PSAT/NMSQT Math: Graphs & Data Interpretation (Problem Solving)
This lesson covers the data-analysis skills tested in the Problem Solving and Data Analysis domain: recognizing whether a situation grows linearly or exponentially, reading graphs made of multiple pieces or intervals, working with histograms, mean, and median, and reading scatterplots and using complementary counting. This free, complete lesson covers all of these skills. 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 quiz below.
1. Identifying Linear vs. Exponential Models
A situation is linear when a quantity changes by the same fixed amount each period, and exponential when it changes by the same fixed percentage (a constant multiplying factor) each period. Reading the wording of a problem carefully, "increases by $10 each week" versus "increases by 10% each week", tells you which model applies.
Worked Example: A person saves $10 every week. Which type of function models their total savings over time?
A fixed dollar amount added each period is linear: S = 10w
1. A person saves $10 every week. Which type of function models their total savings over time? (see worked example above)
Explanation: Adding the same fixed dollar amount each period is the signature of a linear model.
2. A bacterial population doubles every hour. Which function represents the population P after t hours, starting from an initial population of P₀?
Explanation: Doubling each period means multiplying by 2 repeatedly, giving an exponential model, P = P₀ · 2t.
3. A histogram of frequency data shows a distribution that is symmetric and evenly spread. Which pair of median and mean values is most plausible?
Explanation: A roughly symmetric, evenly spread distribution typically has a mean and median that are close to each other, since there's no strong skew pulling one measure away from the other.
2. Reading Piecewise and Multi-Interval Graphs
Some graphs describe a quantity that behaves differently across different intervals, such as a cyclist's speed changing throughout a training session. Read the graph interval by interval, identifying where the described quantity is at its highest, lowest, increasing, or decreasing within each piece.
Worked Example: A cyclist's speed graph shows speed climbing steadily from 0 to 15 minutes, holding steady from 15 to 25 minutes, then climbing again from 25 to 35 minutes to its highest point. When does the cyclist reach maximum speed?
At the end of the 25-to-35-minute interval, since the graph reaches its highest point there
1. Using the cyclist speed graph described above, during which interval is the cyclist's speed constant (not increasing or decreasing)?
Explanation: The description states speed "holds steady" (constant) specifically during the 15-to-25-minute interval.
2. A graph of a car's distance from home over a day shows the distance increasing, then flat (parked), then decreasing back toward 0. What does the flat portion represent?
Explanation: A flat (constant) section on a distance-versus-time graph means the distance isn't changing, so the car isn't moving during that interval.
3. A scatterplot's line of best fit has an equation of y = 2.5x + 10. What does the slope, 2.5, represent in context if x is hours worked and y is total pay in dollars?
Explanation: The slope of a line of best fit represents the rate of change, here the pay earned per additional hour worked.
3. Histograms, Mean, and Median
A histogram shows how frequently values fall into different ranges (bins). To find the mean from a small data set, sum every value and divide by the count. To find the median, order the values and find the middle one (or average the two middle ones for an even count).
Worked Example: Find the mean of the set {2, 4, 6, 6, 8, 11}.
Mean = (2 + 4 + 6 + 6 + 8 + 11) / 6 = 37/6 ≈ 6.17
1. Find the mean of the set {2, 4, 6, 6, 8, 11}. (see worked example above)
Explanation: Sum = 2 + 4 + 6 + 6 + 8 + 11 = 37. Dividing by the count, 6: 37/6 ≈ 6.17.
2. Find the median of the set {8, 4, 6, 11, 8, 2}.
Explanation: Ordered: 2, 4, 6, 8, 8, 11. With 6 values, average the two middle ones: (6 + 8)/2 = 7.
3. Kennedy High School has 400 students, and 90 are in the band. How many students are NOT in the band?
Explanation: This is complementary counting: total minus the given group. 400 − 90 = 310.
4. Scatterplots and Complementary Counting
The slope of a scatterplot's line of best fit is found the same way as any line's slope: rise over run between two clear points on the line. Complementary counting finds "everyone not in a group" by subtracting the group's size from the total.
Worked Example: A line of best fit passes approximately through (0, 20) and (10, 70). Estimate its slope.
m = (70 − 20)/(10 − 0) = 50/10 = 5
1. A line of best fit passes approximately through (0, 20) and (10, 70). Estimate its slope. (see worked example above)
Explanation: Slope = (70 − 20)/(10 − 0) = 50/10 = 5.
2. A club has 150 members. If 55 members play soccer, how many members do NOT play soccer?
Explanation: Total minus the given group: 150 − 55 = 95.
3. A scatterplot shows a clear upward trend, with a line of best fit passing through (2, 8) and (6, 20). What is the slope of the line of best fit?
Explanation: Slope = (20 − 8)/(6 − 2) = 12/4 = 3.
Common Mistakes to Avoid
- Confusing "increases by a fixed amount" with "increases by a fixed percentage." The first is linear; the second is exponential, and mixing them up leads to the wrong type of model entirely.
- Reading a piecewise graph as one continuous trend. Always examine each labeled interval separately, since the behavior (increasing, decreasing, or constant) can differ between them.
- Forgetting to divide by the correct total count when finding a mean. Always divide the sum by exactly how many values are in the data set.
- Forgetting to average the two middle values for an even-sized data set's median. An odd-sized set has one clear middle value, but an even-sized set requires averaging the two central values.
- Subtracting from the wrong total in a complementary counting problem. Always subtract the given group's size from the overall total described in the problem, not from an unrelated number.
Frequently Asked Questions
How do I tell linear and exponential growth apart just from a word problem's phrasing?
Look for the specific wording: a fixed dollar or unit amount added each period ("$10 more each week") signals linear growth, while a fixed percentage or multiplying factor ("increases by 10%" or "doubles") signals exponential growth.
What's the difference between reading a value off a graph and calculating a slope from it?
Reading a value means locating a specific point and reporting its coordinate. Calculating slope means picking two clear points and applying rise over run between them, giving the overall rate of change rather than a single value.
Where can I practice more problems like these?
The Graphs & Data Interpretation quiz in the PSAT/NMSQT Math Question Bank includes 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/NMSQT. You can also browse the full PSAT/NMSQT Math Question Bank.