EST I HS Math · Unit 2 · Data Analysis & Probability
Lesson 2.6 · Scatterplots, Line of Best Fit & Nonlinear Relationships
On the paper: Fewer real questions than any other topic — but the calculator trick below is worth more than any of them
Big idea. A scatterplot's line of best fit is just y = a + bx fitted to real data — a strong correlation is NOT proof of causation, and a curved scatter needs a curved model, not a 'better' straight line.
1Correlation is not causation 🔗
🍼 Baby steps
- 1
strong positive correlation between A and B
October 2020. Both variables rise together — that alone proves nothing about WHY.
- 2
correlated ≠ caused by ≠ proportional
The only statement a scatterplot alone can justify is "they are correlated" — never "A causes B" and never "they rise proportionally" (that needs the line to pass through the origin).
Trap · the classic trap option
2Reading the line of best fit's own equation 📉
🍼 Baby steps
- 1
y = a + bx → b is the SLOPE: average change in y per +1 unit of x
December 2020. A bee population's line of best fit had slope -56.429.
- 2
negative slope → the AVERAGE DECREASE per year, not "per 525.71 years"
The trap option flips the slope into its own reciprocal (1 ÷ 56.429 ≈ 0.0177, ×10,000 ≈ 177 — nowhere close to 525.71, it's just a distractor built to look plausible).
- 3
a (the intercept) is the model's value AT x = 0 — read the axis to know what that year/point actually is
The intercept is not automatically "the starting amount" unless x = 0 really is the start of the data.
Trap · extrapolation is a real danger, not a rule
3When a straight line is the wrong model 📈
🍼 Baby steps
- 1
a scatter that curves upward faster and faster is NOT linear
Forcing a straight "line of best fit" onto curved data still draws a line — it just misses almost every point on purpose.
- 2
a nonlinear relationship needs a nonlinear model: quadratic, exponential, etc.
The shape of the scatter tells you which family of function actually fits — a straight line is only correct when the points genuinely cluster around a line.
🎛 Play with it · fit the line, then break it
Slide the y-values, or load the curved preset — watch the red line of best fit still get drawn even when it's clearly the wrong shape for the data.
line of best fit: y = 0.3 + 2.5x
Try "Curved data" — the red line still gets drawn, but it misses almost every point on purpose. A straight line is the wrong model here; the data needs a curve (quadratic/exponential), not a better straight line.
4Real EST questions — and the one calculator trick worth knowing 📄
A scatter plot shows a strong positive correlation between two variables: A graphed on the x-axis and B graphed on the y-axis. Which of the following statements is justified?
- A.The rise in A caused the rise in B.
Correlation alone never proves which one (if either) caused the other.
- B.The rise in B caused the rise in A.
Same problem, reversed direction.
- C.A & B are correlated, but causation is unknown.
- D.A & B rise proportionally.
"Proportional" would require the line to pass through the origin — not something a correlation alone tells you.
Safe way · ~15 s
- A scatterplot only ever tells you correlation exists and its direction.
- It can never, by itself, prove which variable causes the other.
⏱ 90-second way · ~8 s
- See "strong correlation" → the only safe conclusion is "correlated, causation unknown."
Answer: C
A scatterplot shows a farm's bee population every year since 2012, with a line of best fit and its equation. Which of the following is the best interpretation of the value -56.429 in that equation?
- A.The average increase in the number of bees each year.
Wrong sign — this value is negative, meaning a decrease.
- B.The average decrease in the number of bees every 525.71 years.
Inverts the slope into a made-up unit that doesn't match how slope works.
- C.The average decrease in the number of bees each year.
- D.The number of bees present in the beginning.
That's the intercept (a), not the slope — a different number entirely.
Safe way · ~20 s
- In y = a + bx, b is the slope: change in y per +1 unit of x (here, per year).
- -56.429 is negative → an average DECREASE of 56.429 bees each year.
⏱ 90-second way · ~8 s
- Negative slope in a "y per year" model always reads as "average decrease per year."
Answer: C
A tutor records study hours (x) and quiz scores (y) for 4 students: (1,3), (2,5), (3,8), (4,10). Using the calculator's regression mode instead of computing by hand, what is the slope of the line of best fit?
- A.0.5
That's the intercept (a), not the slope.
- B.1.5
Not what the regression actually returns for this data.
- C.2.4
- D.3.0
Rounds too aggressively — not the calculator's real output.
Safe way · ~90 s by hand
- By hand: mean x = 2.5, mean y = 6.5.
- Slope = Σ(x−x̄)(y−ȳ) ÷ Σ(x−x̄)² = 12 ÷ 5 = 2.4.
⏱ 90-second way · ~25 s on the calculator
- Enter the 4 points into STAT (A+BX) mode once — the calculator does the sums for you.
Answer: C
⏱ The 90-second lane
One question = 90 seconds. These are the moves that finish this topic in one line.
- 1A scatterplot can prove correlation and its direction — never causation, and never "proportional" unless the line passes through the origin.
- 2In y = a + bx, b = slope = average change in y per +1 unit of x; a = the model's value at x = 0.
- 3A negative slope means an average DECREASE per unit of x — read the units the graph actually uses.
- 4Predicting far outside the data you have (extrapolation) gets less reliable the further out you go — a line "reaching zero" is a mathematical artifact, not a guaranteed real event.
- 5A curved scatter needs a curved model — a straight "line of best fit" forced onto curved data will miss almost every point.
- 6The calculator's STAT (A+BX) regression mode computes the line of best fit's a and b directly from a data table — worth using any time you're given more than 2-3 points.
Take with you
- ▸Correlation shown on a scatterplot is never enough, by itself, to prove causation.
- ▸A line of best fit's slope and intercept both have real meaning — read them in the units the graph actually uses.
- ▸When the data curves, change the MODEL (not just re-draw a straight line) — and let the calculator's regression mode do the arithmetic once you know which model to use.