A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
In 2012 the cardiologist Franz Messerli published a short note in the New England Journal of Medicine. For 23 countries he set two numbers side by side: how much chocolate people eat per year, and how many Nobel laureates the country has per 10 million people. The association was strong and positive: a correlation of . Switzerland, which eats the most chocolate, also had the most laureates per person.
Figure 1: An illustrative scatterplot with the same shape as Messerli’s (2012): 23 countries, a strong positive trend, and one country far above it. The points are drawn for this lesson; they are not the study’s data.
Before you read on, make a prediction. Nothing is scored. Should a country that wants more Nobel prizes eat more chocolate? Yes, no, or “it depends on…”?
Pause and write. Write your answer and one sentence of reasoning on paper or in your notes. Keep it: this lesson takes the claim apart one piece at a time (C2, C7, C8 and C9), and the Boss Fight settles it.
The number Messerli reported is called the Pearson correlation coefficient, written . It answers one precise question: how strongly, and in which direction, do two quantitative variables move together along a straight line? It runs from (a perfect falling line) to (a perfect rising line), with meaning no linear trend at all. A number like can be computed exactly right and still be read wrongly. This lesson is about both halves: computing , and knowing what it can and cannot tell you.
After this lesson, you will be able to:
Describe a scatterplot in four dimensions: direction, form, strength, and outliers
Compute the Pearson correlation coefficient from raw data with the computational formula
Compute the coefficient of determination and interpret it as the share of observed response variability accounted for by a fitted line
Distinguish correlation from causation, and recognise when a study design does support a causal claim
Spot when a correct still misleads: a restricted range, one influential point, or read as a slope
This lesson builds on the scatterplot skills from L-DS-2 and the standard deviation and z-score notation from L-DS-5.
Section 2: Prerequisites
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Correlation analysis is built on two earlier skills. Before computing , make sure these are solid.
From L-DS-2: Scatterplots. You can read a scatterplot’s direction (rising, falling, or neither) and form (a straight band or a curve). Today we attach a precise number to that description.
From L-DS-5: Standard deviation and z-scores. and are the sample standard deviations of the - and -values, and a z-score says how many standard deviations a value sits from its mean. The definition of is built from paired z-scores.
Retrieval Checkpoint
Bridge to today: In L-DS-2, describing a scatterplot was qualitative: words like “upward trend” and “tight cluster.” In this lesson we attach a precise number, , to those words. The catch is that measures only the straight-line part of a pattern. A scatterplot can show a strong curved relationship and still have . The number does not replace the picture; it adds to it.
Section 3: Core Concepts
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Ten concepts live in this section. They build in this order:
C1–C3: Reading a scatterplot, computing , and naming its strength.
C4–C6: What does not mean, what adds, and the most common slip ( where belongs).
C7–C8: Correlation versus causation, and the conditions Pearson needs.
C9–C10: What ignores (units, which variable is , the slope), and how a restricted range shrinks it.
C1 — Reading Scatterplots Qualitatively
Before any formula, a scatterplot tells a story in four dimensions. Describe it in this order:
Direction: Do the points trend upward (positive), downward (negative), or neither?
Form: Is the pattern approximately linear (a straight band), or curved?
Strength: How tightly do the points cluster around the trend? Loose cloud = weak; hugging a line = strong.
Outliers: Is any point far from the main pattern?
Mini-example: A scatterplot of hours studied () against exam score () shows points that rise from left to right, fairly tightly around a straight line, with no isolated point. Description: positive direction, linear form, moderate-to-strong strength, no notable outliers.
Eyes are not calibrated instruments: most people read a moderate correlation as weaker than it is. Test yours. Estimate for each of the four plots below, then reveal the exact values. Nothing is scored or saved.
Plot A
Plot B
Plot C
Plot D
Most people read moderate correlations as weaker than they are.
Figure 2: Four unlabelled scatterplots. Type your estimate of under each, then reveal the exact correlations.
Now put names to the shapes. Before you reveal any values in Figure 3, put plots A to E in order from the most negative to the most positive. Then use the button to check your order.
Figure 3: Five scatterplots, A to E. One of them has a single point far from the rest; notice what that point does to when the values are revealed.
C2 — Pearson Correlation Coefficient
The Pearson is the number that quantifies what C1 describes. It is built from one idea: standardise each and each as a z-score, multiply the pairs, add the products, and divide by .
We write for the sample Pearson correlation coefficient, a statistic computed from our data.
Pearson Correlation Coefficient — Definition Form
where and are the sample means, and are the sample standard deviations of and , and is the number of pairs.
Domain: Pearson is defined only when both variables vary: and . If either variable is constant, the denominator is zero and is undefined.
Range when defined:. The sign gives direction; the magnitude gives strength.
The definition form shows the intuition. When is above and is above at the same time, the product is positive and pushes toward . When one is above its mean and the other below, the product is negative and pulls down. With no systematic pattern, the positives and negatives cancel and .
For hand calculations, the computational form avoids computing means and deviations:
Pearson Correlation Coefficient — Computational Form
Computation table: set up five columns, ,,, and ; find the five column sums; substitute.
Why two forms? The definition form shows what means: an average of paired z-score products. The computational form gives the same number with less arithmetic, because you never compute and separately.
Every point casts a vote through its product. Which points vote hardest? In Figure 4, move one point to make as small as you can. Where must it go, and why there?
Goal: move ONE point to make r as small as you can.
Drag a point, or focus it with Tab and use the arrow keys (2 units a press). Once one point has
moved, the others are locked until Reset.
Positive product: pushes r up
Negative product: pulls r down
That point now sits far from both means in an opposite-sign quadrant, so its product is large
and negative.
Figure 4: Eighteen students, homework completed and exam score, with dashed lines at the two means. Blue points make positive contributions to ; orange points pull it down. Drag one point, or select it and use the arrow keys.
Back to the chocolate. Messerli’s is a correctly computed number: the sign and size are what the formula gives for his 23 pairs. Nothing in the rest of the lesson says the arithmetic was wrong. The questions are about what the number means.
C3 — Interpreting — Strength Thresholds
The magnitude tells you how strong the linear relationship is. These conventional guidelines are widely used in introductory statistics:
range
Interpretation
Strong linear relationship
Moderate linear relationship
Weak linear relationship
The sign of tells you direction independently of strength: is a strong negative relationship; is a weak positive relationship.
Mini-example:. Magnitude: falls in , so moderate. Sign: negative, so as increases, tends to decrease. Full interpretation: a moderate negative linear relationship.
These thresholds are conventions, not laws. In medicine or psychology, can be practically important; in a physics calibration, can be disappointingly weak. Always consider the context.
C4 — Does NOT Mean No Relationship
An exact means zero sample linear correlation. A value close to zero means little sample linear association. Neither says anything definite about other kinds of relationship, or about the population correlation.
If you compute and conclude “there is no relationship between and ,” you may be completely wrong. A symmetric U-shaped or arch-shaped pattern, sampled evenly around its turning point, can give even though is perfectly predictable from .Always look at the scatterplot: alone is not enough.
Figure 5: An arch-shaped relationship with . The best straight line is flat, yet every follows exactly from its .
Mini-example: and , so . The scatterplot is a perfect U. Yet , because the products on the right exactly cancel those on the left. The relationship is perfect, but not linear.
C5 — Coefficient of Determination
Squaring gives a quantity with a direct meaning that itself does not have.
Coefficient of Determination
In a simple least-squares regression of on that includes an intercept, . It is the proportion of the observed variability in accounted for by the fitted line, often reported as a percentage.
What does “accounted for” mean? Picture the vertical gaps between each point and the mean of : square them and add them up. Now do the same with the gaps between each point and the fitted line. The line shrinks the total, and is the share it removes. In Figure 6, . Before you choose, decide: will the line shrink the squared gaps by more than 81% or less?
Here r = 0.81. Will the line shrink the sum of squared gaps by more than
81%, or less?
Spread around the mean ȳ
Sum of squared gaps:
Spread around the fitted line
Sum of squared gaps: ?
tighteras observedlooser
Figure 6: The same eight points twice. Left: gaps to the mean of . Right: gaps to the fitted line. The share of the squared spread that the line removes is .
The correct phrasing: “The fitted line using accounts for of the observed variability in .” Never say “ of the relationship” (a relationship is not measured in percent) or “ of ” (it is the variability in , not itself).
Explained variability describes a linear model; it is not a causal claim. A large does not show that changing will change , and the unexplained share is not automatically “random noise”: it can come from other variables, curvature, or measurement.
Mini-example:, so . “The fitted least-squares line using accounts for 64% of the observed variability in . The remaining 36% is not accounted for by this line; it may reflect other variables, non-linear patterns, or measurement variability.”
C6 — Confusing with
This is the most common numerical error. If , the fitted line accounts for of the observed variability in , not 70%.
Never use directly as a percentage of variability accounted for. If , do not say “ explains 70% of .” Square it: , so “the fitted line using accounts for 49% of the observed variability in .” Figure 6 showed why: , but the line removed only 66% of the squared spread.
Squaring shrinks every magnitude strictly between 0 and 1: gives ; gives . Across the moderate band, , the fitted line accounts for anywhere from 25% to just under 64% of the variability, not one fixed share. And when you square, square the full calculator value of , then round.
C7 — Correlation vs. Causation
A strong correlation between and in observational data does not by itself mean causes . The main alternative explanations:
Confounding: a third variable drives both and .
Reverse causation: drives , not the other way around.
Coincidence: the association is a statistical accident, especially with small samples or many comparisons.
When the researchers randomly assign the values of (a dose of fertiliser, a number of quizzes per week), the groups are alike on average in everything else, so a clear association between the assigned and the response does support a causal effect. The design, not the size of , decides.
On hot days more ice cream is sold, and more swimmers get into difficulty at the beach. If we look only at days of similar temperature, will ice-cream sales and swimmers needing help still be strongly correlated? Answer first, then colour Figure 7 by temperature.
36 summer days · overall r = 0.86
14–19 °Cwithin this band r = −0.08
21–26 °Cwithin this band r = 0.21
28–33 °Cwithin this band r = −0.19
Hold temperature roughly fixed and the link almost disappears: heat drives both.
Figure 7: Thirty-six simulated summer days. Overall, ice creams sold and swimmers needing help are strongly correlated. Colour the days by temperature band to see what happens within each band.
Even a large with a small p-value does not establish causation. Under the test’s assumptions, a small p-value says a result this extreme would be unusual if the population correlation were zero; it rules out neither confounders nor reverse causation. A causal conclusion needs a credible causal design, usually random assignment.
Back to the chocolate, piece 1 (who is a data point?): each of Messerli’s 23 points is a country, not a person. An association across countries, an ecological correlation, says nothing about whether the people who eat more chocolate are the ones who win prizes. Piece 2 (a third variable): rich countries can afford both more chocolate and more research, so national wealth is a plausible confounder. Messerli himself called the finding a hypothesis, not proof of cause.
C8 — Conditions for Pearson
Before computing or interpreting Pearson , check three conditions. A scatterplot takes seconds and shows whether they hold.
Conditions for Pearson r
Both variables are quantitative and both vary (not categorical, with and ).
The relationship is approximately linear. If the pattern is clearly curved, Pearson is not the right summary.
No single point dominates. One point far from the rest can create, hide, or reverse an association, especially in a small dataset.
“One country out of 23 can’t matter.” It can. In Messerli’s data, Sweden sat far above the trend: about 32 laureates where its chocolate eating predicted about 14. With Sweden, ; without it, . A single far point moved by 0.07. Report with and without such a point, and find out why it is there before deciding anything.
C9 — What Ignores: Units, Which Variable Is , and the Slope
Because is built from z-scores, it carries no units. Three consequences follow.
Properties of r
Units do not matter. Converting centimetres to inches, or °C to °F, leaves unchanged.
Which variable is does not matter. The correlation of with equals the correlation of with .
is not a slope. It says how tightly the points follow a line, not how much changes per unit of . That rate is the slope of the fitted line, , which does depend on units and on which variable is (L-REG-2).
“, so every extra kilogram of chocolate adds 0.79 laureates.” This reads as a slope. has no units, so it cannot be “laureates per kilogram”. Messerli did estimate a rate (about 0.4 kg of chocolate per person per year for each extra laureate per 10 million people), but that number is a slope from a fitted line, a different quantity, and even a slope from observational data says nothing about what would happen if a country changed its diet. That is back to the chocolate, piece 4.
Mini-example: heights and arm spans of 40 adults give whether height is in centimetres or inches, and whether height or arm span goes on the -axis. The slope of arm span on height is about 1.0 cm per cm in centimetres, but about 2.5 cm per inch if height is in inches: same , different slope.
C10 — Restricted Range
Restricted range
When the data cover only part of the possible range of (because only part of a group was selected, measured or kept), is usually smaller than it would be across the full range, even when the relationship itself has not changed. A narrow slice of leaves little room for the trend to show against the scatter.
A fictional college accepts only applicants who score at least 1300 on its entrance test. Among its accepted students, will between test score and first-year average be larger, smaller, or the same as among all applicants? Decide, then move the cut-off in Figure 8.
All 60 applicants: r = 0.74
Figure 8: Sixty simulated applicants, entrance test score and first-year average. Move the acceptance cut-off and watch among the accepted students only.
“Among our students the test barely predicts first-year grades, so the test is useless.” Every student in those data passed the test, so the range of scores is narrow and is shrunk. To judge a test, you need the relationship across the full range of applicants, for example from a year when everyone was accepted.
Mini-example: in Figure 8, all 60 applicants give . Keep only the 18 who scored at least 1300 and . The trend is the same; the range of is narrower.
Section 4: Worked Examples
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Four examples fade the support deliberately: a visible model, an estimate before computing, a withheld causal decision, and an independent error analysis.
Example 1 — Reading a Scatterplot Qualitatively (Fully Worked)
Problem: A researcher plots body mass index (-axis, kg/m²) against systolic blood pressure (-axis, mmHg) for 80 adults. She describes the plot: “Points generally rise from left to right, clustering fairly tightly around a straight line. Two points in the upper right sit noticeably above the main cloud.” Describe the scatterplot in the four dimensions of C1 and check whether the conditions for Pearson are met.
I notice: the description gives me all four pieces of information I need. I’ll address them in order, then run the conditions check.
Direction: “Points generally rise from left to right” → positive direction.
Form: “Clustering fairly tightly around a straight line” → linear form.
Strength: “Fairly tight” → moderate to strong. Only a computed can pin this down.
Outliers: “Two points in the upper right sit noticeably above the main cloud” → two potential outliers (high BMI, very high blood pressure). These could pull upward.
Conditions check:
Both variables are quantitative ✓
The pattern looks approximately linear ✓
Two upper-right points need investigation: compute with and without them to see how much they matter.
Conclusion: a positive, linear, moderate-to-strong relationship with two potential outliers. Pearson is only provisionally appropriate: report it with and without those points, and investigate their source.
Example 2 — Computing and (Partially Scaffolded)
Problem: Five students record hours studied and test score. Look at the plot and estimate before calculating (its sign, and roughly how strong). Then compute and and interpret .
Student
Hours studied ()
Test score ()
A
2
66
B
4
74
C
5
64
D
7
72
E
8
78
Data in the scatterplot
Hours studied
Test score
2
66
4
74
5
64
7
72
8
78
An estimate first gives you a check: if the arithmetic later produces the opposite sign, or a value near 1 for this loose cloud, you know to inspect the computation.
Pause and write. Record your answer and reasoning on paper or in your notes before continuing. Even if you are unsure, committing your thinking helps you see what you understood and what you need to revisit.
Show Solution
Estimate. The points rise from left to right, but loosely (student C sits low): a positive, moderate correlation, perhaps around 0.6.
Step 1 — Build the computation table.
2
66
132
4
4356
4
74
296
16
5476
5
64
320
25
4096
7
72
504
49
5184
8
78
624
64
6084
: 26
354
1876
158
25196
Step 2 — Compute the numerator.
Step 3 — Compute the bracket terms.
Step 4 — Compute without rounding the intermediate value.
Step 5 — Square the full-precision value and interpret.
Interpretation:: a moderate positive linear relationship, close to the estimate. The fitted line using hours studied accounts for about 41% of the observed variability in test scores; about 59% is not accounted for by this line.
C6 check: square the full calculator value of , not the rounded 0.64 (which would give 0.4096); round only the final result.
Example 3 — Causation Claim from Correlation (Application Twist)
Problem: A news article reports: “A study of 500 teens found that those who spend more time on social media have lower grades. The correlation was (). Researchers conclude that social media use causes lower academic performance.”
Identify the error in the researchers’ conclusion and propose two alternative explanations.
Pause and write. Record your answer and reasoning on paper or in your notes before continuing. Even if you are unsure, committing your thinking helps you see what you understood and what you need to revisit.
Show Solution
What the study supports: a moderate negative linear association. Under the test’s assumptions, is evidence against a population correlation of zero.
Error identified: the researchers turned an observational association into causation. A small p-value does not rule out confounding or reverse causation. A causal claim would need a credible causal design, such as random assignment where that is ethical and practical.
Alternative explanation 1 — reverse causation: students who are struggling may spend more time on social media as an escape.
Alternative explanation 2 — confounding: sleep, workload, or parental supervision might affect both social-media use and grades.
Example 4 — When Hides a Pattern (Independent Error Analysis)
Problem: A student plots the five pairs ,,,, and , computes , and writes:
“Because and , explains none of . The variables are unrelated, so there is no reason to inspect the scatterplot.”
Diagnose every important error. State what the scatterplot would show, what means here, whether Pearson is a useful summary, and what does and does not say.
Pause and write. Record your answer and reasoning on paper or in your notes before continuing. Even if you are unsure, committing your thinking helps you see what you understood and what you need to revisit.
Show Solution
The points lie exactly on the curve : perfectly structured, but not linear.
means the positive and negative linear contributions cancel. It does not mean the variables are unrelated or independent.
Pearson is not a useful one-number summary here, because the linearity condition fails. The scatterplot is essential evidence.
says a straight line accounts for none of the variability in . It does not say has no predictive value: the curve predicts every exactly.
Independent conclusion: report the U-shaped relationship directly, or use a model that represents curvature; do not use Pearson alone to describe these data.
Section 5: Guided Practice
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Six problems to build fluency. Check each choice as you go: the feedback names the step or idea that needs repair.
Problem 1 — Read Direction and Strength from a Plot (C1 + C3)
Problem 2 — Your First Computation of (C2)
The first two steps are worked for you. Finish the computation one step at a time.
Problem 3 — Computing and Interpreting (C5 + C6)
Concept Check — When (C4)
A scatterplot forms a clear U-shaped curve, but the Pearson correlation is approximately . What is the best conclusion?
Problem 4 — Why Do These Two Measures Move Together? (C7)
Sort into groups
Why do these two measures move together?
Each card describes two measures that are associated. Sort it by the explanation that fits best. Read the design: who chose the values of the first measure?
Drag an item into its group, or select it and then choose the group. On a keyboard: select an item and press the group’s number (1–3); 0 sends it back to Unsorted. Each item also has its own “move” buttons.
Unsorted Drop here
Every item is sorted.
Problem 5 — From Back to (C5 + C6)
Problem 6 — What Does Ignore? (C9)
These two analyses are almost identical. Decide what happens in each, then use the feedback to name the one difference that matters.
Structural Difference
Change the units: what moves?
Identify what key word or condition separates Case A from Case B, then select the method for each.
Scenario A
Forty adults have heights in centimetres and arm spans in centimetres; r = 0.90. An analyst converts every height to inches (arm span stays in centimetres) and recomputes r.
Scenario B
Forty adults have heights in centimetres and arm spans in centimetres; r = 0.90. An analyst converts every height to inches (arm span stays in centimetres) and recomputes the slope: centimetres of arm span per unit of height.
Section 6: Independent Practice
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Cold problems from raw data. Look at each plot before you trust any number you compute from it.
Problem 1 — Compute from Raw Data (C1 + C2 + C8)
Complete the table
Five column sums, then r
Fill in every empty cell, then check the whole table once. The cells depend on one another.
Claim 1. A researcher reports and says sleep duration explains 70% of the variability in productivity.
Claim 2. A researcher finds between shoe size and reading ability in children aged 5–12 and claims larger feet cause better reading.
Claim 3. A researcher reports that means advertising spend explains 63% of the variability in complaint rate.
Claim 4. Two studies of the same reading test differ only in who was included.
Structural Difference
Same test, different r: what explains each value?
Identify what key word or condition separates Case A from Case B, then select the method for each.
Scenario A
A researcher tests children aged 6 to 12 and finds r = 0.78 between age and reading speed.
Scenario B
The same researcher tests only children aged 9 and finds r = 0.21 between age (in months) and reading speed.
Problem 4 — Which Points Pull Down? (C2 + C8)
Answer on the Graph
Which students pull r down?
Eighteen students: hours of sleep the night before a test, and the test score. The dashed lines mark the two means. Overall, r = 0.53. Select every student whose own contribution to r is negative.
Select the point or points that answer the question. Select a point again to clear it. Keyboard: arrow keys move through the figure; Enter or Space selects.
Now write the full report for a different study. Read the design carefully: in some studies the values were randomly assigned.
Build the Conclusion
Write the report
Choose a phrase for each blank so that the sentence states the correct conclusion.
Problem 6 — A Responsible Correlation Analysis (C2 + C8 + C1)
Mixed Review — Retrieval from Earlier Lessons
These problems draw on earlier lessons. Attempting them without re-reading is the point: retrieval strengthens long-term memory more than re-reading does.
Review Problem 1 — A z-Score in Context (L-DS-5)
Review Problem 2 — The Conclusion of a Test (L-INF-5)
Build the Conclusion
Finish the test's conclusion
Choose a phrase for each blank so that the sentence states the correct conclusion.
Section 7: Mastery Check
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
No hints and no guided steps unless you ask for them. Four questions to check that the core ideas have landed. Take your time, especially with the Feynman test.
Question 1 — Feynman Test
Here is a scatterplot of daily screen time and sleep-quality score for 16 teenagers. For these data .
Data in the scatterplot
Screen time (hours per day)
Sleep-quality score
1.0
82.0
1.4
76.0
1.9
65.0
2.2
63.0
2.6
69.0
3.0
74.0
3.3
70.0
3.7
62.0
4.1
60.0
4.4
65.0
4.9
65.0
5.3
57.0
5.6
49.0
6.0
51.0
6.5
58.0
7.0
59.0
In 3–5 plain-language sentences, explain to a classmate what and each say about this plot, and why neither number shows that screen time harms sleep.
Pause and write. Record your answer and reasoning on paper or in your notes before continuing. Even if you are unsure, committing your thinking helps you see what you understood and what you need to revisit.
Compare with a model answer
The points fall from upper left to lower right in a fairly tight straight band, so describes a strong negative linear association: teenagers with more screen time tend to have lower sleep-quality scores. Squaring removes the direction: , so a fitted line using screen time accounts for about 64% of the observed variability in sleep-quality scores; the other 36% is not accounted for by that line, which does not make it random noise. Neither number shows cause, because nobody assigned the screen time: stress, school workload or a bedroom phone habit could drive both, or poor sleep could lead to more screen time at night.
Question 2 — Novel Application
Then a different study, already summarised. Complete its report.
Build the Conclusion
Write the report
Choose a phrase for each blank so that the sentence states the correct conclusion.
Question 3 — Error Analysis
Question 4 — What Changes ?
Self-Assessment
Still confusedReady for the Boss Fight
If you’re below 60%: revisit C2 (computing ), C6 ( versus ), C7 (causation) and C10 (restricted range), rework Example 2, and retry Question 2 before the Boss Fight. The Boss Fight needs all of them working together.
Section 8: Boss Fight
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Context: two desks, two claims that sound convincing. Neither tells you which tool to use: deciding that is part of the fight. Each starts with your own complete attempt on paper; open the steps only if you need to find where it went wrong. Lose, and you get a rematch with new data.
🍫 The Chocolate Desk
A columnist wants to tell readers that chocolate wins Nobel prizes. Check the numbers before it runs, and settle the question from the start of the lesson.
📝 The Entrance Test Review
A college committee wants to drop its entrance test because it “barely predicts” first-year grades. Is the evidence what it seems?
🍫 The Chocolate Desk
A second file. Another columnist sends ten more countries. Find the one that matters most.
Answer on the Graph
Which country carries r?
Select the point or points that answer the question. Select a point again to clear it. Keyboard: arrow keys move through the figure; Enter or Space selects.
No point selected yet.
Your note to the editor.
Build the Conclusion
What may the column say?
Choose a phrase for each blank so that the sentence states the correct conclusion.
Then, on paper: go back to the prediction you wrote in the Introduction. In two or three sentences, say whether a country that wants more Nobel prizes should eat more chocolate, and which pieces of this lesson decide it.
Compare with a model answer
No, at least not on this evidence. The correlation of 0.791 is real and correctly computed, but each point is a country, not a person; national wealth could drive both chocolate eating and research; one country (Sweden) moved noticeably on its own; and is not “laureates per kilogram”. A causal answer would need a design in which chocolate is assigned, which no country-level comparison provides.
📝 The Entrance Test Review
A second college. Write the memo for a different committee facing the same question.
Build the Conclusion
Your memo to the committee
Choose a phrase for each blank so that the sentence states the correct conclusion.
Then, on paper: in two sentences, explain to the committee member why “the test barely predicts how our students do” is the wrong evidence for dropping the test, and what evidence would be the right one.
Compare with a model answer
Everyone in your data already passed the test, so their scores cover only the top of the range, and a narrow range of shrinks even when the relationship is unchanged. The right evidence is the relationship across the full range of applicants, such as a year when everyone was accepted. Whether the test is fair to particular groups of applicants is a separate question that a correlation cannot answer.
Section 9: Challenge Problems
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Optional stretch. These problems go beyond the lesson objectives: what “unexplained” variability really is, and why the same can describe very different situations.
Challenge 1 — and Unexplained Variance (C5 + C6)
Two variables have . A student writes: “Since , that means 75% of the variability is random noise.”
(a) Is the student right that 75% is “unexplained”?
(b) What might the 75% “unexplained variance” actually represent?
(c) On paper, write a 2–3 sentence explanation of what “unexplained variance” in means, and what it does not mean.
Compare with a model answer
“Unexplained” means not accounted for by this one straight line using : here 75% of the observed variability in is left over after the line. That share can come from other variables, from curvature the line misses, from measurement, or from genuine individual variation. It does not mean the 75% has no cause, and it does not mean the line is useless.
Challenge 2 — Transfer Beyond a Straight Line (C1 + C4 + C8)
A fresh plot each attempt. Decide what the evidence supports without assuming that one correlation coefficient tells the whole story.
Challenge 3 — Anscombe’s Quartet (C1 + C4 + C8)
Four datasets share nearly the same summary statistics. Inspect each one before deciding whether Pearson is a useful summary.
Here are four small datasets, each with points (Anscombe, 1973):
Dataset I:
10
8
13
9
11
14
6
4
12
7
5
8.0
7.0
7.6
8.8
8.3
10.0
7.2
4.3
10.8
4.8
5.7
Dataset II:
10
8
13
9
11
14
6
4
12
7
5
9.1
8.1
8.7
8.8
9.3
8.1
6.1
3.1
9.1
7.3
4.7
Dataset III:
10
8
13
9
11
14
6
4
12
7
5
7.5
6.8
12.7
7.1
7.8
8.8
6.1
5.4
8.1
6.4
5.7
Dataset IV:
8
8
8
8
8
8
8
19
8
8
8
6.6
5.8
7.7
8.8
8.5
7.0
5.3
12.5
5.6
7.9
6.9
(a) For which dataset(s) is Pearson the most meaningful summary of the relationship?
(b) What does this demonstrate about computing without plotting the data?
You will see these four datasets plotted at the start of L-REG-3. Before then, sketch them yourself: the pictures are the whole point.
Section 10: Solutions Reference
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Complete, step-by-step solutions for the problems in Sections 5–9 are on the solutions page: worked arithmetic, computation tables, common errors, and interpretation guidance.
If you’re stuck: return to Section 3 and find the concept that matches your problem. Example 2 (Section 4) shows a complete computation table: use it as a template whenever you compute by hand.
Coefficient of Determination for simple least-squares regression with an intercept:
Strength Thresholds:
Interpretation
Strong
to
Moderate
Weak
Correct phrasing: “The fitted line using accounts for of the observed variability in .”
Conditions for Pearson : both variables quantitative with nonzero variability; approximately linear form (check the scatterplot); no single dominating point.
Key distinctions:
means no linear relationship, not no relationship of any kind
Correlation is not causation, unless the values of were randomly assigned
has no units, is the same with and swapped, and is not a slope
A restricted range of usually makes smaller
For simple regression with an intercept, is the share of observed response variability accounted for by the fitted line
Whether a population correlation differs from zero is a hypothesis test, taught in L-REG-3.