Scale to 100,000.
$0.0003 \times 100000 = 30$
Per 100,000.
Name the wrong base to avoid.
$0.0003 \times 1000 = 0.3$
Per thousand, a different rate.
Say what the map shows.
$30 \text{ per } 100000$
With the base in the legend.
County B had $90$ cases among $600000$ people. Find its rate.
$\dfrac{90}{600000} \times 100000 = 15$
Per 100,000.
County C had $8$ cases among $20000$. Find its rate.
$\dfrac{8}{20000} \times 100000 = 40$
Per 100,000.
Compare the counts.
$90 > 8$
B has more cases.
Compare the rates.
$40 > 15$
C has more risk.
Find how many times higher C's rate is.
$\dfrac{40}{15} \approx 2.7$
Nearly three times.
A county has $200000$ people. Write its rate for $c$ cases.
$\dfrac{c}{200000} \times 100000$
Count over population, scaled.
Simplify the fraction.
$0.5c$
Half the count.
Find the rate for $60$ cases.
$0.5 \times 60 = 30$
Per 100,000.
Write the function for a county of $25000$.
$4c$
Four per case.
Find its rate for $3$ cases.
$4 \times 3 = 12$
Per 100,000.
Say why the small county swings more.
$\text{each case adds } 4$
Not half a point.
Divide the count by the people.
$\dfrac{120}{300000} = 0.0004$
Per person.
Scale to 100,000.
$0.0004 \times 100000 = 40$
Per 100,000.
Say what to print in the legend.
A county of $500000$ people recorded $250$ cases of a disease in a year. What value should a choropleth map for this county, as cases per 100,000 people?
Complete the worked solution: an urban county had $150$ cases among $500000$ people, and a rural county $24$ cases among $20000$. Find each county's rate per 100,000, and how many times higher the rural rate is.
Find the urban rate.
$\dfrac{\text{count}}{\text{people}} \times 100000 =$ r
Per 100,000.
Find the rural rate.
$\dfrac{\text{count}}{\text{people}} \times 100000 =$ s
Per 100,000.
Compare the rates.
$\dfrac{\text{rural rate}}{\text{urban rate}} =$ t
Times higher.
Say what a count map would show.
$\text{the urban county darker}$
More cases, lower risk.
Match each problem with a choropleth to what causes it.
| mapping counts instead of rates | large areas drawing the eye regardless of people | the choice of class breaks | not reading the legend's unit | |
|---|---|---|---|---|
| every large city is shaded darkest on a map of cases | ||||
| vast, thinly settled counties dominate the picture | ||||
| the same data tell a different story with new classes | ||||
| readers take rates per 100,000 as percents |
County A had $30$ cases among $100000$ people, county B $90$ among $600000$, and county C $8$ among $20000$. Fill in each county's rate per 100,000.
| rate | |
|---|---|
| county A (per 100,000) | |
| county B (per 100,000) | |
| county C (per 100,000) |
A county has $1000000$ residents. Write its rate per 100,000 as a function of its count of cases $c$.
Answer:
A county has $45000$ people over $1500$ square miles. What is its population density, in people per square mile?
Answer: per sq mi
Suppose public-health analysts are mapping heat-related emergency visits. In a county in Louisiana of $70000$ people, there were $35$ visits one summer. What value should the choropleth show, as visits per 100,000 people?
Answer: per 100,000
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A county has $400000$ residents. Write its rate per 100,000 as a function of its count of cases $c$.
Answer:
You can read a choropleth. Explain why a map of case counts mostly shows where people live.
24. Your turn: a county of $300000$ people had $120$ cases. What is its rate per 100,000?, step 3
$\text{cases per 100,000 people}$
The base.