PSAT 8/9 Math: One-Variable Data
One-variable data questions on the PSAT 8/9 test how well you can summarize and compare a single set of values, whether presented as a list, a bar graph, or a box plot. This free, complete lesson covers mean, median, and mode, reading bar graphs and frequency tables, box plots, and how removing a value changes the mean or median. 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, and Mode
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 (or the average of the two middle values, if there's an even count). The mode is the most frequently occurring value.
Worked Example: A data set of 12 values has a sum of 216. If a value of 30 is removed, find the mean of the remaining 11 values.
New sum: 216 − 30 = 186.
New mean: 186 ÷ 11 ≈ 16.9
1. A data set of 12 values has a sum of 216. If a value of 30 is removed, find the mean of the remaining 11 values. (see worked example above)
Explanation: New sum = 216 − 30 = 186. New mean = 186 / 11 ≈ 16.9.
2. Nine students reported reading 4, 5, 5, 6, 8, 10, 12, 14, and 17 books this year. Find the median.
Explanation: With 9 values already in order, the median is the 5th value (middle of 9): 4, 5, 5, 6, [8], 10, 12, 14, 17. The median is 8.
3. A data set is 3, 7, 7, 9, 12, 7, 15. Find the mode.
Explanation: The value 7 appears three times, more often than any other value, so it is the mode.
2. Reading Bar Graphs and Frequency Tables
A bar graph shows counts for different categories, and a frequency table lists counts in rows and columns. To compare categories, read each bar's height or each table cell carefully, and watch for questions that require combining two or more categories.
Worked Example: A bar graph shows that 50 students were born in August and 40 were born in July. Find the difference between these two counts.
50 − 40 = 10 more students were born in August
1. A bar graph shows that 50 students were born in August and 40 were born in July. Find the difference between these two counts. (see worked example above)
Explanation: 50 − 40 = 10.
2. A survey asked how often people exercise. 120 people said "weekly," 45 said "rarely," and 30 said "never." How many more people said "weekly" than said "rarely" or "never" combined?
Explanation: "Rarely" and "never" combined: 45 + 30 = 75. Then 120 − 75 = 45 more people said "weekly."
3. A table shows that a Robotics Club has 18 members and an Art Club has 24 members. What percent of the Art Club's size does the Robotics Club represent?
Explanation: 18/24 = 0.75, which is 75%.
3. Box Plots
A box plot summarizes a data set using five key values: the minimum, the first quartile, the median, the third quartile, and the maximum. The line inside the box marks the median.
Worked Example: A box plot summarizes 15 values, and the line inside the box (the median marker) is at 5. What is the median of this data set?
The median is 5, read directly from the box plot's middle line.
1. A box plot summarizes 15 values, and the line inside the box is at 5. What is the median of this data set? (see worked example above)
Explanation: The line inside a box plot's box always marks the median, so the median is 5.
2. A box plot shows a minimum of 10, a maximum of 90, and a median at 45. Which statement must be true?
Explanation: By definition, the median splits an ordered data set so that about half the values fall at or below it and half fall at or above it. The mean, mode, and full range of values aren't directly shown by the median alone.
3. A box plot of timberland acreage across 13 counties has its median line positioned closest to 8,000. Which value is most likely the median?
Explanation: Among the choices, 8,831 is the value closest to the described median position near 8,000.
4. How Removing a Value Changes Mean or Median
Removing a value from a data set changes the mean predictably (subtract it from the sum, divide by the new count), but can change the median in less obvious ways, especially if the removed value was the median itself or an extreme value.
Worked Example: Nine students' book counts are 4, 5, 5, 6, 8, 10, 12, 14, 17. Remove the greatest value (17) and find the new median of the remaining 8 values.
Remaining values in order: 4, 5, 5, 6, 8, 10, 12, 14.
With 8 values, the median is the average of the 4th and 5th: (6 + 8)/2 = 7
1. Nine students' book counts are 4, 5, 5, 6, 8, 10, 12, 14, 17. Remove the greatest value and find the new median of the remaining 8 values. (see worked example above)
Explanation: After removing 17, the remaining 8 values are 4, 5, 5, 6, 8, 10, 12, 14. The median is the average of the two middle values (6 and 8): (6+8)/2 = 7.
2. Set A is 5, 7, 9, 11, 13 (an odd count of 5 values). Set B is 4, 6, 8, 10, 12, 14 (an even count of 6 values). How do their medians compare?
Explanation: Set A's median (middle of 5 values) is 9. Set B's median (average of the two middle values, 8 and 10) is also 9. Both medians equal 9.
3. A data set of 5 values has a mean of 20, so the sum is 100. If a new value of 40 is added (making 6 values total), find the new mean.
Explanation: New sum = 100 + 40 = 140. New mean = 140 / 6 ≈ 23.33.
Common Mistakes to Avoid
- Forgetting to reorder values before finding the median, the median only works correctly on data that's been sorted from least to greatest.
- Using the wrong rule for an even number of values, the median of an even-sized data set is the average of the two middle values, not either one alone.
- Confusing the mean, median, and mode, remember that mean involves a sum and division, median is positional, and mode is about frequency.
- Forgetting to update the count (denominator) when finding a new mean after adding or removing a value, the total number of values changes too.
- Assuming removing the maximum or minimum always changes the median the same way, the effect depends on the specific data set and must be recalculated.
Frequently Asked Questions
How do I quickly recalculate a mean after adding or removing a value?
Adjust the sum first (add or subtract the value), then adjust the count (add or subtract 1), and divide the new sum by the new count. There's no need to recompute everything from scratch.
What does a box plot actually show?
A box plot displays five numbers: the minimum, the first quartile (25th percentile), the median, the third quartile (75th percentile), and the maximum. The box itself spans from the first to third quartile, with a line marking the median.
Where can I practice more problems like these?
The One-Variable Data 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.