Week 4 · Chapter

Chapter 4 — Exploring Relationships Between Two Variables

Introduction to Statistics Free sample edition

The module's primary reading · ~30–40 minutes · Objective 3 (describe the relationship between two variables using scatterplots, the correlation coefficient, and two-way tables, and explain why association alone does not establish causation) · Read it first this week; the lecture, the tutorial, the quiz, and the Data Lab all build on it.

In plain language

Some numbers seem to move together, like hot days and cold drinks sold, and this week you learn to check whether they really do and how closely. You will draw a scatterplot (a graph with one dot for each individual, placed by its two measurements). You will read a correlation (one number that says how tightly the dots follow a straight line). You will also read a table that sorts people or animals two ways at once. Last, you will learn why "these move together" is not the same as "one causes the other."

Why you already care about this

Between 2007 and 2009, researchers on three islands of Antarctica's Palmer Archipelago measured 342 penguins. For each bird they recorded two lengths and a weight, among other things. Here is a question any curious person would ask of that field log: do penguins with longer flippers tend to weigh more? You cannot answer it by looking at flipper lengths alone, or at weights alone. You have to look at both numbers for each bird at the same time.

You ask questions like this all the time. Does a longer commute go with less sleep? Do neighborhoods with more trees feel cooler in summer? Does a phone lose resale value as it gets older? Each question pairs two measurements and asks how they move together. And each one tempts you to jump from "they move together" to "one causes the other." That jump is where most bad headlines come from.

This chapter gives you the tools a careful analyst uses before saying anything. You will draw the pairs, describe the pattern in four words, attach one honest number to it, and then ask what else might be going on.

By the end of this chapter you will be able to:

name which of two variables explains and which responds in a question you are asked;
describe a scatterplot's direction, form, strength, and unusual points;
read a correlation of 0.8712 and say what it does and does not tell you;
turn the counts in a two-way table into marginal and conditional percentages;
name a lurking variable that could explain a pattern without any cause and effect.

Carry this sentence with you: the correlation sees straight lines — not curves, and not causes.

Where we are

Already covered: Week 1 separated observational studies from experiments; Week 2 drew one variable at a time and introduced the two-way table; Week 3 put numbers on center and spread.
This week: two variables at once — scatterplots, the correlation r, marginal and conditional distributions, and the difference between association and causation.
How they connect: a scatterplot is a dot plot with a second axis, and a conditional distribution is a relative-frequency table for one slice of a two-way table.
Where we're going next: Week 5, Probability Foundations, opens a new unit on chance and reads probabilities straight out of the same two-way table.

Each earlier week hands you one tool for today. Week 1 taught you that the penguin log is an observational study, where researchers record what they see without assigning anything, and that only a randomized experiment supports a cause-and-effect claim. Week 2 taught you to read a two-way table cell by cell, with its row and column totals, and to turn a count into a percentage of a total. Week 3 gave you the mean and the standard deviation for one variable. Today you put two variables side by side and ask how they relate.

1. Name the roles: explanatory and response variables

Picture a coffee cart owner on a college quad. Every afternoon she writes down two numbers: the day's high temperature and how many iced drinks she sold. She suspects the heat helps explain the sales. The temperature is the number she thinks does the explaining. The iced-drink count is the number she wants to understand. That difference in roles is the first thing to settle in any two-variable question.

An explanatory variable is, in a two-variable question, the variable that may help explain or predict the other one; it is plotted on the horizontal axis of a scatterplot and written x.
A response variable is, in a two-variable question, the variable whose values you want to explain or predict; it is plotted on the vertical axis of a scatterplot and written y.

So x (the explanatory variable) goes across the bottom, and y (the response variable) goes up the side. Each individual gives one pair of values, written (x, y), and each pair becomes one dot.

Worked example — choosing roles in the penguin question. The question is: do penguins with longer flippers tend to weigh more?

Find the two variables. Flipper length, in mm, and body mass, in g.
Ask which one explains. The question asks whether flipper length helps us predict weight, so flipper length is the explanatory variable, x.
The other one responds. Body mass is the response variable, y.
Count the pairs. The field log has both measurements for 342 penguins, so n (the number of pairs) is 342.

In words: we will plot 342 dots, flipper length across and body mass up.

Misconception check. Students often think "explanatory" means "proven cause." It does not. Calling flipper length the explanatory variable is a choice about which number we use to predict the other. It makes no claim that a longer flipper makes a penguin heavier. The cure: read "explanatory" as "the one we start from," and save the word "cause" for Section 7.

Sometimes neither variable is the natural starting point. A runner's weekly miles and resting heart rate could be read in either order. Then you may pick either one as x, and you should say which you picked.

2. Draw the scatterplot: direction, form, strength, and unusual points

A scatterplot turns a list of pairs into a picture. Think of a phone resale site. Each listing has the phone's age in months and its asking price. Plot each listing as a dot, age across and price up. You would expect the dots to slope downward, because older phones tend to sell for less.

A scatterplot is a graph of two quantitative variables measured on the same individuals, with one point for each individual placed at its (x, y) pair of values.

Once the dots are down, describe the picture with four words, always in this order.

Direction is whether the points on a scatterplot tend to rise from left to right (a positive association) or fall from left to right (a negative association).
A positive association is a relationship in which larger values of one variable tend to go with larger values of the other, so a scatterplot's points rise from left to right.
A negative association is a relationship in which larger values of one variable tend to go with smaller values of the other, so a scatterplot's points fall from left to right.
Form is the overall shape of the pattern on a scatterplot, such as a straight-line (linear) band, a curve, or separate clusters of points.
A linear relationship is a relationship between two quantitative variables whose scatterplot points cluster around a straight line.
Strength is how tightly the points on a scatterplot follow their overall pattern; points that hug a line or curve show a strong relationship, and loosely scattered points show a weak one.
An unusual point is a point that sits far from the overall pattern; it deserves a second look, the same way an outlier did in Week 2.

Worked example — reading the penguin scatterplot. Plot the 342 penguins with flipper length across and body mass up.

Set the axes from the data. Flipper lengths run from 172 mm to 231 mm, and body masses run from 2,700 g to 6,300 g. So the horizontal axis spans about 170 to 235 mm and the vertical axis about 2,500 to 6,500 g.
Direction. The points rise from lower left to upper right. Birds with short flippers sit low; birds with long flippers sit high. The direction is positive.
Form. The points fall in a band that follows a straight line rather than bending. The form is linear.
Strength. The band is fairly narrow for its length, so the association looks strong. Section 3 puts a number on that.
Unusual points. Scan for any dot far from the band. That check is yours to make in the Data Lab, where you draw this plot yourself.

In words: penguins with longer flippers tend to weigh more, and the pattern is close to a straight line.

Misconception check. Students often describe a scatterplot by its steepness: "the dots climb fast, so the relationship is strong." Steepness depends on the units you chose for the axes, so it says nothing about strength. Stretch the vertical axis and the same dots look steeper. The cure: judge strength by how tightly the dots hug the pattern, never by how steep the pattern looks.

In practice (your tools). Load the field log into Google Sheets with the same =IMPORTDATA(...) formula you used in Week 1. Flipper length lands in column E and body mass in column F. Select E1:F345, then use Insert, then Chart. In the Chart editor, under Setup, set Chart type to Scatter chart. Sheets puts the left column, flipper length, on the horizontal axis, which is what you want. The two rows with NA are text, so they draw no dot. In Excel, select the same two columns and use Insert, then Insert Scatter (X, Y), then the first Scatter option.

3. Put one number on a straight line: the correlation coefficient

Words like "strong" and "fairly narrow" leave room to argue. Two people can look at the same cloud of dots and disagree. A single number settles the argument for straight-line patterns.

The correlation coefficient is a unitless number, written r, between negative 1 and positive 1 that measures the direction and strength of the straight-line relationship between two quantitative variables; it does not detect curved patterns.

Read r (the correlation) in two parts. Its sign gives the direction: a plus sign means a positive association, and a minus sign means a negative one. Its size, ignoring the sign, gives the strength. A value near 1 or near −1 means the dots hug a straight line. A value near 0 means no straight-line pattern at all. An r of exactly 1 or −1 means every dot sits on one line.

A rough guide to reading the size of r, ignoring its sign
Size of r, ignoring the sign	How the dots look	A fair word for the strength
close to 1	tightly packed around a line	strong
around one half	a clear tilt with plenty of scatter	moderate
close to 0	no straight-line tilt	weak or none

Honesty note: those cut points are a guide, not a law. Different fields use different words for the same r, so always report the number along with your word.

The symbol r belongs to a sample. If you could measure every penguin of these species, the correlation for that whole population would be written ρ, the Greek letter that stands in for r. This week you compute r from the 342 penguins in the log, never ρ.

Worked example — the correlation for the penguins. The software computes r from all 342 pairs.

Get the number. For flipper length and body mass, r (the correlation) is 0.8712.
Read the sign. The sign is positive, which matches the rising dots from Section 2.
Read the size. 0.8712 is close to 1, so the dots hug a straight line closely.
Say it in words. The association between flipper length and body mass is strong, positive, and linear.

So a field biologist who knows only a penguin's flipper length can make a good guess about its weight. In Week 15 you will build the line that turns that guess into a number.

Misconception check. Students often read a correlation of 0.8712 as "87% of penguins follow the pattern" or "flipper length explains 87% of weight." Neither is right. The correlation is not a percentage of anything. It is a measure of how tightly the dots hug a line on a scale from −1 to 1. The cure: report r as a plain number, and describe it with direction, form, and strength words.

In practice (your tools). With the field log in Google Sheets, type =CORREL(E2:E345,F2:F345) in an empty cell. Set it to show four decimals with Format, then Number, then Custom number format, and it shows 0.8712. The two NA rows are skipped, because both of their cells are text. In Excel, the same =CORREL formula gives the same value. Swap the two ranges and the answer does not change, because r treats x and y the same way.

4. Know what r can and cannot do

The correlation is useful, but it has four habits you need to know. It always runs from −1 to 1. It has no units. It only sees straight lines. And one unusual point can drag it a long way. Here is each habit with a concrete case.

Worked example — changing the units leaves r alone. A field team in another country records flipper length in centimeters instead of millimeters.

Convert. Divide every flipper length by 10. The range 172 mm to 231 mm becomes 17.2 cm to 23.1 cm.
Recompute r. The dots move to new axis labels, but their pattern is exactly the same.
Result. r stays 0.8712, because r carries no units.

In words: a team measuring in centimeters and a team measuring in millimeters report the same correlation. That is why researchers in different countries can compare their correlations directly.

Worked example — a perfect pattern with a correlation of 0. A board game scores each roll of a die by how far it lands from the middle. Let x be the face that lands, and let y be the score. The game sets y = (x − 3.5)², the squared distance of the face from the die's mean of 3.5.

Score each face. Face 1 scores (1 − 3.5)² = 6.25. Face 2 scores 2.25. Face 3 scores 0.25. Face 4 scores 0.25. Face 5 scores 2.25. Face 6 scores 6.25.
Picture the dots. The six points dip from 6.25 down to 0.25 and climb back to 6.25. That is a perfect U shape.
Compute r. The left half falls and the right half rises by exactly the same amounts, so the two halves cancel. The correlation r (for these six pairs) comes out to exactly 0.

In words: the score is completely decided by the face, yet the correlation is zero. A player who reads a correlation of 0 as "the face has nothing to do with the score" would be badly wrong.

Misconception check. Students often think "r near 0 means no relationship." It means no linear relationship. The die scores have the strongest possible relationship, a perfect curve, and r cannot see it. The cure: always draw the scatterplot before you read r, because only the picture shows you a curve.

Worked example — one point flips the sign. Pair each month's position in the year, 1 through 12, with its number of days in a common year.

All twelve months. r is about 0.27. That is a weak positive association.
Spot the unusual point. February, month 2, has 28 days. Every other month has 30 or 31.
Remove February. r for the other 11 months is about −0.13. That is a weak negative association.

In words: one unusual month flipped the sign of the correlation. A calendar never changes, but the summary number did, because one point moved it. Statisticians say r is not resistant: like the mean, it uses the size of every value, so one far-off point can pull it a long way.

Honesty note: neither 0.27 nor −0.13 is a "true" correlation for the calendar. Month position and month length have no meaningful straight-line link at all. The lesson is about r's sensitivity, not about months.

Pause and reflect — choose one. Pick either of the 2 prompts; they count the same.

(a) Think of two things in your life that rise and fall together in a curve, such as room temperature and comfort; describe the shape in words and say what the correlation might miss.
(b) Think of a dataset where one unusual entry could fool a summary number, such as one huge tip in a server's week or one snow day in a month of commutes; say what you would check before trusting r.

5. Read a two-way table two ways: marginal and conditional distributions

In Week 2 you counted the penguins by species and island at once. Here is that two-way table again, with its totals.

The 344 penguins in the Palmer field log, counted by species (rows) and island (columns), with totals
Species	Biscoe	Dream	Torgersen	Total
Adelie	44	56	52	152
Chinstrap	0	68	0	68
Gentoo	124	0	0	124
Total	168	124	52	344

A two-way table holds two kinds of story. One story lives in the margins, the totals along the edges. The other lives inside a single column or a single row. A grocery manager reading a table of shoppers by aisle and time of day faces the same two choices: look at the whole store, or look at one aisle.

A marginal distribution is the distribution of one variable in a two-way table on its own, read from the row totals or the column totals and divided by the grand total.
A conditional distribution is the distribution of one variable among only the individuals in one category of the other variable, read from a single row or column of a two-way table and divided by that row's or column's total.

Worked example — the two marginal distributions. Read each total along an edge of the table and divide it by the grand total, 344.

Species, from the row totals. Adelie 152 ⁄ 344 ≈ 44.2%, Chinstrap 68 ⁄ 344 ≈ 19.8%, Gentoo 124 ⁄ 344 ≈ 36.0%. These add to 100.0%.
Island, from the column totals. Biscoe 168 ⁄ 344 ≈ 48.8%, Dream 124 ⁄ 344 ≈ 36.0%, Torgersen 52 ⁄ 344 ≈ 15.1%. These add to 99.9% because of rounding.

In words: across the whole log, a bit under half the penguins are Adelie, and a bit under half were seen on Biscoe.

Worked example — the conditional distribution of species on each island. Now look down one column at a time and divide by that column's total, not by 344.

Biscoe (168 birds). Adelie 44 ⁄ 168 ≈ 26.2% and Gentoo 124 ⁄ 168 ≈ 73.8%. These add to 100.0%.
Dream (124 birds). Adelie 56 ⁄ 124 ≈ 45.2% and Chinstrap 68 ⁄ 124 ≈ 54.8%.
Torgersen (52 birds). Adelie are 52 ⁄ 52 = 100%.

In words: on Biscoe, about three in four penguins are Gentoo, while on Torgersen every penguin in the log is Adelie.

Conditional distribution of species within each island, as percentages of that island's total
Species	Biscoe	Dream	Torgersen
Adelie	26.2%	45.2%	100%
Chinstrap	0%	54.8%	0%
Gentoo	73.8%	0%	0%
Total	100.0%	100.0%	100%

Misconception check. Students often divide by the wrong total. For "what share of Biscoe penguins are Gentoo?" they compute 124 ⁄ 344 and get 36.0%. But that answers a different question: what share of all penguins are Gentoo? The cure: say the group you are looking inside out loud first. "Of the Biscoe penguins" means the denominator is 168.

In practice (your tools). In Google Sheets, build the pivot table from Week 2: rows by species, columns by island, values by species summarized by COUNTA. Then, in the Values box, change "Show as" from Default to "% of column." Each island's column now shows its conditional distribution. In Excel, right-click any count in the PivotTable, choose Show Values As, then % of Column Total.

6. Compare conditional distributions to spot an association

A table becomes interesting when the conditional distributions disagree. Imagine a hiking club that sorts its members by preferred trail and by home neighborhood. If every neighborhood picked trails in the same proportions, knowing someone's neighborhood would tell you nothing about their trail. If the proportions differ, the two variables are linked.

An association is a relationship between two variables in which knowing the value of one changes what you expect for the other; for categorical variables, the conditional distributions differ from one another.

Worked example — species and island are associated. Compare the Gentoo share in the whole log with the Gentoo share on one island.

The whole log. Gentoo are 36.0% of all penguins, from the marginal distribution.
One island. Gentoo are 73.8% of the Biscoe penguins, from the conditional distribution.
Compare. 36.0% versus 73.8%: on Biscoe the Gentoo share takes a 2-fold jump versus the whole log, since 73.8 ⁄ 36.0 ≈ 2.
Conclude. Knowing the island changes your guess about the species, so species and island are associated.

In words: if a field assistant told you a penguin came from Biscoe, you would bet on Gentoo; if she said Torgersen, you would bet on Adelie.

Notice what this did not need. There was no correlation r here, because species and island are categories, not numbers. You cannot compute r for labels. For two categorical variables, comparing conditional distributions does the job that r does for two quantitative ones.

Misconception check. Students often decide two categorical variables are "unrelated" because the grand total looks balanced. Island totals of 168, 124, and 52 say nothing about the link between island and species. Only the conditional distributions can show an association. The cure: compare one column's percentages with another's, or with the marginal percentages, before you decide.

7. Separate association from causation: lurking and confounding variables

Here is a pattern that shows up in many places with warm summers. On days when ice-cream sales are high, more people drown. Should a town ban ice cream to save swimmers? No. Hot weather sends more people to buy ice cream, and hot weather sends more people into the water. The heat drives both.

Causation is a relationship in which changing one variable actually produces a change in the other; only a well-designed randomized experiment can establish it.
A lurking variable is a variable that is not among the variables being studied but that influences them, and can create or hide an association between them.
A confounding variable is a variable whose effect on the response cannot be separated from the effect of the explanatory variable, because the two are tangled together in the data.

Worked example — naming the lurking variable. For each pattern, ask what third variable could push both numbers up together.

Ice cream and drownings. Both rise in hot weather. The lurking variable is temperature.
Shoe size and reading level in children. Children with bigger feet tend to read at a higher level. Bigger feet do not teach reading. Older children have bigger feet and have had more years of school. The lurking variable is age.
Island and body mass in the penguin log. Every Gentoo lives on Biscoe, and Gentoo average 5,076.02 g against 3,700.66 g for Adelie. So any comparison of body mass by island mixes in species. Species is a confounding variable for that comparison: we cannot tell an island effect apart from a species effect.

In words: in each case a third variable could produce the pattern with no cause and effect between the two we measured.

Worked example — what about flippers and weight? The correlation r (for the 342 penguins) is 0.8712, which is strong. Does a longer flipper cause a heavier penguin?

Check the study design. The penguin log is an observational study. Researchers measured the birds; they did not assign flipper lengths.
Look for a lurking variable. Larger birds overall tend to have both longer flippers and more mass. Species could also play a part, since the species differ in size.
Conclude. The data show a strong association. They cannot show causation, because no experiment was run.

Caveat: a strong correlation is a reason to ask about causes, not an answer. A researcher who wants to claim cause and effect needs a randomized experiment, or a long chain of careful evidence from many studies.

Misconception check. Students often think "a strong enough correlation proves causation." Strength does not help. The ice-cream pattern could be strong, and it would still be driven by the weather. The cure: before saying "causes," ask two questions. Was there a randomized experiment? And what third variable could drive both?

Pause and reflect — choose one. Pick either of the 2 prompts; they count the same.

(a) Find a headline or social media post that links two things, such as a food and a health outcome or an app and a grade; name one lurking variable that could explain the link.
(b) Think of a pattern you have noticed at work, at school, or at home where two things rise together; describe the experiment you would need to test whether one causes the other.

Key terms

Explanatory variable — In a two-variable question, the variable that may help explain or predict the other one; it is plotted on the horizontal axis of a scatterplot and written x. (first used: 1. Name the roles: explanatory and response variables)
Response variable — In a two-variable question, the variable whose values you want to explain or predict; it is plotted on the vertical axis of a scatterplot and written y. (first used: 1. Name the roles: explanatory and response variables)
Scatterplot — A graph of two quantitative variables measured on the same individuals, with one point for each individual placed at its (x, y) pair of values. (first used: 2. Draw the scatterplot: direction, form, strength, and unusual points)
Direction — Whether the points on a scatterplot tend to rise from left to right (a positive association) or fall from left to right (a negative association). (first used: 2. Draw the scatterplot: direction, form, strength, and unusual points)
Positive association — A relationship in which larger values of one variable tend to go with larger values of the other, so a scatterplot's points rise from left to right. (first used: 2. Draw the scatterplot: direction, form, strength, and unusual points)
Negative association — A relationship in which larger values of one variable tend to go with smaller values of the other, so a scatterplot's points fall from left to right. (first used: 2. Draw the scatterplot: direction, form, strength, and unusual points)
Form — The overall shape of the pattern on a scatterplot, such as a straight-line (linear) band, a curve, or separate clusters of points. (first used: 2. Draw the scatterplot: direction, form, strength, and unusual points)
Linear relationship — A relationship between two quantitative variables whose scatterplot points cluster around a straight line. (first used: 2. Draw the scatterplot: direction, form, strength, and unusual points)
Strength — How tightly the points on a scatterplot follow their overall pattern; points that hug a line or curve show a strong relationship, and loosely scattered points show a weak one. (first used: 2. Draw the scatterplot: direction, form, strength, and unusual points)
Correlation coefficient — A unitless number, written r, between negative 1 and positive 1 that measures the direction and strength of the straight-line relationship between two quantitative variables; it does not detect curved patterns. (first used: 3. Put one number on a straight line: the correlation coefficient)
Marginal distribution — The distribution of one variable in a two-way table on its own, read from the row totals or the column totals and divided by the grand total. (first used: 5. Read a two-way table two ways: marginal and conditional distributions)
Conditional distribution — The distribution of one variable among only the individuals in one category of the other variable, read from a single row or column of a two-way table and divided by that row's or column's total. (first used: 5. Read a two-way table two ways: marginal and conditional distributions)
Association — A relationship between two variables in which knowing the value of one changes what you expect for the other; for categorical variables, the conditional distributions differ from one another. (first used: 6. Compare conditional distributions to spot an association)
Causation — A relationship in which changing one variable actually produces a change in the other; only a well-designed randomized experiment can establish it. (first used: 7. Separate association from causation: lurking and confounding variables)
Lurking variable — A variable that is not among the variables being studied but that influences them, and can create or hide an association between them. (first used: 7. Separate association from causation: lurking and confounding variables)
Confounding variable — A variable whose effect on the response cannot be separated from the effect of the explanatory variable, because the two are tangled together in the data. (first used: 7. Separate association from causation: lurking and confounding variables)

Summary

Name the roles first: the explanatory variable goes across, and the response variable goes up.
Describe every scatterplot by direction, form, strength, and unusual points.
The correlation r runs from negative 1 to 1 and measures only straight-line patterns.
Changing units never changes r, but one unusual point can change it a lot.
Draw the scatterplot before you trust r, because r cannot see a curve.
A marginal distribution divides by the grand total; a conditional one divides by one row's or column's total.
Two categorical variables are associated when their conditional distributions differ.
Association is not causation: look for a lurking variable, and ask whether there was a randomized experiment.

Check your understanding

Work these before looking at the answers — they are ungraded, and the answers (with reasoning) are at the bottom of the page.

A campus café asks 200 customers what they ordered and when. In the morning, 72 ordered coffee and 18 ordered tea. In the afternoon, 55 ordered coffee and 55 ordered tea. Find the marginal distribution of drink, find the conditional distribution of drink for each time of day, and say whether drink and time of day are associated.
A bakery finds a correlation of −0.64 between oven temperature in degrees Fahrenheit and baking time in minutes for its bread loaves. Describe the direction and strength in words, and say what r becomes if the temperatures are rewritten in degrees Celsius.
A coach records hours of practice and injuries for a season and finds a correlation of 0.02. A teammate says, "So practice time has nothing to do with injuries." What should you check before agreeing, and why?
A news story reports that cities with more fire stations have more fires each year. Name a lurking variable, and explain how it could produce the pattern.

Answers (worked reasoning — read after attempting)

First the totals. Coffee: 72 + 55 = 127. Tea: 18 + 55 = 73. Morning: 72 + 18 = 90. Afternoon: 55 + 55 = 110. Grand total: 127 + 73 = 200, and 90 + 110 = 200, so the two routes agree. The marginal distribution of drink divides by 200: coffee 127 ⁄ 200 = 63.5% and tea 73 ⁄ 200 = 36.5%. The conditional distribution in the morning divides by 90: coffee 72 ⁄ 90 = 80.0% and tea 18 ⁄ 90 = 20.0%. In the afternoon it divides by 110: coffee 55 ⁄ 110 = 50.0% and tea 55 ⁄ 110 = 50.0%. The two conditional distributions differ, 80.0% coffee versus 50.0%, so drink and time of day are associated.
The sign is negative, so the direction is negative: hotter ovens go with shorter baking times. The size, 0.64, is moderate, so the dots show a clear downward tilt with plenty of scatter. Rewriting Fahrenheit as Celsius is a change of units, and r has no units, so r stays −0.64.
Check the scatterplot first. An r near 0 only rules out a straight-line pattern. Injuries could be high for players who barely practice and for players who practice the most, with fewer in the middle. That U shape would give an r near 0 even though the two are strongly related. The teammate made the "r near 0 means no relationship" mistake.
City size is a lurking variable. Bigger cities have more buildings and more people, so they have more fires. Bigger cities also build more fire stations to cover more ground. City size drives both counts, so the fire stations need not cause any fires.

Next up in this module: the Readings & Resources page, the lecture tutorial, the practice set, and the quiz — and next week each conditional distribution you read here becomes a probability rule.

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