The least-squares line is the one making the sum of squared residuals as small as possible.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The slope in context
It is the predicted change in y for a one-unit increase in x — predicted, not guaranteed.
What r² means
The proportion of the variation in y explained by the linear relationship with x.
Do not extrapolate
A line fitted over one range says nothing beyond it. That is where absurd predictions come from.
The slope has a contextual meaning
The slope predicts the change in the response for a one-unit increase in the explanatory variable. Exam answers must state it in context with units, and must say "predicted" rather than asserting an actual change.
Why least squares
The line is chosen to minimise the sum of squared residuals. Squaring penalises large misses heavily and makes the solution unique, which is why this criterion won out over the alternatives.
What r² means
r² is the proportion of variation in the response explained by the linear relationship with the explanatory variable. It is not the proportion of points on the line and not a measure of causation.
Extrapolation is unreliable
A regression line describes the range of the data. Predicting far outside it assumes the relationship continues, which nothing in the data supports. Exam questions test this deliberately.
Step 2: Try It Yourself
Tap and try it out.
The slope is 1: for every 1 across, the line goes 1 up.
Step 3: Watch an Example
One step at a time.
Watch Rosa Interpret a Fit
A regression of weight on height gives slope 2.5 and r² = 0.64.
- Step 1
The slope says each extra unit of height predicts 2.5 more units of weight.
Step 4: Your Turn
Practice makes it stick.
From r to r²
Problem 1 of 2
r = 0.6. What is r²?
Unexplained
Problem 2 of 2
r² = 0.75. What proportion of the variation is not explained?
Reading a Regression
1 of 8
r = 0.5. What is r²?
2 of 8
ŷ = 3x + 4. What is y predicted at x = 2?
3 of 8
r² = 0.9. Percent of variation explained?
4 of 8
A residual is actual minus predicted. Actual 12, predicted 10. Residual?
5 of 8
Does a strong r² prove causation? 1 for yes, 0 for no.
6 of 8
r = −0.8. Is the slope negative? 1 for yes, 0 for no.
7 of 8
Data covers x from 1 to 10. Is predicting at x = 50 sound? 1 for yes, 0 for no.
8 of 8
Match each quantity to what it tells you.
Tap a card on the left to start.
Step 5: Quick Check
Show what you know.
Question 1 of 1
r = 0.9. What is r²?
What You Learned
- The least-squares line minimises the sum of squared residuals.
- The slope is a predicted change per unit, and r² is the proportion of variation explained.
- A line says nothing outside the range of the data it was fitted to.