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Four different stories produce exactly the same count of two things moving together, and the count picks none of them.
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.
You will take a finding that two things move together and write out the four things that could be true behind it: each direction of cause, a third thing causing both, and coincidence. You will draw the common-cause picture with its arrows, and say what the count on its own has actually settled.
You can say what a piece of evidence reaches and where it stops. A count of two things moving together is the piece of evidence people over-read more often than any other, and this lesson is the list of what it leaves open.
| Term | What it means |
|---|---|
| Association | A pattern in how measured variables vary together across observed cases. |
| Common cause | A factor that influences both variables whose association is being considered. |
| Reverse causation | An explanation in which the proposed outcome helps produce the proposed cause. |
| Intervention | A deliberate change to a variable, distinguished from merely observing its value. |
Two measured quantities are associated when their values show a pattern together across the cases examined. That pattern can be useful for description or prediction. It does not, by itself, identify the mechanism producing it.
Four useful possibilities to consider are: A influences B; B influences A; another factor C influences both; or the observed pattern is a chance feature of the collected sample. These are starting hypotheses, not an exhaustive list of mutually exclusive worlds. Real systems can contain several causes, feedback in both directions, selection effects, and measurement errors at once.
A causal arrow has a stronger meaning than a line showing association. A → B proposes that changing A can affect B through a relevant process. C → A and C → B proposes a common cause. Drawing one of these diagrams makes a hypothesis explicit; it does not prove that hypothesis. The next task is to ask which further observations or comparisons would distinguish the competing explanations.
Another way: picture
Draw A → B, B → A, and C → A together with C → B as separate candidate diagrams. Add a note that a sample pattern might occur by chance. Keep observed association distinct from each proposed causal explanation.
Another way: steps
Name the measured variables and the cases. Describe the association without saying causes. List plausible directions and common causes. Then ask what extra evidence would favor one explanation or rule out an important alternative.
A statement that two things go together needs a unit of observation. Are the cases people, schools, streets, days, or repeated measurements of one device? 'Places with more lamps have more reported incidents' compares places. It does not say what happens to one particular street after a lamp is installed. 'This street had more lamps and fewer incidents after renovation' compares times in one place. The two designs supply different information, so identify the cases before drawing a causal conclusion.
Then define the variables. A count of reported incidents is not identical to a count of all incidents that actually occurred. More reports might reflect more events, better detection, more residents, or easier reporting. A raw count is also different from a rate per resident or per visit. A large busy district can have more reports than a small quiet one even if each person's risk is lower. An unexplained association between totals may largely reflect the size of the populations being compared.
Describe the direction of the association in ordinary language. Across these cases, larger values of A tend to accompany larger values of B, or larger values of A tend to accompany smaller values of B. An association can also be nonlinear: both very low and very high values of a setting might accompany poor performance. Not every pattern is a straight rising line. A graph can help reveal the form before anyone tries to explain it.
Next ask about timing. A cause must precede the effect it produces, but timing alone is insufficient. A new schedule may begin before attendance rises while an advertising campaign begins at the same time. The temporal order rules out some simple reverse stories, yet leaves common influences and other changes to examine. Conversely, existing problems may prompt an intervention: busy intersections receive more signals because traffic is already heavy. A positive association between signals and congestion can therefore arise partly from the response to congestion.
Be careful about levels of aggregation. Schools with more teachers often also have more desks because they have more students. That does not show that placing unused desks in an empty classroom will cause teachers to appear. It also does not establish that a particular student's learning depends only on the total number of desks in the school. A pattern among whole schools cannot be transferred automatically to individual students or to the effect of a particular action.
An accurate first sentence might be: 'Among these three schools, total desks and total teachers increase with enrolment.' This statement preserves the observed quantities, group, and common size factor. It avoids pretending that the descriptive pattern has already settled the causal arrow. Once the observation is stated carefully, the possible explanations become easier to compare.
Suppose three schools follow an announced allocation rule: one desk per student and one teacher per 25 students in the exact enrolment groups used here. A school with 200 students receives 200 desks and eight teachers; one with 400 receives 400 desks and sixteen teachers; one with 600 receives 600 desks and twenty-four teachers. Desks and teachers move together perfectly in this small constructed example, but the stated policy identifies enrolment as the input driving both allocations.
The common-cause diagram has enrolment pointing to desks and enrolment pointing to teachers. It does not need an arrow from desks to teachers. Adding such an arrow would make a new causal claim that the stated allocation rule does not supply. If someone proposes that insufficient desks can delay teacher deployment in real schools, that is a further mechanism worth investigating. It is not established by this table alone, and it should not be silently inserted into the stipulated model.
A reverse-causation story changes which variable responds. Suppose a city council installs additional lamps after receiving many reports of incidents. The earlier reports can help cause later lamp installation. A table showing more lamps in high-report areas is then compatible with lamps being a response rather than the original source of the reports. The diagram should use time-sensitive meanings when necessary: earlier reports → later lamps. Without that distinction, a single arrow can conceal the chronology.
A common cause is not proved merely because one can name it. 'Busy streets' may be a plausible factor affecting both lamp allocation and report counts, but its actual role needs evidence. Measure relevant activity, compare suitable streets, or inspect the policy that allocated lamps. Inventing a possible confounder shows that an inference is not yet secure; it does not prove the competing explanation true. The discipline cuts both ways: neither the attractive direct story nor the attractive alternative gets a free pass.
Chance also deserves a place in the inquiry. If many unrelated pairs are examined in a small dataset, some will look unusually similar by accident. A pattern chosen after searching hundreds of pairs needs different scrutiny from a prediction specified before collecting new observations. More data, replication, and a clear measurement plan can help, though a large sample alone does not remove systematic confounding or biased selection. Size and design solve different problems.
Finally, association can be useful even before its cause is settled. A variable may help predict another under stable conditions. However, prediction from observing A is different from predicting what will happen if we intervene to change A. The desk count may predict teacher allocation because both track enrolment. Buying extra unused desks does not reproduce an enrolment increase. This distinction prepares the next lesson's question: what evidence tells us that changing the proposed cause would change the outcome?
A causal diagram is a compact statement of a proposed structure. An arrow from A to B does not by itself say that the effect is large, immediate, beneficial, or the same in every case. It also does not say that A is the only cause of B. Those are further claims requiring information about the mechanism, conditions, and size of the effect. A small diagram deliberately leaves some of that information out so the direction can be inspected clearly.
Labels therefore matter. 'Reported incidents this year' and 'lamps installed next year' give a clearer hypothesis than simply crime and lamps. The fuller labels help prevent a later response from being mistaken for an earlier cause. When an exercise supplies a particular model, draw that model exactly and distinguish it from your beliefs about the real system. You may think another cause exists, but adding its arrow changes the answer being represented.
After drawing, read each edge as a sentence and compare it with the stated explanation. Then read any omitted direct edge as a limit of this model, rather than a universal declaration that the two quantities could never affect each other. This final check catches reversed arrows, extra mechanisms, and overly broad conclusions before the diagram is used to justify an action.
A fictional school district allocates one desk per student and one teacher per 25 students for the three enrolment sizes in this exercise. The first school has 200 students, 200 desks, and eight teachers. The second has 400 students, 400 desks, and sixteen teachers. The third has 600 students, 600 desks, and twenty-four teachers. A newsletter observes that schools with more desks also have more teachers.
That association is correctly described. Doubling enrolment from 200 to 400 doubles both desk allocation and teacher allocation. Increasing enrolment from 400 to 600 adds 200 desks and eight teachers. But the district's stated rule supplies the common input: enrolment. The supported diagram is enrolment → desks and enrolment → teachers. The table does not show that desks themselves recruit teachers.
To see the difference, imagine delivering 50 spare desks to the 200-student school while keeping enrolment and the staffing rule unchanged. The school now holds 250 desks, but the rule still allocates 200 ÷ 25 = eight teachers. The extra desk delivery breaks the old predictive relationship without changing the staffing input. Observing a school's normal desk allocation and intervening on its desk stock are different operations.
The practical lesson is to identify which rule or process connects the quantities. A planning team can use the enrolment rule to forecast resources for a stated student count. It should not use the descriptive desk-teacher association as a promise that buying furniture will increase staffing. The example makes a causal distinction visible with simple numbers and a fully stated mechanism, so the arrows can be checked rather than guessed.
Treating a close match as proof of a direct cause. A common cause can also create a close match.
Treating four diagrams as an exhaustive list. Real systems can combine causes, feedback, selection, and measurement effects.
Treating a possible confounder as established. Naming an alternative shows a question remains; evidence is needed to support the alternative.
Confusing a predictor with an intervention target. A desk count can predict teacher allocation without extra unused desks causing extra teachers.
Identify the unit of observation.
Weeks in one town.
The pattern compares periods, not individual purchases or people.
Identify the two recorded variables.
Ice-cream sales and reported sunburn cases.
A descriptive claim must say what was actually counted.
State the observed pattern.
Higher-sales weeks also have more reported sunburn cases.
This wording reports association without asserting a mechanism.
Propose a common influence to investigate.
Sunny weather may affect outdoor activity, exposure, and sales.
A factor outside the pair can help explain their joint movement.
Limit the causal conclusion.
The association alone does not establish ice cream causing sunburn.
The proposed direct arrow needs evidence distinguishing it from alternatives.
State the cross-street association.
Streets with more lamps have more reported incidents.
This describes the recorded pattern rather than its direction of causation.
Inspect a possible allocation process.
Earlier incident reports may prompt later lamp installation.
The proposed outcome can influence the apparent cause at a later time.
Draw that candidate arrow.
Earlier reports → later lamps.
The arrow follows the hypothesized response process.
Name another candidate influence.
Busy street activity may influence both lamp allocation and reporting.
A common cause remains possible alongside the reverse-direction story.
Specify what remains to check.
Inspect dates, allocation records, and comparable street activity.
The diagram states a hypothesis rather than establishing which process actually occurred.
Write the stipulated rules.
One desk per student; one teacher per 25 students.
These rules identify the common input for both allocations.
Apply them at 200 students.
200 desks; 200 ÷ 25 = 8 teachers.
Both quantities are calculated from enrolment.
Apply them at 400 students.
400 desks; 400 ÷ 25 = 16 teachers.
The same input change raises both outputs.
Draw the common-cause model.
Enrolment → desks; enrolment → teachers.
No direct desk-to-teacher link is needed to represent the stated policy.
Change only desk stock at the smaller school.
Deliver 50 spare desks: 250 desks, still 200 students.
This intervention leaves the staffing input unchanged.
Calculate the unchanged staffing allocation.
200 ÷ 25 = 8 teachers.
An observational predictor need not be the variable whose intervention changes the outcome.
Identify the observed pair across an age-mixed group.
Shoe size and reading score.
The association compares children at potentially different stages.
State the proposed common developmental factor.
Age can relate to both physical growth and accumulated reading instruction.
This supplies a candidate influence outside the measured pair.
Keep the direct intervention claim separate.
Four explanations of one finding: the streets with the most lamps have the most reported crime. Match each to its shape.
| The first causes the second | The second causes the first | Something else causes both | Coincidence | |
|---|---|---|---|---|
| Lamps draw people out at night, and that brings crime. | ||||
| Streets where crime is reported get extra lamps put up. | ||||
| Busy central streets get more lamps and have more people about to report things. | ||||
| It happens to have come out that way in these eleven streets. |
For this example, a district allocates one desk per student and one teacher per 25 students. A school has 400 students. A later delivery adds 50 spare desks without changing enrolment. Complete the allocation calculations.
Apply the teacher-allocation rule.
Students ÷ 25 = teachers teachers.
Enrolment is the staffing input under the stipulated rule.
Apply the desk-allocation rule.
One desk per student gives desks allocated desks.
Both allocations respond to the same student count.
Add only the furniture delivery.
Allocated desks + 50 = stock desks in stock.
The delivery changes desk stock without changing the staffing input.
Somebody counts and finds that the children with the biggest shoes are the children who read best. Taken on its own, what does the count settle?
The weeks with the most ice cream sold are the weeks with the most sunburn. Draw the common-cause explanation: put in every arrow it says, and no others.
This task has no paper form; do it on a device.
Four explanations of one finding: the students who eat breakfast have the better grades. Say which of the four shapes each explanation has.
| Where the arrows start | Which of the four? | |
|---|---|---|
| Breakfast steadies the morning, so the work goes better | at the breakfast | |
| Students who do well are given breakfast as a treat | at the grades | |
| A settled home routine produces both the breakfast and the grades | outside the pair | |
| It came out that way in this year group and in no other | nowhere |
The children with the biggest shoes are the children who read best. Draw the common-cause explanation of that, with every arrow it says and no others.
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 bakery's stated rule prepares two loaves and reserves one bag for each confirmed order. Across days, total loaves and reserved bags move together because the confirmed order count determines both. Construct this common-input model only; do not add a direct loaf-to-bag or bag-to-loaf cause.
This task has no paper form; do it on a device.
You can name all four explanations of a pair that moves together, and draw the one where a third thing moves both. Tell somebody why bigger shoes and better reading is the same shape of argument as breakfast and better grades. Next: what would settle which of the four it is.
13. Age, shoe size, and reading experience, step 3
Buying larger shoes does not follow as a way to improve reading from this association.
A common-factor explanation does not establish the shoes-to-reading arrow.