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An explanation of a distribution is tested against a rival: map rates, not counts, and find evidence the explanations predict differently.
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By the end of this lesson you will be able to test an explanation of a distribution against evidence and a rival explanation, using rates, slopes and group comparisons.
Every lesson in this course described a distribution and explained it: population, migration, cities, jobs, development, inequality, hazards and climate. You can find rates, means and slopes. This lesson sets out the method behind all of those explanations, and shows how to test one against a rival.
| Term | What it means |
|---|---|
| Distribution | How something is spread across places. |
| Pattern | A regular shape in a distribution, such as clustering or a change with distance. |
| Explanation | A proposed reason for a pattern, which makes a testable prediction. |
| Rival explanation | A different reason that fits the same pattern. |
| Correlation | Two things tending to rise or fall together across places. |
| Confounding factor | A third factor that drives two things, so they rise together without one causing the other. |
| Rate | A count divided by a population, such as cases per ten thousand people. |
An explanation of a distribution is tested in steps:
$$\text{rate} = \dfrac{\text{count}}{\text{population}} \times 10000.$$
Another way: picture
Picture a map showing that ice-cream sales and drownings both peak in the same beach towns. Ice cream does not cause drowning. Hot weather brings crowds to the beach, where they buy ice cream and swim. The heat is the confounding factor, and an honest explanation has to find it.
Another way: steps
Prepare a short illustrated report answering this question: why might heat-illness visit rates differ across these fictional districts? Use the following invented same-summer evidence. North has 120 visits among 60000 residents, 30 percent tree cover and 20 percent outdoor workers. South has 90 visits among 15000 residents, 10 percent tree cover and 40 percent outdoor workers. East has 80 visits among 40000 residents, 15 percent tree cover and 15 percent outdoor workers. A visit is an event, so several visits may belong to one person. These are teaching data, not a real health study.
First calculate visits per 10000 residents and check that your rates are 20, 60 and 20. Draw a count-symbol sketch and a rate choropleth with a legend; use arbitrary adjacent district shapes, label the maps schematic and do not invent precise locations. Explain why the largest count and highest rate belong to different districts. Propose a mechanism linking heat exposure and one indicator to the pattern, then test it against a rival explanation using the third district. Does East fit a simple claim that tree cover alone determines visit rates? Explain why the supplied observations cannot isolate a causal effect.
Compare the original district view with a combined North-South region: 210 visits among 75000 residents gives 28 per 10000. Explain what that aggregation hides. State the year as a hypothetical common summer, the population denominator, the visit-count definition and two uncertainties: for example repeated visits, unequal reporting, unmeasured age differences or exposure among commuters. Request a new comparison that would distinguish your explanation from its rival. End with a bounded service-planning implication, preserving both total visits and rates rather than treating every resident as equally affected.
Before the independent report, practice a smaller argument: fictional districts with equal income but different tree cover can help test an income-based rival to a shade explanation. If the pattern persists, that weakens this rival but does not rule out every other cause. This is how to build a qualified claim, not a license to announce causation from a map.
Human rubric: score each criterion 0 (absent or materially wrong), 1 (partly supported but missing a required link), or 2 (accurate and explicit). Representation: both maps have meaningful legends and identify schematic boundaries, the common period and denominator. Quantitative evidence: the three rates and pooled rate are correct and the count-rate reversal is explained. Explanation: a plausible mechanism uses observations from at least two districts, and East is used to test a rival or limit a single-factor claim. Scale and uncertainty: the report explains the loss from aggregation and gives two relevant limitations without assigning an area average to an individual. Inquiry and consequence: a discriminating follow-up and a bounded service implication follow from the evidence. A reviewer should request revision on any zero and record reasons and feedback for every score; a total alone is insufficient. This report is taught but not machine-assessable. Automatic answers elsewhere do not approve the report, establish mastery of original inquiry or constitute a human review.
Consider two invented regions observed during one year. Harbor records 240 emergency calls among 12000 people, and Mesa records 120 among 3000. A count-symbol map makes Harbor larger, but a rate choropleth makes Mesa darker: their rates are 20 and 40 calls per thousand. Harbor may require more total response capacity, while Mesa has more calls relative to population. These conclusions can both be true.
Do not repair every map by dividing by population automatically. For crop yield the relevant denominator may be harvested area; for a commuter share it is employed residents; for crash exposure it may be distance traveled. Choose the denominator to fit the question and explain its limits. Residents are not necessarily all the people exposed to traffic in a commuter center. Compare the same period and boundary, and retain counts beside rates when planning a service. A denominator is part of a geographical claim, not just a calculation trick.
It is easy to look at a map and see a reason. Counties with more of one thing have more of another, and the link seems obvious. But many different reasons can produce the same pattern, and the first one that comes to mind is often wrong.
Geography, like every science, treats an explanation as a claim to be tested. The test asks whether the explanation predicts something the evidence could contradict, and whether a rival explanation predicts something different.
A map of counts, such as cases of a disease per county, mostly shows where people live. Big counties have more of everything. To see whether something is unusually common, divide each count by its population.
A county with $120$ cases among $60000$ people has $20$ per ten thousand; one with $90$ cases among $15000$ people has $60$, three times as high, though its count is smaller. Always map rates when the question is about how common something is.
When two things rise together across places, they are correlated. A correlation suggests an explanation but does not prove one. Either could cause the other, or a third factor could drive both.
Counties with more doctors have more deaths, not because doctors are dangerous but because both follow population. Counting rates removes that factor; other confounders need other tests.
For every explanation, ask what else could produce the same pattern. Counties with more fresh-food stores have less obesity; perhaps food access matters, but perhaps richer counties have both more stores and healthier residents.
Naming a rival is not a way of dismissing the first explanation. It is the step that tells you what evidence you need: something the two explanations predict differently.
The strongest tests compare places alike in the rival factor. To test food access against income, compare counties with similar incomes but different numbers of stores. If the pattern remains, income alone cannot explain it.
If the pattern vanishes, the rival was doing the work. Either result teaches something, which is how a good test differs from evidence that could only ever agree.
Time can separate explanations too. If towns near a new highway grew faster, ask whether they were growing before the highway came. Compare each town's growth before and after it opened, and compare with similar towns the highway bypassed.
A cause must come before its effect. Evidence arranged in time is often the clearest way to show which came first.
Many geographic patterns change with distance: fewer people commute to a city from farther away, and land prices fall away from a center. A slope measures the change per mile: the change in the value divided by the change in distance.
A share falling from $60$ to $15$ percent between $10$ and $40$ miles changes by $-45$ points over $30$ miles, or $-1.5$ points per mile. The slope describes the pattern; explaining it still needs a tested reason.
A simple test compares the means of two groups that differ in the factor being tested. If towns with sidewalks average $46$ percent of students walking to school and similar towns without average $35$, sidewalks are associated with an $11$-point difference.
The comparison is fair only if the groups are alike in other ways: size, distance to school, income, climate. The more alike they are, the more the difference can be credited to the factor being tested.
Checking an answer. A rate map should not simply look like a population map. A test is useful only if the two explanations predict different results.
Dividing by population is allowed because the question is how common something is, and a share of the people answers that; the count answers how big the place is.
Holding a rival factor fixed is allowed because, when compared places share that factor, it cannot account for differences between them. What remains must come from something else, which is the logic behind every controlled comparison.
A pattern across counties does not automatically hold for individuals. If counties with more immigrants have higher incomes, that does not show that immigrants earn more; the county pattern might come from where jobs attract everyone.
Reading a pattern for areas as a fact about the people in them is called the ecological fallacy. A careful explanation states the scale at which it holds and does not stretch it to another.
How a map is drawn changes what it seems to show. Large, thinly populated counties catch the eye even when few people live there. Different class breaks can make the same data look alarming or calm.
So an explanation should rest on the numbers behind a map, not only its look. Checking the rates, the class breaks and the scale guards against explaining a pattern that the drawing created.
Even a well-tested explanation has limits. It may hold in one region but not another, or at one scale but not another. There may be rival explanations nobody has thought of yet.
A careful conclusion says what the evidence supports, how strongly, and what it cannot rule out. That is not weakness; it is what lets the next study build on this one.
Every lesson in this course followed this method. Population change was split into its flows; urbanization was separated from urban growth; development was measured more than one way; disasters were split into hazard, exposure and vulnerability; climate impacts were described region by region.
In each case the move was the same: describe the pattern with the right measure, propose a reason, and check it against a rival. That habit is the most useful thing geography teaches.
The most common slip is explaining a map of counts when a map of rates is needed. Another is treating a correlation as proof that one thing causes the other.
A third is choosing evidence that fits both explanations and calling it a test. A fourth is stretching a pattern among places into a claim about every person living in them.
In the early 1900s pellagra, a disease causing rashes, diarrhea and confusion, struck hundreds of thousands of people across the American South. Its pattern pointed to poor mill towns, orphanages and asylums, and many doctors explained it as an infection spreading in crowded places.
Joseph Goldberger of the U.S. Public Health Service proposed a rival explanation: a poor diet of cornmeal, molasses and fat pork. He noticed that staff in the same institutions, living among the sick but eating better food, did not get pellagra, which an infection could not explain. In 1915, at a prison farm in Mississippi, volunteers fed that restricted diet developed the disease.
Goldberger tested his explanation by holding the rival factor fixed: people sharing the same crowded places but eating differently. Later research identified the missing nutrient as niacin, and fortifying flour ended pellagra in the United States. The story shows how a pattern that seemed to explain itself had a different cause.
The Great Plains are famous as Tornado Alley, but studies of tornado deaths have found that more people per tornado die in the Southeast, in states such as Alabama, Mississippi and Tennessee. Geographers and meteorologists have proposed several explanations.
One is housing: a larger share of Southeastern residents live in mobile homes, which offer little protection. Another is timing: a larger share of Southeastern tornadoes strike at night, when people are asleep and miss warnings. A third is terrain and trees, which hide approaching storms from view.
Researchers test these by comparing like with like: deaths in mobile homes and other homes for tornadoes at the same time of day, or nighttime and daytime tornadoes in the same region. The evidence suggests several factors add together, which has led to Southeastern programs for community storm shelters and for warnings that wake people.
It is natural to see two things rising together on a map and conclude that one causes the other. But a third factor can drive both, and several different explanations can fit the same pattern. Ice-cream sales and drownings peak together because of hot weather, not because of each other.
Only evidence that the explanations predict differently can settle which is right. Name a rival, hold its factor fixed, and see whether the pattern remains. A conclusion reached that way can be trusted in a way that a first impression cannot.
County A has $200$ cases among $100000$ people. Find its rate per ten thousand.
$\dfrac{200}{100000} \times 10000 = 20$
Per ten thousand.
County B has $75$ cases among $15000$ people. Find its rate.
$\dfrac{75}{15000} \times 10000 = 50$
Per ten thousand.
Compare the rates.
$\dfrac{50}{20} = 2.5$
B's rate is two and a half times A's.
Say what the counts alone suggested.
$200 > 75$
The wrong county looked worse.
At $8$ miles from a city, $66$ percent of workers commute in. Record the point.
$(8,\ 66)$
Distance and share.
At $28$ miles, $26$ percent do. Record the point.
$(28,\ 26)$
Farther out.
Find the change in the share.
$26 - 66 = -40$
Percentage points.
Find the slope.
$\dfrac{-40}{28 - 8} = -2$
Points per mile.
Name a rival reason for the fall.
$\text{farther towns have their own jobs}$
Not only distance.
Neighborhoods with more trees are cooler. State the explanation.
$\text{shade cools the streets}$
A proposed cause.
Name a rival explanation.
$\text{less pavement in leafy areas}$
Another cause that fits.
Design a test that tells them apart.
$\text{shade and sun on the same street}$
Pavement held fixed.
Suppose shade is $5$ °F cooler on the same street. Read the result.
$\text{supports the shade explanation}$
Pavement cannot explain it.
Say what it cannot rule out.
$\text{pavement adds to the pattern too}$
Both may matter.
State the conclusion with its limit.
$\text{shade cools, other factors may add}$
Supported, not complete.
Divide the cases by the people.
$\dfrac{60}{40000}$
Cases per person.
Multiply by ten thousand.
$\dfrac{60}{40000} \times 10000$
Per ten thousand.
Evaluate the rate.
Geographers notice that counties with universities have higher median household incomes. One explanation is that universities create well-paid jobs. A rival explanation is that the universities were built in counties that were already large and wealthy. Which evidence would best test the first explanation against the rival?
Complete the worked solution: to test whether sidewalks lead more students to walk or bike to school, a study compares three towns with sidewalks, where $21$, $26$ and $34$ percent do, with three similar towns without, where $18$, $22$ and $17$ percent do. Find each group's mean and the difference between them.
Find the mean for towns with sidewalks.
$\dfrac{\text{sum of three}}{3} =$ m
The first group.
Find the mean for towns without.
$\dfrac{\text{sum of three}}{3} =$ n
The second group.
Subtract the second mean from the first.
$\text{with} - \text{without} =$ g
Percentage points.
Say what would make the test fair.
$\text{towns alike in size, distance and income}$
Then sidewalks are the main difference.
Match each term to its meaning.
| how something is spread across places | a proposed reason for that spread | a different reason that fits the same spread | a third factor that drives two things so they rise together | |
|---|---|---|---|---|
| pattern | ||||
| explanation | ||||
| rival explanation | ||||
| confounding factor |
County A recorded $120$ cases of a disease among $60000$ people; county B recorded $90$ cases among $15000$ people. Fill in each county's rate per ten thousand people, and how many times higher B's rate is than A's.
| value | |
|---|---|
| county A's rate per 10,000 | |
| county B's rate per 10,000 | |
| B's rate divided by A's |
A county has $200000$ people. Write its rate per ten thousand people as a function of the number of cases $c$.
Answer:
In a town $4$ miles from a city, $72$ percent of workers commute into the city; in a town $24$ miles away, $32$ percent do. Treating the pattern as a straight line, by how many percentage points does the share change per mile? Give a fall as a negative number.
Answer: points per mile
Suppose a city in Texas, with $300000$ residents, recorded $150$ emergency visits for heat illness in one summer. What was its rate per ten thousand residents?
Answer: per 10,000
Two maps report library visits in the same year. River has 6 thousand visits and 6 thousand residents. Hill has half as many visits and one quarter as many residents. Mark every supported sentence reconciling the count and visits-per-resident maps, including the limit on interpreting access.
This task has no paper form; do it on a device.
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A county has $150000$ people. Write its rate per ten thousand people as a function of the number of cases $c$.
Answer:
You can test an explanation. Explain why two things rising together on a map does not show that one causes the other.
28. Your turn: a county of $40000$ people records $60$ cases. What is its rate per ten thousand?, step 3
$15$
Cases per ten thousand.