Scatterplots and Line of Best Fit

What a Scatterplot Shows

A scatterplot plots one point for each observation using two variables — one on the horizontal axis, one on the vertical axis. The cloud of points reveals whether the two variables tend to move together. The SAT expects you to describe that tendency, called the association, and to reason with a line of best fit drawn through the cloud.

There are three associations to recognize at a glance. A positive association means the points rise from lower-left to upper-right — as $x$ increases, $y$ tends to increase. A negative association means the points fall from upper-left to lower-right. No association means the points show no consistent direction. Association can also be described as strong (points hug an imaginary line tightly) or weak (points scatter loosely), and as linear or nonlinear (curved).

The Line of Best Fit

The line of best fit (or trend line) is the straight line that comes closest to all the points at once. It is a model: it summarizes the trend so you can make predictions. Its equation has the familiar form $y = mx + b$, and the SAT loves to ask you to interpret the two constants in the context of the data.

The slope $m$ tells you the predicted change in $y$ for each one-unit increase in $x$ — always stated with the data's units. The $y$-intercept $b$ is the predicted value of $y$ when $x = 0$; sometimes it is meaningful and sometimes it is only a mathematical artifact outside the data's range.

Predicting and Interpreting

Suppose a scatterplot relates hours studied ($x$) to a $100$-point quiz score ($y$), and the line of best fit is $y = 6x + 52$.

Interpret the slope. The slope $6$ means the model predicts each additional hour of study is associated with about a $6$-point higher score.

Interpret the intercept. The intercept $52$ predicts a student who studied $0$ hours scores about $52$.

Predict a value. For a student who studied $5$ hours:

$y = 6(5) + 52 = 30 + 52 = 82$

The model predicts about an $82$.

Residuals — Model vs. Reality

A residual is the vertical gap between an actual point and the line:

$\text{residual} = \text{actual } y - \text{predicted } y$

If an actual student who studied $5$ hours scored $88$, the residual is $88 - 82 = 6$: that student did $6$ points better than the model predicted. Points above the line have positive residuals — the model underpredicted them. Points below the line have negative residuals — the model overpredicted them. The best-fit line is precisely the line that makes these residuals as small as possible overall.

Reading Values Off the Graph

Many SAT questions never give the equation; they hand you the plotted line and ask you to read a value or count points meeting a condition — for example, "how many points lie above the line" or "for how many observations did the line overpredict." Overprediction means the actual point is below the line. Work carefully off the gridlines.

Finally, be wary of extrapolation: predictions far outside the plotted range of the data are unreliable, because there is no evidence the trend continues. Using $y = 6x + 52$ to predict the score for $40$ hours of study would output an impossible $292$ on a $100$-point quiz — a reminder that the model is only trustworthy near the data it summarizes. And a strong association never proves causation: two variables can move together because a third factor drives both.