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.
Around 1900, the statistician Karl Pearson and the mathematician Alice Lee collected the heights of 1,078 fathers and their adult sons. The fathers averaged 67.7 inches; the sons averaged 68.7 inches.
A father is 74 inches tall (6 ft 2 in). Predict his son’s adult height.
Pause and write. Write your prediction in inches, with one sentence of reasoning, on paper or in your notes. You will check it in C2, and the same families come back in C7.
Most people predict about 74 inches, or more: tall fathers have tall sons. The data say something more interesting, and it is the reason this lesson is called regression. Francis Galton, who first noticed the pattern in family heights, called it “regression toward the mean”. This lesson uses Pearson and Lee’s 1903 family measurements for what they show about prediction. Galton and Pearson also promoted eugenics, ideas that statistics later helped discredit.
In L-REG-1 you measured how strongly two variables move together with the correlation . Here you turn into a prediction line, and ask five questions about it: which line is best, how steep it is, where it sits, how far each point is from it, and what changes when you predict the other way round.
After this lesson, you will be able to:
Explain what makes the least-squares line “best”, and why passing through the most points is not the test.
Compute and , and explain the slope as ” standard deviations of per standard deviation of ”, which is why extreme cases regress toward the mean.
Predict with , say what the slope means “on average”, and check whether the intercept means anything for the observed range.
Compute residuals , use , and build the separate line that predicts from .
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.
What you need coming in, and why it matters today:
From L-REG-1 — correlation: is the average product of z-scores, , between −1 and +1. Today the slope of the prediction line comes straight out of .
From L-DS-5 — z-scores: counts standard deviations from the mean. You will make the first prediction in standard units.
From L-DS-4 — standard deviations and : their ratio converts a prediction in standard units back into inches, points or dollars.
From L-DS-3 — means and : every least-squares line passes through , which gives you a check on your arithmetic.
Quick check — can you recall this from L-REG-1?
What does the Pearson correlation coefficient measure?
Bridge to today: in L-REG-1, the correlation between height and weight was the same as between weight and height: swapping and left unchanged. A slope is not symmetric like that. The line that predicts weight from height is not the line that predicts height from weight, so before every regression you must decide which variable is being predicted.
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.
Seven ideas, five questions about one line:
Which line? (C1) The least-squares criterion.
How steep? (C2) The slope in standard units, and regression to the mean.
Where? (C3) The intercept, and the point of means every least-squares line passes through.
What the line says (C4–C5) The slope sentence, and whether the intercept means anything.
How far off? (C6) Residuals.
Which way round? (C7) Predicting from needs its own line.
C1 — Which Line? The Least-Squares Criterion
Any straight line through a scatter plot makes a prediction (read “y-hat”) for every . The residual of a point is its vertical miss, . The least-squares line is the line whose squared residuals add up to the smallest possible total.
Try it before the definition. Figure 1 starts with a flat line. Turn it with Slope and move it with Shift until you think it fits the eight students best, and write down its SSE (the total of the squared residuals). Then try Line B, which passes exactly through four of the points. Only then press Reveal least squares.
Slope turns the line about the point of means (4.5, 71.13).
Shift moves the whole line up or down.
The eight students in Figure 1
Hours studied
Exam score
1
55
2
60
3
68
4
65
5
75
6
80
7
78
8
88
Figure 1: Eight students' hours studied and exam scores. Fit a line by eye, write down
its SSE (the sum of squared residuals), then reveal the least-squares line and compare. Line B
passes exactly through four of the points.
The Least-Squares Criterion
For data with at least two different -values(), the least-squares regression line is the one line that makes the sum of squared errors as small as possible:
where is an observed value and is the line’s prediction at the same . No other straight line has a smaller SSE for the same data.
Why squares? Squaring makes every miss count as positive, so a point 3 above the line and one 3 below do not cancel. It also weighs a big miss more than two small ones: a miss of 6 adds 36, two misses of 3 add 18. The line that wins avoids large misses.
”The best line goes through as many points as possible.” It feels right: a point on the line is a perfect prediction. But a line can be perfect for four students and badly off for the other four. In Figure 1, Line B passes exactly through four points and still misses by a total of 87 squared points; the least-squares line touches none of them and totals about 55.15. A line is judged by all its misses together, never by its hits.
C2 — How Steep? The Slope in Standard Units
In standard units the least-squares prediction is simple: a case that is standard deviations from the mean of is predicted to be only standard deviations from the mean of . Converting back to the original units gives the slope .
Write both z-scores out and solve for the prediction:
The number multiplying is the slope.
Slope of the Least-Squares Line
where is the correlation, the standard deviation of the response and that of the predictor (). The slope has the sign of . One standard deviation across in moves the prediction only standard deviations in .
Example: the 74-inch father. Fathers: in, in. Sons: in, in. .
He is standard deviations above the fathers’ mean.
His son is predicted only standard deviations above the sons’ mean:
inches.
In slope form, inches of son per inch of father. Compare 71.9 with the number you wrote in the Introduction.
Because , every prediction is fewer standard deviations from the mean than the case it is made from. Nothing pulls tall families down. A father’s height is partly what he passes on and partly chance factors his son does not share; the chance part does not repeat, so the son is expected to be tall, but less extreme.
Regression to the Mean
When , a case that is extreme on is predicted to be less extreme on , measured in standard deviations: .
Predict before you look. In Figure 2, each dot is a father and his son. Take the fathers in one one-inch strip, say 70 to 71 inches, and average their sons’ heights. Will those strip averages follow the dashed SD line (one SD up in sons for every SD up in fathers) or the solid least-squares line? Choose, then show them.
Prediction: in each one-inch strip of fathers, take the average height of their sons. Which line will those averages follow?
SD line (slope 1.03)
Least-squares line (slope 0.514)
Average son in a one-inch strip of fathers
Average son's height in each one-inch strip of fathers (simulated)
Fathers' strip (in)
Fathers in strip
Average son (in)
Least-squares prediction (in)
62 to 63
10
66.7
66.0
63 to 64
14
66.5
66.5
64 to 65
23
66.9
67.0
65 to 66
34
67.4
67.6
66 to 67
44
68.1
68.1
67 to 68
45
68.8
68.6
68 to 69
50
68.8
69.1
69 to 70
26
69.4
69.6
70 to 71
31
70.6
70.1
71 to 72
20
69.7
70.6
72 to 73
10
72.2
71.2
Figure 2:
The graph of averages. Each dot is one father–son pair; diamonds (after you predict) mark the average son in each one-inch strip of fathers.
Simulated data: 320 pairs drawn to match the published summaries of Pearson and Lee's 1903 family
measurements (1,078 fathers and sons), not the original records.
The averages climb only standard deviations of sons for each standard deviation of fathers. That is what the slope measures: the least-squares line runs through the average at each , and it is flatter than the SD line whenever .
The inverted ratio: the formula is , with the spread of the response on top. Writing gives a different, wrong slope. A memory aid: is in units of per unit of , so goes on top.
”The extreme case regressed because something caused it.” When a top scorer’s next score is lower, it is tempting to look for a cause: pressure, overconfidence, a coach’s praise. But regression to the mean needs no cause. With , an extreme first value is partly luck that does not repeat, so the next value is expected to be closer to the mean even if nothing changed. Before crediting or blaming anything, compare the new value with the regression prediction, not with the old value. The Boss Fight is about exactly this.
C3 — Where? The Intercept and the Point of Means
Once the slope is known, the intercept places the line so that it passes through the point of means: . Every least-squares line goes through , which gives a quick check on your arithmetic.
Intercept of the Least-Squares Line
Substituting into gives , so the point is always on the line. Use the unrounded slope when you compute .
Example (continued)., so . Check: .
Passing through is necessary, not sufficient: every line through that point passes the check, whatever its slope. Example 4 and Independent Practice Problem 3 put that to work.
C4 — What the Slope Says
The slope is a rate: for each one-unit increase in , the predicted changes by units, on average. “Predicted” and “on average” are part of the meaning, because individuals scatter around the line.
A sentence that says it fully: “For each additional [unit of ], the predicted [] [increases or decreases] by [units of ], on average.”
Example. For each additional inch of father’s height, the predicted son’s height increases by 0.514 inches, on average.
”If a father is one inch taller, his son will be 0.514 inches taller” drops “predicted” and “on average”. The line describes how the average son’s height shifts across fathers; any one son can be well above or below it. A slope sentence without those words claims a guarantee the data do not give, and it reads like a cause, which a regression on observational data cannot show.
C5 — What the Intercept Says
The intercept is the prediction at . It means something only when lies inside the observed range of ; otherwise it simply positions the line.
Two cases. A tutoring centre’s line is , for = hours of tutoring per week (observed 0 to 6) and = test score. Students with no tutoring were observed, so 61.5 is a supported prediction for them.
A shoe store’s line is , for = adult height in cm (observed 150 to 195) and = foot length in cm. At it predicts a foot of −3.6 cm: the intercept only positions the line.
Reading a line far outside the observed range of , in either direction, is called extrapolation. L-REG-3 shows why it is risky; here the rule is simply to check the range before you read the intercept.
”The intercept is always the prediction at , so I report it.” Mathematically it is; as a claim about the world it is supported only where the data are. Check the observed range first. If is outside it, say “the intercept positions the line; was not observed.”
C6 — How Far Off? Residuals
A residual is how far an observation sits from the line, measured vertically: , observed minus predicted. A positive residual means the point is above the line (the model under-predicted it); a negative one means it is below (the model over-predicted it).
Residual
For the least-squares line with an intercept, computed without rounding, the residuals add to zero: . The misses above the line and below it balance. The SSE of C1 is the sum of their squares, .
Example. A 74-inch father’s son is 70 inches tall. , so : the son is about 1.93 inches below the line.
Before you trust any line: a least-squares line is a fair summary only when both variables are numbers, the pattern is roughly straight, and no single point steers it. L-REG-3 checks these properly, with residual plots.
C7 — Which Way Round? Predicting from
Predicting from is a different least-squares problem: it minimises the horizontal misses, so its slope is , not . The two lines cross at the point of means and coincide only when .
Predict before you look. Figure 3 shows the same families with the line that predicts sons from fathers. Where will the line that predicts fathers from sons run?
Prediction: a separate least-squares line predicts a father’s height from his son’s. Drawn on these same axes, where will it run?
SD line (slope 1.03)
Son from father (slope 0.514)
Father from son
Figure 3:
Two lines for one cloud: sons predicted from fathers, and fathers predicted from sons. Both cross at the point of means; the SD line lies between them.
Simulated data: 320 pairs drawn to match the published summaries of Pearson and Lee's 1903 family
measurements (1,078 fathers and sons), not the original records.
Two Regression Lines
Predicting from : slope .
Predicting from : slope , and the line passes through :.
In general ; in fact .
The other half of the hook. Predicting fathers from sons: , and , so .
A 74-inch son’s father is predicted at about inches. Tall fathers have shorter sons, on average, and tall sons have shorter fathers, on average. Both are regression to the mean, seen from the two sides.
”To predict from , solve the first equation for .” Solving for gives a slope of , which predicts a 74-inch son’s father at about 78 inches: more extreme than the son, the opposite of what the data show. Build the second line with .
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.
Example 1 — The Line Figure 1 Revealed (Fully Worked)
The eight students in Figure 1 studied between 1 and 8 hours. Their summary statistics are , hours, hours, points, points. Find the least-squares line, interpret it, and find how far each student is from it.
Step 1 — Slope. I notice is positive and close to 1, so I expect a positive slope. The units will be points per hour, so goes on top.
I keep all four decimals for the next step.
Step 2 — Intercept. I use the unrounded slope and both means.
Step 3 — Equation. This is the line Figure 1 revealed.
Step 4 — Check at the point of means. I substitute and expect .
Step 5 — The slope sentence. I say “predicted” and “on average”, because students scatter around the line.
For each additional hour studied, the predicted exam score increases by about 4.37 points, on average.
Step 6 — The intercept check. I look at the observed range before reading . The students studied 1 to 8 hours, so was not observed: 51.46 positions the line, and I would not report it as the score of a student who did not study.
Step 7 — One residual. Student D studied 4 hours and scored 65. I predict first, then subtract observed minus predicted.
D is 3.94 points below the line: the model over-predicted D.
Step 8 — All eight residuals. I repeat Step 7 for every student and square each residual.
Student
(h)
A
1
55
55.83
−0.83
0.69
B
2
60
60.20
−0.20
0.04
C
3
68
64.57
3.43
11.76
D
4
65
68.94
−3.94
15.53
E
5
75
73.31
1.69
2.86
F
6
80
77.68
2.32
5.39
G
7
78
82.05
−4.05
16.38
H
8
88
86.42
1.58
2.51
Total
0.00
≈ 55.15
The residuals add to 0, as C6 promised. The exact SSE is about 55.15; the rounded squares in the table add to 55.16 only because each one was rounded. No other line through these points has a smaller SSE, and this one passes through none of them.
Example 2 — Temperature and Hot Drinks (Partially Scaffolded)
A café records the day’s outdoor temperature (, °C) and hot drinks sold (). On the observed days the temperature ranged from 8 °C to 31 °C. Summary statistics: ,,,,.
Before opening the solution, predict the sign of the slope. Then compute and , write the equation, check the point of means, and interpret the slope and, if it is supported, the intercept.
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
Step 1 — Slope. The slope is negative because .
Step 2 — Intercept.
Step 3 — Equation and check.
Step 4 — Interpret. For each additional °C, the predicted number of hot drinks sold decreases by 2.16, on average. The intercept would be the prediction at 0 °C, but the observed days ran from 8 °C to 31 °C, so 123.2 only positions the line.
Example 3 — Prediction and Residual: Equation Given
For adults who exercise between 10 and 70 minutes a day, the least-squares line for daily exercise minutes () and resting heart rate (, beats per minute) is .
(a) Predict the resting heart rate of a person who exercises 45 minutes a day.
(b) That person’s actual resting heart rate is 68 beats per minute. Compute the residual and say whether the point is above or below the line.
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
(a)
(b)
The residual is positive, so the point is above the line: the model under-predicted this person’s resting heart rate.
Example 4 — Which Is the Regression Line?
Two analysts fit a line to the same data set, in which and .
Analyst A:
Analyst B:
Check each line at , decide what that check can establish, and say what additional evidence would identify the least-squares line.
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 Verification
Analyst A:
Analyst B:
Both lines pass through , so the point-of-means check alone cannot tell them apart: you need additional information. For example, the raw data would let you compute and compare the two SSE values; supplied SSE values would allow a direct comparison; or ,, and would determine the slope through . Passing through is necessary, not sufficient.
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.
Each problem gives feedback step by step, and a fresh one is generated when you finish. In this section the numbers are tidy, so the arithmetic stays light.
Problem 1 — Build the Equation and Check It (C2 + C3)
Problem 2 — Predict an Extreme Case (C2)
Work in standard units first, as C2 did for the 74-inch father.
Problem 3 — Say What the Slope Means (C4)
Build the Conclusion
Say what the slope means
Choose a phrase for each blank so that the sentence states the correct conclusion.
Problem 4 — One Residual (C6)
Problem 5 — Where Do the Misses Go? (C1 + C6)
Complete the table
Predictions, residuals and SSE
Fill in every empty cell, then check the whole table once. The cells depend on one another.
Problem 6 — On Your Own, with Steps on Request (C2 + C3)
Try the whole prediction first; open the steps only if you need them.
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.
From here on the numbers are untidy, as real summaries are, and the observed range of is stated. Keep unrounded (at least four decimal places) and round only the answers you enter.
Problem 1 — Prediction and Residual (C2 + C3 + C6)
Problem 2 — Diagnose the Write-Up (C2 + C3 + C4 + C6)
Problem 3 — Could This Be the Least-Squares Line? (C2 + C3)
Sort into groups
Could this be the least-squares line?
A study reports r = 0.60, a standard deviation of 4 for x and 10 for y, and means x̄ = 12 and ȳ = 50. Sort each proposed line.
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 4 — Is the Intercept Meaningful? (C5)
These two cafés are deliberately almost identical. Decide for each one, then use the feedback to name the single fact that changes the answer.
Structural Difference
Same equation, two cafés
Identify what key word or condition separates Case A from Case B, then select the method for each.
Scenario A
A café in Québec City records iced-coffee sales on days from −8 °C to 24 °C. Its least-squares line is ŷ = 41.7 + 2.35x (x = temperature in °C, y = iced coffees sold). The manager writes: “At 0 °C we predict about 42 iced coffees.”
Scenario B
A café in Miami records iced-coffee sales on days from 12 °C to 33 °C. Its least-squares line is ŷ = 41.7 + 2.35x (x = temperature in °C, y = iced coffees sold). The manager writes: “At 0 °C we predict about 42 iced coffees.”
Problem 5 — Predict the Other Way Round (C7)
Type the formula
Write the line that predicts x from y
We read this as:Your formula will appear here as you type
Symbols you can use: Write a product as * or a space, a power as ^, a square root as sqrt( ).
Problem 6 — Synthesis (C2 + C3 + C4)
Retrieval — Is a Straight Line the Right Summary?
Sort into groups
Is a least-squares line a fair summary?
Each card describes a data set someone wants to summarise with a least-squares line. Sort it.
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 (1–2); 0 sends it back to Unsorted. Each item also has its own “move” buttons.
Unsorted Drop here
Every item is sorted.
Mixed Review — Retrieval from Earlier Lessons
These problems draw on earlier lessons. Attempting them without rereading is the point: retrieval strengthens long-term memory more than rereading does.
Review Problem 1 — Sample Standard Deviation (L-DS-4)
Then, on paper: explain why a sample standard deviation divides by rather than .
Compare with a model answer
The deviations are measured from the sample mean , which is computed from the same data, so they are a little smaller, on average, than deviations from the true mean would be. Dividing by instead of enlarges the estimate just enough that is an unbiased estimator of the population variance (Bessel’s correction).
Review Problem 2 — and (L-REG-1)
Then, on paper: a sports scientist reports between weekly training volume (km) and 100 m race time (seconds) for 30 sprinters. (a) Interpret .(b) A coach concludes: “More training causes slower sprint times.” Identify the flaw and name one specific confounding variable.
Show Solution
(a): the fitted linear relationship accounts for about 42% of the variation in race times. The other 58% is not accounted for by this model, and does not say what produced it.
(b) The flaw is inferring causation from correlation. One confounder is event specialty: sprinters who also race 200 m or 400 m log more weekly kilometres and are, on average, slower over 100 m than pure 100 m specialists. Specialty drives both variables, producing a positive even if extra kilometres do not slow anyone down.
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.
Question 1 — Feynman Test
Explain to a friend who missed this lesson why the line is called the least-squares line, why the line through the most points is not the best one, and what its slope says “on average”. Use plain language, mention residuals, and distinguish a prediction from a guarantee. Aim for three or four sentences on paper or in your notes.
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
Every line misses most points by some amount; each vertical miss, observed minus predicted, is a residual. The least-squares line is the one whose squared residuals add up to the smallest total, so it is judged on all its misses at once. A line through many points can still miss the others badly and lose; the winner often touches no point at all. Its slope says how much the predicted response changes for each one-unit increase in , on average: individuals scatter around the line, so it is a prediction pattern, not a guarantee or a cause.
Question 2 — Which Line Is Least Squares? (C1)
Question 3 — The Equation from Untidy Summaries (C2 + C3)
Question 4 — Does the Intercept Mean Anything? (C5)
Question 5 — A Residual from a Printed Line (C6)
Question 6 — The Other Way Round (C7)
Self-Assessment
Still confusedReady for the Boss Fight
If you’re below 60%: reread C1 and C2 with Figures 1 and 2, then redo Guided Practice Problems 2 and 5. Those two hold the ideas the rest of the lesson builds on: the smallest SSE, and the slope as standard deviations.
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: in both missions, something seems to have worked. Neither tells you which tool to use: choosing it is part of the fight. Start with your own complete attempt and open the steps only if you need to find where it went wrong. Lose, and you get a rematch with new numbers.
✈️ The Praise Paradox
A flight coach has proof that praise backfires and yelling works. Is it proof of anything?
🩺 The Clinic
A supplement company reports a drop in blood pressure. How much of it did the supplement earn?
✈️ The Praise Paradox
A second logbook entry. A new week, two new cadets, the same claim. Write the reply.
Build the Conclusion
Reply to the coach
Choose a phrase for each blank so that the sentence states the correct conclusion.
Then, on paper: go back to your prediction for the 74-inch father’s son. In two or three sentences, explain how the coach made the same mistake, and what a fair test of praise would need.
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
Predicting a 74-inch son for a 74-inch father, and blaming praise for a top cadet’s weaker second landing, are the same mistake: both expect an extreme value to repeat. With below 1, an extreme first measurement is partly luck, so the next one is expected to be closer to the average whatever anyone does. A fair test would randomly assign cadets to praise or to no feedback and compare the two groups’ second landings with each other, not with their first.
🩺 The Clinic
The clinic, the company and every number are fictional. This is a statistics exercise, not medical advice.
A second announcement. The company runs the same study again. Write the reply.
Build the Conclusion
Reply to the company
Choose a phrase for each blank so that the sentence states the correct conclusion.
Then, on paper: in two or three sentences, explain why enrolling only people with a high first reading guarantees an impressive-looking drop, and what the company should do instead.
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
A single blood-pressure reading is part a person’s usual level and part day-to-day variation. Choosing people because one reading was high selects many who were having a high day, and their next reading is expected to be lower even with no treatment: regression to the mean. The company should randomly assign people chosen the same way to the supplement or to a look-alike pill, and compare the two groups’ second readings.
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 go beyond the lesson’s outcomes. Nothing later in the course depends on them.
Challenge 1 — Two Lines, One Product
For a data set with ,,, and , find the line that predicts from and the line that predicts from . Compare the second slope with the reciprocal of the first, and show how the product of the two slopes relates to .
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
Predicting from :
Predicting from :
The reciprocal of is about , not : one line minimises vertical misses, the other horizontal ones. Their product is
so the two lines coincide only when .
Challenge 2 — Sensitivity of the Slope
A fresh comparison changes the correlation while holding the ratio fixed. Find how strongly the slope responds and explain what does, and does not, change when the association becomes stronger.
Challenge 3 — Why Is the Slope in Standard Units Equal to ?
C2 used the fact that in standard units the least-squares line is . Prove it. Consider every line through the origin in standard units, , write its sum of squared misses as a function of , and find the that makes it smallest. Use and . What is the smallest sum, and what does it say about ?
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 Proof
The square is smallest, zero, at , so the least-squares slope in standard units is . (Both variables have mean 0 in standard units, so the best line passes through the origin.) The smallest sum is : the fraction of the variation in the line leaves unexplained is , which is why is the fraction it accounts for.
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.
If you’re stuck: the solutions page works through every problem in Sections 5 to 9, including how each generated problem is solved, with the wrong turns the feedback names.