PSAT/NMSQT Math: Graphs & Data Interpretation (Advanced Math)
This lesson covers the data-analysis skills tested in the Advanced Math domain: describing a data set with mean, median, and mode, understanding how changing the data affects each measure, comparing spread with range and standard deviation, and reading scatterplots and other graphs. 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. Mean, Median, Mode, and How Data Changes Affect Them
The mean is the sum of all values divided by how many there are. The median is the middle value when the data is ordered. The mode is the value that appears most often. Adding, removing, or changing a data point can shift some of these measures while leaving others unchanged.
Worked Example: Find the mode of the set {5, 7, 7, 9, 12, 12, 12, 15}.
12 appears three times, more than any other value, so the mode is 12
1. Find the mode of {5, 7, 7, 9, 12, 12, 12, 15}. (see worked example above)
Explanation: 12 appears three times, more than any other value in the set, making it the mode.
2. The data set {3, 5, 5, 8, 11} has a new value of 2 added, becoming {2, 3, 5, 5, 8, 11}. Which measures change: the mean, the mode, or the median?
Explanation: Adding a new value changes the total (mean) and shifts the middle position (median) since the set now has 6 values instead of 5. The mode, 5 (still the most frequent), stays the same.
3. A data set has mean 20. If every value in the set is increased by 5, what is the new mean?
Explanation: Adding the same constant to every value shifts the mean by that same constant: 20 + 5 = 25.
2. Range and Standard Deviation
The range is the difference between the highest and lowest values in a data set. Standard deviation measures how spread out the values are from the mean, on average; a larger standard deviation means the data is more spread out, while a smaller one means the data is more tightly clustered.
Worked Example: Data set A: {10, 12, 11, 13, 9}. Data set B: {1, 20, 5, 15, 4}. Both have the same mean of 11. Which has the larger standard deviation?
Set B has values much farther from the mean than Set A, so Set B has the larger standard deviation
1. Find the range of the data set {14, 8, 22, 5, 19}.
Explanation: Range = highest − lowest = 22 − 5 = 17.
2. Two data sets have the same mean. Set A: {10, 12, 11, 13, 9}. Set B: {1, 20, 5, 15, 4}. Which has the larger standard deviation? (see worked example above)
Explanation: Set B's values range much farther from the shared mean of 11 than Set A's do, giving Set B the larger standard deviation, even with an identical mean.
3. A data set is tightly clustered close to its mean. What does this suggest about its standard deviation?
Explanation: Values that stay close to the mean produce a small standard deviation, since standard deviation measures the average distance from the mean.
3. Scatterplots and Correlation
A scatterplot shows individual data points for two variables, and a line of best fit summarizes their overall trend. A line with positive slope shows positive correlation (as one variable increases, so does the other); a line with negative slope shows negative correlation (as one increases, the other decreases).
Worked Example: A scatterplot's line of best fit shows a downward trend between distance from the coast and average temperature. What does this suggest?
As distance from the coast increases, average temperature tends to decrease, a negative correlation
1. A scatterplot's line of best fit shows a downward trend between distance from the coast and average temperature. What does this suggest? (see worked example above)
Explanation: A downward-trending (negative slope) line of best fit indicates a negative correlation: as one variable increases, the other tends to decrease.
2. A scatterplot shows points scattered with no clear upward or downward trend, and a nearly horizontal line of best fit. What does this suggest about the two variables?
Explanation: A nearly flat line of best fit with scattered points suggests the two variables have little to no linear relationship with each other.
3. Why does a strong correlation between two variables not necessarily mean one causes the other?
Explanation: Correlation only describes an observed pattern between two variables; it doesn't rule out an unrelated third factor, reversed causation, or coincidence as the real explanation.
4. Reading Bar Graphs and Margin of Error
Reading a bar graph or similar chart means carefully matching each bar's height to its label and units before doing any calculation. Margin of error describes a range of plausible values around a reported statistic, the true value is estimated to fall within the reported value plus or minus the margin.
Worked Example: A survey estimates a 9% defect rate with a margin of error of 2.4%. What is the plausible range for the true defect rate?
9% − 2.4% = 6.6% to 9% + 2.4% = 11.4%
1. A survey estimates a 9% defect rate with a margin of error of 2.4%. What is the plausible range for the true rate? (see worked example above)
Explanation: Subtract and add the margin of error to the reported value: 9% − 2.4% = 6.6%, and 9% + 2.4% = 11.4%.
2. A bar graph shows the number of stale chip bags found across 4 quality inspections: 3, 5, 2, and 6. What is the average number found per inspection?
Explanation: Mean = (3 + 5 + 2 + 6)/4 = 16/4 = 4.
3. A stopping-distance graph shows a car needs about 60 feet to stop from 30 mph, and about 120 feet to stop from 45 mph. What is the difference in stopping distance between these two speeds?
Explanation: Reading both values off the graph and subtracting: 120 − 60 = 60 feet.
Common Mistakes to Avoid
- Confusing which measure changes when data is added, removed, or shifted. Always re-examine mean, median, and mode individually rather than assuming they all change together.
- Assuming two data sets with equal means must have equal spread. Standard deviation and range are independent of the mean, and must be evaluated separately.
- Assuming correlation proves causation. A strong correlation on a scatterplot never by itself proves that one variable causes changes in the other.
- Misreading a bar graph's axis scale or units, leading to an answer that's off by a consistent multiplying factor.
- Applying margin of error incorrectly, such as adding it only in one direction instead of creating a full range both above and below the reported value.
Frequently Asked Questions
Does adding a value to a data set always change the mean?
Only if the added value differs from the current mean. If you add a value exactly equal to the existing mean, the mean itself doesn't change, though the median and mode still might.
How do I compare standard deviations without calculating them exactly?
Visually or numerically compare how far the data points typically sit from the mean. A data set with values clustered closely around the mean has a smaller standard deviation than one with values spread widely apart.
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.