Mistake Master
The points happened; the line predicts
A scatterplot carries two stories at once: what each subject actually did, and what the trend expects. The arithmetic is one substitution. The points slip away when a plotted observation answers a prediction question, when the slope and intercept swap jobs, and when the gap between a point and the line gets replaced by one of its ingredients. Keep the two stories separate.
§1
What this topic is about
▸
A scatterplot shows real observations; its line of best fit shows the trend through them. Every question here turns on keeping the two apart: predictions come from the line, observations from the points, the line's slope and intercept each mean one specific thing, and the gap between a point and the line is its own quantity.
§2
The line predicts; the points happened
▸
To predict, substitute into the line's equation. To report what a subject actually did, read its point. A plotted point above the line beat its prediction; below, it fell short.
- "According to the line of best fit" means substitute into the equation, always.
- A single point is one observation, never a parameter of the line.
- Points above the line have actual $>$ predicted; below, actual $<$ predicted.
Worked example. The fit line for plant height is $y = 2x + 10$ (cm vs weeks). One plant is plotted at $(8, 30)$. What height does the line predict at $8$ weeks?
$2(8) + 10 = 26$ cm.
The $(8, 30)$ plant is an observation $4$ cm above the trend; it does not change what the line predicts.
§3
Slope and intercept each mean one thing
▸
In context, the slope is the predicted CHANGE in $y$ per one unit of $x$, carrying both units. The intercept is the prediction at $x = 0$. Swapping their jobs is this topic's most common error.
- Slope: "each additional [x-unit] adds about [slope] [y-units] to the prediction."
- Intercept: "at zero [x-units], the predicted [y] is [intercept]."
- Slopes add per unit; they never multiply the output.
Worked example. For $y = 2x + 10$, interpret the $2$ and the $10$.
Each added week predicts $2$ more centimeters of height.
At week $0$, the predicted height is $10$ cm. Giving the $10$ the growth-per-week job, or the $2$ the starting-height job, is the classic swap.
§4
The gap between a point and the line
▸
Actual minus predicted measures how far an observation beat or missed the trend. It needs both numbers, and it is neither of them.
- Compute the prediction first, then subtract in the order the question sets.
- "Exceeds the prediction by" means actual minus predicted.
- The gap's sign says above or below the line.
Worked example. With $y = 2x + 10$, a plant is plotted at $(12, 38)$. By how much does it exceed the prediction?
The line says $2(12) + 10 = 34$; the plant measured $38$.
$$38 - 34 = 4 \text{ cm}.$$ Reporting $38$ or $34$ hands in an ingredient instead of the answer.
§5
Slopes from two points
▸
When the line's equation is not given, two points on the LINE recover its slope: rise over run, changes only.
Predict with the line, read observations from the points, give the slope and intercept their own jobs, and treat the actual-minus-predicted gap as its own quantity.
§6
Three patterns that cost real points
▸
Three patterns recur on scatterplot questions. They are the same ones the diagnostic routes on.
A point answers a line question.
The question says "according to the line of best fit" and a nearby plotted point gets read instead, or a single observation gets treated as the trend.
Fix. The phrase "the line predicts" always means substitute into the equation. Points are what happened; the line is what is expected.
The slope and intercept trade jobs.
The intercept becomes a rate, the slope becomes a starting value, or a prediction drops one of them entirely.
Fix. Attach units: the slope carries y-units PER x-unit; the intercept is plain y-units at zero. A prediction always uses both numbers.
An ingredient answers a gap question.
Asked how far a point sits from the line, the actual value or the predicted value gets reported, or the subtraction flips sign.
Fix. The gap needs both values and a subtraction in the stated order. Compute prediction, compute gap, and report the gap.
Ten quick checks across the patterns: predicting from the line, interpreting slope and intercept, slopes from two points, and the actual-minus-predicted gap. Pick or type your answer, then check. Progress is saved.