Two-Variable Data & Scatterplots | Free SAT Math Course

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<title>Two-Variable Data and Scatterplots - Free SAT Math Lesson | The School of Mathematics</title>

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<h1>Two-Variable Data and Scatterplots: Free SAT Math Lesson</h1>

"Two-variable data: models and scatterplots" is an officially named skill in College Board's Problem-Solving and Data Analysis domain, and it tests how well you can read a relationship between two quantities and use it to make predictions. This free, complete lesson covers reading scatter plots to identify positive, negative, or no association, the line of best fit and what its slope and y-intercept mean in context, nonlinear quadratic and exponential models, and the difference between interpolation and extrapolation. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback. Everything here is free, and you can keep practicing afterward with the full SAT Math Question Bank linked below.

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<h2>What's covered in this lesson</h2>

<ol>

<li><a href="#scatter">Reading Scatter Plots and Associations</a></li>

<li><a href="#nonlinear">Nonlinear Models</a></li>

<li><a href="#predictions">Interpolation, Extrapolation, and Residuals</a></li>

<li><a href="#mistakes">Common Mistakes to Avoid</a></li>

<li><a href="#faq">Frequently Asked Questions</a></li>

</ol>

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<h2 id="scatter">1. Reading Scatter Plots and Associations</h2>

<p>A <strong>scatter plot</strong> displays the relationship between two numerical variables as a set of plotted points. When the points generally trend upward from left to right, the variables have a <strong>positive association</strong>. When the points trend downward, they have a <strong>negative association</strong>. When there's no clear upward or downward pattern, the variables have <strong>no association</strong>.</p>

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<p class="figure-caption"><strong>Positive association</strong></p>

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<p class="figure-caption"><strong>Negative association</strong></p>

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<p class="figure-caption"><strong>No association</strong></p>

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<p><strong>Worked Example:</strong> A scatter plot plots weight (ounces) on the x-axis and price (dollars) on the y-axis for five products: A(4, 6), B(8, 8), C(10, 15), D(6, 12), E(12, 18). Which product has the lowest unit price?</p>

<p>Unit price = <span class="step-math">price ÷ weight</span>. A: 6/4 = 1.5. B: 8/8 = 1. C: 15/10 = 1.5. D: 12/6 = 2. E: 18/12 = 1.5.</p>

<p>Product B has the lowest unit price, at $1 per ounce.</p>

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<p class="prompt">1. In a scatter plot, as the x-values increase, the y-values tend to decrease. What type of association does this describe?</p>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-1',false)">A) Positive association</button>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pa-1',true)">B) Negative association</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-1',false)">C) No association</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-1',false)">D) Nonlinear association</button>

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<p><span class="label">Explanation:</span> As one variable increases while the other decreases, that's a negative association.</p>

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<p class="prompt">2. A scatter plot shows points scattered with no clear upward or downward trend. What type of association does this describe?</p>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-2',false)">A) Positive association</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-2',false)">B) Negative association</button>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pa-2',true)">C) No association</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-2',false)">D) Strong association</button>

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<p><span class="label">Explanation:</span> Without a clear upward or downward trend, the variables show no association.</p>

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<p class="prompt">3. If the slope of a line of best fit is positive, what type of association does the data show?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pa-3',true)">A) Positive association</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-3',false)">B) Negative association</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-3',false)">C) No association</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-3',false)">D) Cannot be determined</button>

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<p><span class="label">Explanation:</span> A positive slope means y increases as x increases, which is exactly what a positive association describes.</p>

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<p class="prompt">4. A scatter plot shows a strong upward trend with points tightly clustered around a line. A second scatter plot shows an upward trend, but with points loosely scattered around the line. Which shows a stronger positive association?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pa-4',true)">A) The tightly clustered scatter plot</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-4',false)">B) The loosely scattered scatter plot</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-4',false)">C) They show equally strong associations</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-4',false)">D) Cannot be determined without calculating a value</button>

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<p><span class="label">Explanation:</span> The more tightly the points cluster around the trend line, the stronger the association, regardless of whether it's positive or negative.</p>

</div>

<p class="prompt">5. A scatter plot plots weight (ounces) on the x-axis and price (dollars) on the y-axis for five products: A(4, 6), B(8, 8), C(10, 15), D(6, 12), E(12, 18). Which product has the lowest unit price?</p>

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<p><span class="label">Explanation:</span> This is the worked example above: Product B has the lowest price-to-weight ratio, $1 per ounce.</p>

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<p>The <strong>line of best fit</strong> (or regression line) is a line drawn to approximate the overall trend of a scatter plot as closely as possible. Once you have its equation, you can interpret the slope and y-intercept in the context of the situation, and use the equation to make predictions.</p>

<p><strong>Worked Example:</strong> A line of best fit relating study hours (x) to test score (y) is y = 3x + 62. Interpret the slope and y-intercept, and predict the score for 8 hours of studying.</p>

<p>The slope, 3, means each additional hour of studying is associated with an increase of about 3 points on the test. The y-intercept, 62, is the predicted score for a student who studies 0 hours.</p>

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<p class="prompt">1. A line of best fit is y = 2.5x + 40, where x is advertising spend (in thousands of dollars) and y is sales (in thousands of dollars). What does the slope represent?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pb-1',true)">A) For each additional $1,000 in advertising, sales increase by about $2,500</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pb-1',false)">B) For each additional $1,000 in sales, advertising increases by $2,500</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pb-1',false)">C) Total sales when advertising is $0</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pb-1',false)">D) The maximum possible sales</button>

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<p><span class="label">Explanation:</span> The slope describes how much y (sales, in thousands) changes for each one-unit increase in x (advertising, in thousands): about $2,500 more in sales for every additional $1,000 spent.</p>

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<p class="prompt">2. Using the line y = 2.5x + 40, what is the predicted sales (in thousands of dollars) when advertising spend is $10,000 (x = 10)?</p>

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<p><span class="label">Explanation:</span> y = 2.5(10) + 40 = 25 + 40 = 65, or $65,000.</p>

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<p class="prompt">3. A line of best fit is y = −4x + 150, relating hours of TV watched per week (x) to weekly exercise minutes (y). What does the y-intercept represent?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pb-3',true)">A) Predicted exercise minutes when TV watching is 0 hours</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pb-3',false)">B) The slope of the line</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pb-3',false)">C) Predicted TV hours when exercise is 0 minutes</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pb-3',false)">D) The maximum exercise minutes possible</button>

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<p><span class="label">Explanation:</span> The y-intercept is the predicted value of y when x = 0, so it's the predicted exercise minutes for someone who watches 0 hours of TV.</p>

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<p class="prompt">4. Using y = −4x + 150, what is the predicted weekly exercise minutes for someone who watches 12 hours of TV?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pb-4',true)">A) 102 minutes</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pb-4',false)">B) 198 minutes</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pb-4',false)">C) 48 minutes</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pb-4',false)">D) 138 minutes</button>

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<p><span class="label">Explanation:</span> y = −4(12) + 150 = −48 + 150 = 102 minutes.</p>

</div>

<p class="prompt">5. A scatter plot's line of best fit has a slope very close to 0. What does this suggest about the relationship between the two variables?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pb-5',true)">A) Little to no linear relationship</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pb-5',false)">B) A strong positive relationship</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pb-5',false)">C) A strong negative relationship</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pb-5',false)">D) A perfect relationship</button>

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<p><span class="label">Explanation:</span> A slope near 0 means changes in x are associated with very little change in y, suggesting little to no linear relationship between the variables.</p>

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<h2 id="nonlinear">3. Nonlinear Models</h2>

<p>Not every scatter plot follows a straight-line trend. A <strong>quadratic model</strong> fits data that falls then rises (or rises then falls), forming a U-shape or an upside-down U. An <strong>exponential model</strong> fits data where the rate of change itself grows (or shrinks) as x increases, producing a curve that bends increasingly steeply.</p>

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<p class="figure-caption"><strong>Quadratic model</strong> – falls then rises</p>

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<p class="figure-caption"><strong>Exponential model</strong> – slow, then rapid, growth</p>

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<p class="prompt">1. A scatter plot shows y-values that decrease, reach a minimum, then increase as x increases. What type of model best fits this data?</p>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-1',false)">A) Linear</button>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pc-1',true)">B) Quadratic</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-1',false)">C) Exponential</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-1',false)">D) No model fits</button>

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<p><span class="label">Explanation:</span> A pattern that falls to a minimum and then rises matches the U-shape of a quadratic model.</p>

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<p class="prompt">2. A scatter plot shows y-values that increase slowly at first, then increase much more rapidly as x increases. What type of model best fits this data?</p>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-2',false)">A) Linear</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-2',false)">B) Quadratic</button>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pc-2',true)">C) Exponential</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-2',false)">D) No model fits</button>

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<p><span class="label">Explanation:</span> A slow-then-rapid increase, where the rate of growth itself is increasing, is the signature of exponential growth.</p>

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<p class="prompt">3. A population of bacteria is modeled by P(t) = 500(1.2)<sup>t</sup>. Is this a linear, quadratic, or exponential model?</p>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-3',false)">A) Linear</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-3',false)">B) Quadratic</button>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pc-3',true)">C) Exponential</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-3',false)">D) None of these</button>

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<p><span class="label">Explanation:</span> A function of the form a · b<sup>t</sup>, with the variable in the exponent, is exponential.</p>

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<p class="prompt">4. The height of a ball thrown in the air is modeled by h(t) = −16t² + 40t + 5. Is this a linear, quadratic, or exponential model?</p>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-4',false)">A) Linear</button>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pc-4',true)">B) Quadratic</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-4',false)">C) Exponential</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-4',false)">D) None of these</button>

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<p><span class="label">Explanation:</span> A function with a squared variable term, at² + bt + c, is quadratic.</p>

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<h2 id="predictions">4. Interpolation, Extrapolation, and Residuals</h2>

<p>Using a model to predict a value within the range of the original data is called <strong>interpolation</strong>, and it's generally reliable. Using a model to predict a value outside that range is called <strong>extrapolation</strong>, and it's less reliable, since there's no guarantee the same relationship continues to hold. A <strong>residual</strong> is the difference between an actual observed value and the value a model predicts: residual = actual − predicted.</p>

<p class="prompt">1. A researcher collects data for x-values between 0 and 20 and creates a line of best fit. Using this line to predict a y-value at x = 50 is an example of:</p>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pd-1',false)">A) Interpolation</button>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pd-1',true)">B) Extrapolation</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pd-1',false)">C) Correlation</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pd-1',false)">D) Causation</button>

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<p><span class="label">Explanation:</span> Since x = 50 falls outside the original data range (0 to 20), this prediction is extrapolation.</p>

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<p class="prompt">2. Using a line of best fit to predict a y-value at an x-value that falls within the range of the original data is called:</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pd-2',true)">A) Interpolation</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pd-2',false)">B) Extrapolation</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pd-2',false)">C) Regression</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pd-2',false)">D) Residual</button>

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<p><span class="label">Explanation:</span> Predicting within the range of the observed data is interpolation.</p>

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<p class="prompt">3. Why is extrapolation generally considered less reliable than interpolation?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pd-3',true)">A) The relationship between the variables may not continue to hold outside the observed data range</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pd-3',false)">B) Extrapolation always requires a calculator</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pd-3',false)">C) Interpolation is only used for exponential models</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pd-3',false)">D) There is no real difference in reliability</button>

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<p><span class="label">Explanation:</span> A model is built from observed data, and there's no guarantee the same trend continues beyond the range where it was actually measured.</p>

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<p class="prompt">4. A line of best fit predicts a company's revenue based on years of operation, y = 50x + 200 (in thousands of dollars). If the actual revenue in year 5 was $410,000, but the line predicts $450,000, what is the residual (actual minus predicted)?</p>

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<p><span class="label">Explanation:</span> Residual = actual − predicted = $410,000 − $450,000 = −$40,000. The negative sign shows the actual value was below the model's prediction.</p>

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<h2 id="mistakes">Common Mistakes to Avoid</h2>

<li><strong>Confusing a strong association with a large slope.</strong> The strength of an association is about how tightly points cluster around the trend line, not how steep that line is. A shallow line with tightly clustered points shows a stronger association than a steep line with widely scattered points.</li>

<li><strong>Misreading which variable is on which axis.</strong> The slope's units and interpretation depend on which variable is x and which is y. Always check the axis labels before interpreting a slope or intercept.</li>

<li><strong>Treating extrapolated predictions as equally reliable as interpolated ones.</strong> A model that fits the observed data well can still fail badly outside that range, real-world relationships often change, level off, or reverse beyond where they were measured.</li>

<li><strong>Mixing up the sign convention for a residual.</strong> Residual = actual − predicted, not predicted − actual. A positive residual means the actual value was higher than predicted; a negative residual means it was lower.</li>

<li><strong>Assuming a correlation implies causation.</strong> A strong association between two variables shows they tend to move together, but it doesn't prove that one causes the other.</li>

</ul>

<h2 id="faq">Frequently Asked Questions</h2>

<h3>How do I know whether to use a linear, quadratic, or exponential model for a scatter plot?</h3>

<p>Look at the overall shape. A roughly straight-line trend calls for a linear model. A U-shape or upside-down U calls for a quadratic model. A curve that keeps bending more steeply in one direction, without turning back, calls for an exponential model.</p>

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<h3>Does a positive association mean one variable causes the other to increase?</h3>

<p>Not necessarily. Association describes a pattern in the data, while causation is a claim about why that pattern exists. Two variables can be strongly associated because one influences the other, because both are influenced by a third factor, or purely by coincidence.</p>

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<h3>What's the practical difference between interpolation and extrapolation on a test question?</h3>

<p>If the x-value you're asked to predict falls between the smallest and largest x-values in the original data, that's interpolation, generally considered safe. If it falls outside that range, that's extrapolation, and some questions specifically test whether you recognize that the prediction is less trustworthy.</p>

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<h3>Where can I practice more problems like these?</h3>

<p>The <a href="https://theschoolofmathematics.com/quiz/sat-two-variable-data-models-and-scatterplots-quiz-1">Two-Variable Data quizzes</a> in the SAT Math Question Bank include additional original problems on this topic, along with quizzes covering every other Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry and Trigonometry skill on the Digital SAT.</p>

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<a class="cta-btn cta-primary" href="https://theschoolofmathematics.com/quiz/sat-two-variable-data-models-and-scatterplots-quiz-1">Practice Two-Variable Data Free</a>

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