Back to the on-screen lesson ·
Proportional symbols size circles by area, so the radius grows with the square root of the value, and dot maps show counts with one dot for every so many.
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 size proportional symbols fairly, build a dot map from counts, and choose the map that suits a kind of data.
You know a choropleth should shade rates, not counts. But counts are often exactly what a map needs to show: how many passengers use each airport, where the farms are. This lesson shows the two maps built for counts, and the arithmetic of drawing them fairly.
| Term | What it means |
|---|---|
| Proportional-symbol map | A map with a symbol at each place sized in proportion to a count. |
| Dot map | A map with one dot for every so many people or events, placed where they occur. |
| Dot value | The number each dot stands for. |
| Isoline map | A map of lines joining equal values of a smooth surface, like temperature. |
| Square root | The number that multiplied by itself gives a value; the square root of 9 is 3. |
Counts belong on symbols and dots, not on shading.
Scaling a circle's radius directly by its value exaggerates big values badly.
Another way: picture
Picture two pizzas, one twice as wide as the other. The big one does not feed two people for every one the small one feeds; it feeds four, because its area is four times larger. A proportional circle that doubles in width shows four times the value.
Another way: steps
When a circle's radius grows, its area grows much faster.
| Value ratio | Radius factor | Area factor |
|---|---|---|
| 4 | 2 | 4 |
| 9 | 3 | 9 |
| 25 | 5 | 25 |
| 100 | 10 | 100 |
Sizing by area keeps the area factor equal to the value ratio. Sizing the radius by the value would make a city a hundred times bigger look ten thousand times bigger.
Choose a reference: a circle of radius $1$ cm for $10000$. For a value of $90000$, the ratio is $9$, its square root is $3$, and the circle's radius is $3$ cm.
Its area is then $9$ times the reference circle's, matching the value. Most mapping software does this automatically, but the setting must say area, not radius.
Studies of map readers found that people tend to underestimate the size of larger circles even when areas are drawn correctly. Some cartographers therefore enlarge big circles slightly beyond true area.
Whichever method is used, a good map includes a legend of nested reference circles, so readers can compare against known sizes rather than guess.
A dot map shows a count by scattering dots across an area, each worth the same number. With one dot for $500$ people, a county of $12000$ gets $24$ dots.
Unlike a choropleth, a dot map shows where within a county the people are: clustered in towns and along roads, thin in forests and mountains. The Census Bureau has published dot maps of the whole country with one dot per person.
If the dot value is too small, dots merge into black blobs in crowded places; if too large, sparse places get no dots and vanish from the map. A common rule is to choose a value so that dots just begin to touch in the densest area.
The dot value must be printed in the legend: a map with one dot per $1000$ people looks very different from one with a dot per $100$.
Some values change smoothly across space: temperature, rainfall, elevation. They are mapped with isolines, lines joining places of equal value, like contour lines for height or isobars for pressure.
Isolines suit continuous surfaces, not counts in districts. A choropleth of temperature would falsely suggest that temperature changes abruptly at county lines.
Each kind of data has a map that suits it. Counts at points suit proportional symbols. Counts spread across areas suit dot maps. Rates for areas suit choropleths. Smooth surfaces suit isolines.
Choosing the wrong kind is one of the commonest ways a map misleads without a single wrong number.
Checking an answer. Square the radius factor: it must equal the value ratio. The dots times the dot value must give back the count.
Sizing by area is allowed, and needed, because readers compare the ink a symbol covers, which is its area. Taking the square root is exact, because a circle's area is pi times the radius squared, and pi cancels in a ratio.
Dividing by a dot value is allowed because each dot is an equal group.
In crowded regions, big circles overlap and hide smaller ones. Cartographers draw circles transparent or with outlines, place the largest underneath, or use smaller reference sizes.
Where overlap is severe, a different map, such as a graduated-color point map or an inset, may serve better.
Scaling a circle's area to its value is correct, but readers do not judge areas exactly. Studies of map reading, beginning with the cartographer James Flannery's in the 1970s, found that people tend to underestimate the size of large circles compared with small ones.
Some mapmakers therefore enlarge big circles slightly beyond true area, and many add a legend of nested reference circles so readers can compare instead of guess. Others print the values beside the largest symbols. Whatever the choice, the legend must say what one circle's size stands for, or the map invites every reader to judge by eye alone.
The most common slip is scaling a circle's radius by the value instead of by its square root. Another is choosing a dot value so large that small places disappear.
A third is putting counts on a choropleth. A fourth is leaving the dot value or the reference circles out of the legend.
After recent censuses, researchers built maps of the United States with one dot for every person counted, more than three hundred million dots in all. Zoomed out, the dots merge into a glowing picture of where Americans live: the dense Northeast corridor, the cities of the Midwest, the thin scatter across the Great Plains and the West.
Zoomed in, the same map shows neighborhoods, highways lined with homes, and empty national forests and parks. A choropleth of counties could never show that detail, because it gives each county one shade whatever the pattern inside.
Dot maps like these have a long history in the United States. Nineteenth-century census atlases used them for crops, livestock and population, and they remain one of the clearest ways to show how many and where at once.
Maps of air travel in the United States often draw a circle at each airport sized by passengers. Atlanta, Dallas-Fort Worth, Denver and Chicago O'Hare, among the busiest in the world, get the largest circles, with regional airports as small circles scattered between.
Drawn by area, the circles let a reader see that the busiest hub handles perhaps several times the passengers of a mid-sized airport. Drawn by radius, the same data would make the hub look many times more dominant than it is, and the regional airports would shrink to near invisibility.
Because big circles overlap in crowded regions such as the Northeast, mapmakers draw them semi-transparent and include a legend of nested circles so that readers can compare sizes against reference values.
It seems natural to make a circle for a value four times as big four times as wide. But a circle four times as wide covers sixteen times the area, and readers see the area, so the map exaggerates the big value fourfold.
Size proportional symbols so that their areas match the values: the radius grows with the square root of the value ratio. Four times the value means twice the width.
A city has $9$ times the people of the reference. Relate area to radius.
$\text{area} \propto r^2$
Pi times radius squared.
Set the area factor.
$9$
Area matches the value.
Find the radius factor.
$\sqrt{9} = 3$
Three times wider.
Name the wrong answer to avoid.
$9 \text{ times wider}$
Area would be 81 times.
One dot stands for $1000$ people. A county has $45000$. Divide.
$\dfrac{45000}{1000} = 45$
Dots.
Another county has $8000$. Divide.
$\dfrac{8000}{1000} = 8$
Dots.
A tiny county has $300$. Divide.
$\dfrac{300}{1000} = 0.3$
Less than one dot.
Say what happens to it.
$\text{no dot, unless rounded up}$
It may vanish.
Say how to fix it.
$\text{a smaller dot value}$
If crowded areas allow.
A reference circle of radius $0.5$ cm stands for $1000$ daily passengers. An airport has $25000$. Find the ratio.
$\dfrac{25000}{1000} = 25$
Times the reference.
Find the radius factor.
$\sqrt{25} = 5$
Five times wider.
Find the radius.
$0.5 \times 5 = 2.5\ \text{cm}$
Centimeters.
Another airport has $4000$. Find its radius.
$0.5 \times \sqrt{4} = 1\ \text{cm}$
Twice the reference.
Check the area ratio between them.
$\left(\dfrac{2.5}{1}\right)^2 = 6.25$
Matches $25000 \div 4000$.
Say what a radius-scaled map would show.
$\text{a ratio of } 39 \text{ in area}$
Wildly exaggerated.
Relate the area to the value.
$\text{area factor} = 16$
Area matches the value.
Take the square root.
$\sqrt{16} = 4$
The radius factor.
Check by squaring.
On a proportional-symbol map, circle areas are proportional to values. One city has $100$ times as many people as another. How many times wider should its circle be?
Complete the worked solution: on a proportional-symbol map, a circle of radius $0.4$ cm stands for $1000$. Find how many times larger $25000$ is, how many times wider its circle is, and its radius.
Find the value ratio.
$\dfrac{\text{larger}}{\text{reference}} =$ f
Times larger.
Find the radius factor.
$\sqrt{\text{value ratio}} =$ g
Times wider.
Find the radius.
$\text{reference radius} \times \text{factor} =$ n
Centimeters.
Say what readers should see.
$\text{an area the value's size}$
Fair comparison.
Match each kind of data to the kind of map that suits it.
| a proportional-symbol map | a dot map | a choropleth | an isoline map | |
|---|---|---|---|---|
| passengers at each airport | ||||
| farms scattered across a state | ||||
| unemployment rate for each county | ||||
| average July temperature everywhere |
On a dot map, one dot stands for $2000$ people. County A has $30000$ people, county B $14000$, and county C $90000$. Fill in the dots each county needs.
| dots | |
|---|---|
| county A (dots) | |
| county B (dots) | |
| county C (dots) |
On a dot map, one dot stands for $100$ people. Write the number of dots for an area as a function of its population $p$.
Answer:
On a proportional-symbol map, a circle of radius $1$ cm stands for $10000$. What radius, in centimeters, should the circle for $250000$ have?
Answer: cm
Suppose a newspaper map of American cities uses a circle of radius $0.5$ cm for $100000$ people, with circle areas proportional to population. What radius, in centimeters, should a city in Pennsylvania of $1600000$ people get?
Answer: cm
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
On a dot map, one dot stands for $1000$ people. Write the number of dots for an area as a function of its population $p$.
Answer:
You can draw fair symbol and dot maps. Explain why a city with four times the people gets a circle only twice as wide.
23. Your turn: a value is $16$ times the reference. How many times wider is its circle?, step 3
$4^2 = 16$
Back to the value ratio.