SAT-54: Scatterplots — Lines of Best Fit and Trends

SAT-54: Scatterplots — Lines of Best Fit and Trends

Description: A scatterplot shows many (x, y) points to reveal a relationship between two quantities. A line of best fit is a straight line drawn through the cloud of points to summarize the trend. The SAT asks you to read the line, find its slope, and make predictions.

Correlation — what the cloud is telling you

• Positive correlation: as x goes up, y goes up (points rise left-to-right).
• Negative correlation: as x goes up, y goes down (points fall).
• No correlation: no clear pattern.

(Oʻzbekcha: musbat bogʻlanish — x oshganda y ham oshadi; manfiy — x oshganda y kamayadi.)

The line of best fit

It is the straight line that comes closest to all the points (roughly equal points above and below). You read it just like any line: it has a slope and a y-intercept, and its equation is y = (slope)·x + (intercept). (Oʻzbekcha: eng mos chiziq nuqtalarga eng yaqin oʻtadigan toʻgʻri chiziq.)

What slope and intercept mean here

• The slope = how much y changes for each +1 in x (the rate, with units).
• The y-intercept = the predicted y when x = 0 (the starting value).

Worked Example 1 — describe the relationship

A scatterplot of "hours studied" vs "test score" rises from lower-left to upper-right. Describe it.

• As hours increase, score increases → positive correlation. More study tends to mean a higher score.

Worked Example 2 — interpret the slope

A best-fit line is score = 5·(hours) + 60. What does the 5 mean, and the 60?

• Slope 5: each extra hour of study predicts about 5 more points.
• Intercept 60: a student who studies 0 hours is predicted to score 60.

(Oʻzbekcha: nishab (5) — har bir qoʻshimcha soat uchun ball oʻzgarishi; kesma (60) — boshlangʻich qiymat.)

Worked Example 3 — predict a value

Using score = 5·(hours) + 60, predict the score for 4 hours.

• score = 5(4) + 60 = 20 + 60 = 80.

Interpolation (predicting inside the data range) is reliable; extrapolation (far outside the range) is risky — the trend may not continue. (Oʻzbekcha: maʼlumot oraligʻidan tashqarida bashorat qilish xavfli.)

Correlation is not causation

This is one of the SAT's favorite ideas. A strong correlation means two quantities move together — it does not prove that one causes the other. Ice-cream sales and drowning numbers rise together, but ice cream doesn't cause drowning; hot weather drives both. So if a question asks what you can conclude from a scatterplot, "there is an association/relationship" is safe, while "X causes Y" is usually a trap unless the data came from a controlled experiment (see SAT-62). (Oʻzbekcha: bogʻlanish (korrelyatsiya) sababiy bogʻlanishni isbotlamaydi — birga oʻzgarishi sabab degani emas.)

Reading residuals (how good is the fit?)

The vertical gap between a real data point and the best-fit line is called the residual. Small residuals everywhere mean the line fits well; large, patterned residuals mean a straight line may be the wrong model. You don't usually compute them, but knowing the word helps you read SAT answer choices.

Practice 1

A best-fit line is cost = 0.5·(miles) + 3 for a taxi. What does the 0.5 mean?