SAT: Scatterplots & Modeling Cheat Sheet

The core ideas of SAT: Scatterplots & Modeling distilled into a single, scannable reference — perfect for review or quick lookup.

PiqCue — piqcue.com/sat-scatterplots-modeling/cheatsheet

Quick Reference

Line of Best Fit (Least-Squares Regression)

The straight line $\hat{y} = a + bx$ that minimizes the sum of squared residuals for a scatterplot. It summarizes the overall trend in the data.

Slope in Context

The slope $b$ in a regression equation $\hat{y} = a + bx$ represents the predicted change in the response variable $y$ for each 1-unit increase in the explanatory variable $x$. The SAT always asks you to interpret slope using the units of both variables.

Y-Intercept in Context

The constant $a$ in $\hat{y} = a + bx$ is the predicted value of $y$ when $x = 0$. On the SAT, you must state its meaning in the scenario and note whether $x = 0$ makes practical sense.

Residual

The difference between an observed value and its predicted value: $e = y - \hat{y}$. A positive residual means the actual value is above the line; a negative residual means the actual value is below.

Residual Plot

A graph of residuals vs. the explanatory variable $x$. If residuals scatter randomly around 0, a linear model is appropriate. A curved pattern suggests a nonlinear model would fit better.

Correlation Coefficient ($r$)

A number from $-1$ to $1$ that measures the strength and direction of a linear relationship. Values near $\pm 1$ indicate strong linear association; values near 0 indicate weak or no linear association.

Coefficient of Determination ($R^2$)

$R^2$ is the proportion (0 to 1) of the variance in $y$ that is explained by the linear model. It equals the square of $r$.

Linear vs. Exponential Growth

A linear model has constant additive change: $y = a + bx$. An exponential model has constant multiplicative (percent) change: $y = a \cdot b^x$ with $b > 0, b \neq 1$. For large $x$ with $b > 1$, exponential growth always overtakes linear growth.

Interpolation vs. Extrapolation

Interpolation predicts $y$ for an $x$-value within the observed data range -- generally reliable. Extrapolation predicts $y$ for an $x$ outside the data range -- less reliable because the trend may not continue.

Correlation vs. Causation

A strong correlation between two variables does not prove that one causes the other. There may be lurking (confounding) variables or the relationship may be coincidental. The SAT tests this distinction with answer choices that use causal language.

Key Terms at a Glance

Scatterplot:A graph that displays ordered pairs $(x, y)$ as dots, showing the relationship between two quantitative variables.

Line of Best Fit:The least-squares regression line $\hat{y} = a + bx$ that minimizes the sum of the squared residuals for a set of data points.

Residual:The difference $e = y - \hat{y}$ between an observed value and the value predicted by a model.

Residual Plot:A graph of residuals vs. $x$ (or vs. predicted values). Used to assess whether a linear model is appropriate.

Correlation Coefficient ($r$):A measure from $-1$ to $1$ of the strength and direction of a linear association between two variables. $r > 0$ = positive, $r < 0$ = negative, $|r|$ near 1 = strong.

Coefficient of Determination ($R^2$):The proportion (0 to 1) of the variance in $y$ explained by the model. Equals $r^2$ for simple linear regression.

Extrapolation:Using a model to predict $y$ for an $x$-value outside the range of the observed data. Less reliable than interpolation.

Interpolation:Using a model to predict $y$ for an $x$-value within the range of the observed data. Generally more reliable than extrapolation.

Exponential Growth/Decay:A model of the form $y = a \cdot b^t$ where $b > 1$ gives growth and $0 < b < 1$ gives decay. Characterized by constant percent change per unit time.

Confounding Variable:A variable not included in the model that affects both $x$ and $y$, potentially creating a misleading association between them.

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