PSAT 8/9 Math: Two-Variable Data and Scatterplots
Two-variable data questions explore how one quantity relates to another, using scatterplots, lines of best fit, and linear models. This free, complete lesson covers lines of best fit and predictions, describing association, evaluating linear models, and classifying linear versus exponential patterns. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback and full explanations. Everything here is free, and you can keep practicing afterward with the linked quizzes below.
1. Lines of Best Fit and Predictions
A line of best fit is a straight line drawn through a scatterplot that best represents the overall trend of the data. To predict a value, find the input on the line and read off the corresponding output (or substitute into the line's equation, if given).
Worked Example: A scatterplot's line of best fit is given by y = 0.8x − 13.5, where x is advertising spend (in hundreds of dollars) and y is sales (in units). Predict sales when x = 50.
Substitute x = 50: y = 0.8(50) − 13.5.
y = 40 − 13.5 = 26.5 units
1. A line of best fit is y = 0.8x − 13.5. Predict y when x = 50. (see worked example above)
Explanation: y = 0.8(50) − 13.5 = 40 − 13.5 = 26.5.
2. A scatterplot shows temperature versus beach visitors. Using the line of best fit, the predicted number of visitors at 32°C is approximately 468. Rounded to the nearest 10, what is the predicted value?
Explanation: Rounding 468 to the nearest 10 means looking at the ones digit (8), which rounds up: 470.
3. A graph compares braking distance (feet) to speed (mph). At 30 mph, braking distance is about 50 feet; at 50 mph, it's about 150 feet. How many more feet does it take to stop at 50 mph than at 30 mph?
Explanation: 150 − 50 = 100 feet farther.
2. Describing Association
A scatterplot can show a positive association (both variables increase together), a negative association (one increases as the other decreases), or no association (no clear pattern). Remember that association doesn't necessarily mean one variable causes the other to change.
Worked Example: A scatterplot shows that as exercise minutes increase, calories burned also tend to increase. What kind of association does this describe?
A positive association (both variables increase together)
1. A scatterplot shows that as exercise minutes increase, calories burned also tend to increase. What kind of association does this describe? (see worked example above)
Explanation: Since both quantities increase together, this is a positive association.
2. A scatterplot shows that as the number of student absences increases, final grades tend to decrease. What is the most reasonable interpretation?
Explanation: A downward trend shows a negative association, but a scatterplot alone cannot establish that one variable directly causes the other, other factors could be involved.
3. A scatterplot of shoe size versus vocabulary test scores for adults shows points scattered with no visible upward or downward pattern. What does this suggest?
Explanation: Scattered points with no visible pattern indicate no clear association between the variables.
3. Evaluating Linear Models
A linear model in context, like y = mx + b, can be evaluated the same way as any linear function: substitute the given input and compute the output. The slope and intercept represent real-world rates and starting values.
Worked Example: A plant's height is modeled by h = 2x + 5, where x is days since planting and h is height in centimeters. Predict the height after 4 days.
Substitute x = 4: h = 2(4) + 5.
h = 8 + 5 = 13 cm
1. A plant's height is modeled by h = 2x + 5, where x is days since planting. Predict the height after 4 days. (see worked example above)
Explanation: h = 2(4) + 5 = 8 + 5 = 13 cm.
2. A worker's earnings are modeled by y = 5x + 40, where x is hours worked and y is total dollars earned. Find the earnings for 6 hours.
Explanation: y = 5(6) + 40 = 30 + 40 = \$70.
3. A tank drains according to the model V = 500 − 25t, where t is minutes and V is volume in gallons. Find the volume after 8 minutes.
Explanation: V = 500 − 25(8) = 500 − 200 = 300 gallons.
4. Classifying Linear vs. Exponential Patterns
In a table of values, a linear pattern has a constant difference between consecutive outputs. An exponential pattern has a constant ratio between consecutive outputs instead. Checking differences first is usually the fastest way to classify a pattern.
Worked Example: A function g(x) has values g(0) = 5, g(1) = 8, g(2) = 11, g(3) = 14. Classify this function.
Check the differences: 8 − 5 = 3, 11 − 8 = 3, 14 − 11 = 3. The difference is constant (3).
Since the differences are constant and positive, this is an increasing linear function.
1. A function g(x) has values g(0) = 5, g(1) = 8, g(2) = 11, g(3) = 14. Classify this function. (see worked example above)
Explanation: The output increases by a constant difference of 3 each time, which is the signature of a linear (not exponential) function, and since it's increasing, it's an increasing linear function.
2. A function f(x) has values f(0) = 3, f(1) = 6, f(2) = 12, f(3) = 24. Classify this function.
Explanation: The differences (3, 6, 12) aren't constant, but the ratios are: 6/3 = 2, 12/6 = 2, 24/12 = 2. A constant ratio signals an exponential pattern, and since the values grow, it's increasing exponential.
3. A function h(x) has values h(0) = 100, h(1) = 50, h(2) = 25, h(3) = 12.5. Classify this function.
Explanation: The ratio between consecutive terms is constant: 50/100 = 0.5, 25/50 = 0.5, 12.5/25 = 0.5. A constant ratio less than 1 signals a decreasing exponential pattern.
Common Mistakes to Avoid
- Assuming association proves causation. A scatterplot can show that two variables move together, but it cannot alone prove that one causes the other to change.
- Substituting into the wrong variable of a linear model, always double check which variable represents the input you're given.
- Checking only the differences (not the ratios) when classifying a pattern, an exponential pattern won't have constant differences, you must check for a constant ratio instead.
- Misreading the direction of a trend on a scatterplot or line graph, carefully confirm whether the pattern is increasing or decreasing before answering.
- Forgetting to round correctly when a question asks for a prediction "to the nearest 10" or similar, always apply the specific rounding instruction given.
Frequently Asked Questions
How can I tell linear and exponential patterns apart quickly?
Check the differences between consecutive output values first. If they're constant, it's linear. If the differences aren't constant, check the ratios between consecutive outputs instead, a constant ratio signals an exponential pattern.
Why doesn't a scatterplot prove that one variable causes changes in another?
Two variables can be associated because of a third, unmeasured factor, or simply by coincidence. A scatterplot only shows that two variables tend to change together, additional evidence (like a controlled experiment) is needed to establish cause and effect.
Where can I practice more problems like these?
The Two-Variable Data and Scatterplots quizzes in the PSAT 8/9 Math Question Bank include additional original problems on this topic, along with quizzes covering every other Heart of Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry and Trigonometry skill tested on the PSAT 8/9. You can also browse the full PSAT 8/9 Math Question Bank.