ISEE Upper Level Mathematics Achievement Quiz: Bar Graphs And Box Plots

What this quiz covers

This quiz focuses on Bar Graphs And Box Plots, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Upper Level Mathematics Achievement.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. The median will increase and the interquartile range will remain unchanged
2. The median will remain the same and the third quartile will increase significantly
3. Both the first quartile and third quartile will shift, but the median may stay approximately the same (correct answer)
4. The minimum and maximum values will change, but all quartile values will remain identical

Explanation: When you encounter box plot questions involving added data points, focus on how new values affect the positions of quartiles rather than just looking at the raw numbers being added. Let's analyze how adding 15, 27, and 36 impacts each part of the box plot. The original dataset has Q1 = 18, median = 24, and Q3 = 30. When you add three points, you're increasing the dataset size, which shifts where the quartile positions fall mathematically. The new points (15, 27, 36) are distributed across different sections: 15 falls between the minimum and Q1, 27 falls between the median and Q3, and 36 falls between Q3 and the maximum. This distribution means both Q1 and Q3 will likely shift as the quartile positions recalculate with more data points. However, since 27 is relatively close to the original median of 24, the median may remain approximately the same, especially if the dataset is reasonably large. Answer C correctly captures this nuanced effect. Answer A incorrectly assumes the median will definitely increase and the IQR won't change - but adding points at different positions typically affects the IQR. Answer B wrongly focuses only on Q3 changing significantly while claiming the median stays exactly the same. Answer D makes the impossible claim that quartiles remain identical when adding data points - quartile positions must recalculate with additional data. Remember: when data points are added to a dataset, quartile positions shift mathematically, and you need to consider where the new points fall relative to existing quartiles to predict the direction of change.

Question 2

1. Between the third quartile and the maximum value, closer to the maximum (correct answer)
2. Exactly at the third quartile position on the box plot
3. Between the median and the third quartile within the box
4. At the maximum whisker value since 90th percentile equals the maximum

Explanation: Box-and-whisker plots organize data into quartiles, with each section representing 25% of the data. The key insight is understanding what the 90th percentile means: a student scoring there performed better than 90% of all test-takers, placing them in the top 10%. Let's map out this plot: the box spans from the first quartile (70, representing the 25th percentile) to the third quartile (85, representing the 75th percentile), with the median at 78 (50th percentile). The whiskers extend to the minimum (45) and maximum (95) values. Since the 90th percentile falls between the 75th percentile (third quartile at 85) and the 100th percentile (maximum at 95), a score at the 90th percentile would be positioned in the upper whisker region. More specifically, it would be closer to the maximum value than to the third quartile, since 90 is much closer to 100 than to 75. Answer choice A correctly identifies this positioning. Answer choice B is wrong because the third quartile represents exactly the 75th percentile, not the 90th. Answer choice C incorrectly places the score within the box, but the 90th percentile must fall outside the box since the box only contains data from the 25th to 75th percentiles. Answer choice D assumes the 90th percentile equals the maximum, but the maximum represents the 100th percentile, not the 90th. Remember: each section of a box plot represents 25% of the data, so use this to quickly estimate where any percentile should fall.

Question 3

1. The new score would extend the left whisker to 45 without changing the box structure
2. The new score would be plotted as an outlier point, and the whisker would extend to the lowest non-outlier value (correct answer)
3. The new score would shift the first quartile downward but remain within the normal whisker range
4. The new score would be ignored in the box plot since it represents an invalid data measurement

Explanation: When you encounter box plot questions involving outliers, you need to understand how the 1.5×IQR rule works and how outliers affect the display structure. First, calculate the IQR: $Q3 - Q1 = 88 - 74 = 14$. The outlier boundaries are $Q1 - 1.5 \times IQR = 74 - 21 = 53$ (lower bound) and $Q3 + 1.5 \times IQR = 88 + 21 = 109$ (upper bound). Since the new score of 45 is below 53, it's an outlier. When a data point is an outlier, box plots follow a specific protocol: the outlier gets plotted as a separate point (usually a dot or asterisk), and the whisker extends only to the most extreme non-outlier value. Since 45 is an outlier, it would be shown as a point, and the left whisker would still extend to 62 (the original minimum, which isn't an outlier). Choice A is wrong because outliers don't extend whiskers—they're plotted separately. Choice C incorrectly suggests the quartiles would change; adding one data point to a dataset rarely shifts quartiles significantly, and outliers don't affect the box structure anyway. Choice D is incorrect because outliers are valid measurements that must be displayed, not ignored. The correct answer is B: the score becomes a plotted outlier point while the whisker extends to the lowest non-outlier value. Remember this pattern: outliers always appear as separate points on box plots, never as whisker extensions. Calculate the 1.5×IQR boundaries first, then determine what happens to the display structure.

Question 4

1. City A likely has more extreme outliers due to its lower median temperature value
2. City B likely has more extreme outliers since its larger IQR indicates greater overall variation
3. City A likely has more extreme outliers since its smaller IQR with the same range suggests more data in the tails (correct answer)
4. Both cities have identical outlier patterns since they share the same total temperature range

Explanation: When analyzing box-and-whisker plots, understanding the relationship between the interquartile range (IQR) and the overall range helps you identify where data concentrations occur. The IQR captures the middle 50% of data, while the range shows the spread from minimum to maximum values. City A has a smaller IQR (8°F) compared to City B (12°F), but both cities share the same total range. This means City A's middle 50% of temperatures are more tightly clustered around the median, leaving more "room" in the tails of the distribution for extreme values. Since the same total range must be distributed across the data, City A likely has more extreme outliers extending toward the minimum and maximum temperatures. Choice A incorrectly focuses on the median value itself. A lower median doesn't inherently create more outliers—it's the distribution pattern that matters. Choice B makes the common error of assuming larger IQR means more outliers, but larger IQR actually indicates the middle data is more spread out, potentially leaving less extreme variation in the tails. Choice D ignores the crucial relationship between IQR and range—having the same range doesn't guarantee identical outlier patterns when the central tendencies differ. Study tip: Remember that outliers appear in the tails of distributions. When comparing datasets with equal ranges, the one with the smaller IQR (tighter middle clustering) typically has more extreme values pushed toward the minimum and maximum, creating more potential outliers.

Question 5

1. 16-28, 28-35, 35-42, 42-68 years for the four age categories respectively (correct answer)
2. 16-25, 25-35, 35-45, 45-68 years for the four age categories respectively
3. 16-30, 30-40, 40-50, 50-68 years for the four age categories respectively
4. 16-27, 27-36, 36-43, 43-68 years for the four age categories respectively

Explanation: When you encounter box plots with questions about dividing data into equal groups, remember that the five-number summary (minimum, Q1, median, Q3, maximum) already divides your data into four quartiles, each containing exactly 25% of the participants. The beauty of box plots is that they're specifically designed to show these quartile divisions. Q1 marks where the bottom 25% ends, the median shows where the bottom 50% ends, and Q3 indicates where the bottom 75% ends. So if you want four categories with 25% of participants each, you simply use these quartile boundaries as your category limits. Choice A correctly uses the quartile boundaries: 16-28 (bottom 25%), 28-35 (second 25%), 35-42 (third 25%), and 42-68 (top 25%). These ranges correspond exactly to the four quartiles shown in the box plot. Choice B (16-25, 25-35, 35-45, 45-68) ignores the actual data distribution and creates arbitrary, rounded boundaries that wouldn't contain equal numbers of participants. Choice C (16-30, 30-40, 40-50, 50-68) makes the same mistake with different rounded numbers. Choice D (16-27, 27-36, 36-43, 43-68) appears to approximate the quartile values but uses incorrect boundaries that would create unequal group sizes. Study tip: When a question asks you to divide data into equal percentages and you're given quartile information, always use the exact quartile boundaries (Q1, median, Q3) as your category divisions. The quartiles are mathematically designed to create these equal groups.

Question 6

1. League A likely has more players within one standard deviation due to its smaller interquartile range (correct answer)
2. League B likely has more players within one standard deviation due to its higher median height
3. Both leagues have identical distributions since their ranges are similar despite different medians
4. Neither league can be determined without knowing the exact positions of all outlier values

Explanation: When you encounter questions about data distribution and variability, focus on how different measures of spread relate to each other. The interquartile range (IQR) and standard deviation both measure variability, and they're positively correlated—smaller IQR typically indicates smaller standard deviation. League A has a smaller IQR (4 inches vs. 6 inches), which suggests its data is more tightly clustered around the center. When data is less spread out, more values fall within one standard deviation of the mean. Since both leagues have similar overall ranges, the key difference is how the data is distributed within that range. League A's smaller IQR indicates less variability in the middle 50% of players, suggesting the entire distribution is more concentrated. Choice A correctly identifies this relationship between IQR and concentration of data. Choice B incorrectly assumes that a higher median affects the proportion of players within one standard deviation—but the median's position doesn't determine how spread out the data is. Choice C makes the error of assuming similar ranges mean identical distributions, ignoring the crucial difference in IQRs that reveals different clustering patterns. Choice D unnecessarily complicates the problem by focusing on outliers, when the IQR comparison already provides sufficient information about the distribution's concentration. Remember this connection: smaller IQR generally means smaller standard deviation, which means more data points fall within one standard deviation of the mean. On statistics questions, always consider how different measures of variability relate to each other.

Question 7

1. Coffee Shop A is more consistent with better overall performance than Coffee Shop B
2. Coffee Shop B has higher peak performance but Coffee Shop A has more predictable daily sales (correct answer)
3. Both shops have identical median performance, but Coffee Shop B shows greater sales variability
4. Coffee Shop A has lower minimum sales but Coffee Shop B has more extreme outlier potential

Explanation: When analyzing box plots and five-number summaries, you need to compare both central tendency (median performance) and variability (consistency) between datasets. The five-number summary gives you (minimum, Q1, median, Q3, maximum), which tells the complete story of distribution shape and spread. Let's examine what each shop's data reveals. Coffee Shop A has a range of $140 (320-180) and an interquartile range (IQR) of $60 (280-220). Coffee Shop B has a range of $180 (340-160) and an IQR of $90 (290-200). Shop B's median ($250) exceeds Shop A's median ($240), indicating better overall performance. However, Shop A shows much less variability in both range and IQR, meaning more predictable daily sales. Choice B correctly identifies that Shop B has higher peak performance ($340 maximum vs $320) while Shop A has more predictable sales due to its smaller spread measures. Choice A is wrong because while Shop A is more consistent, Shop B actually has better overall performance with a higher median. Choice C incorrectly states the medians are identical ($240 ≠ $250), though it correctly notes Shop B has greater variability. Choice D misses the main comparison points—while Shop A does have a higher minimum ($180 vs $160), the key insight is about predictability versus peak performance, not outlier potential. Remember: When comparing distributions, always analyze both center (median) and spread (range, IQR) separately. Higher medians indicate better performance, while smaller spreads indicate greater consistency.

Question 8

1. Company X provides more predictable response times due to its lower median value overall
2. Company Y provides more predictable response times due to its smaller interquartile range of values (correct answer)
3. Both companies provide equally predictable service since their median values are very similar
4. Predictability cannot be determined without knowing the minimum and maximum response times for each company

Explanation: When analyzing box plots to determine predictability or consistency in data, you need to focus on measures of spread rather than central tendency. Predictability in response times means less variation - customers can more reliably expect similar service times. The key measure here is the interquartile range (IQR), which captures the spread of the middle 50% of the data. Company X has an IQR of $60 - 30 = 30$ seconds, while Company Y has an IQR of $55 - 40 = 15$ seconds. Company Y's smaller IQR means their response times are more tightly clustered around the median, making their service more predictable. Choice A incorrectly focuses on the median value. While Company X does have a lower median (45 vs 50 seconds), a lower average response time doesn't indicate predictability - it just means faster service on average. Predictability is about consistency, not speed. Choice C makes the error of comparing medians to assess predictability. The medians being similar (45 vs 50 seconds) tells us about typical response times, but nothing about how much variation customers should expect. Choice D suggests you need the minimum and maximum values, but this is incorrect. While the full range could provide additional context, the IQR already gives you the essential information about spread in the central portion of the data, which is more reliable than extreme values. Remember: when questions ask about predictability, consistency, or reliability in data, look for measures of spread like IQR or standard deviation, not measures of central tendency like mean or median.

Question 9

1. 86 points
2. 88 points (correct answer)
3. 92 points
4. 94 points

Explanation: When you encounter box-and-whisker plot questions involving quartiles and interquartile ranges, focus on the relationships between these key values rather than getting lost in the individual numbers. Start by finding Class A's interquartile range (IQR): $84 - 72 = 12$ points. Since Class B's IQR is 50% larger, calculate $12 \times 1.5 = 18$ points for Class B's IQR. Now you can find Class B's third quartile. You know Class B has a median of 78 (same as Class A) and a first quartile of 70. Since IQR equals third quartile minus first quartile: $18 = Q_3 - 70$, so $Q_3 = 88$ points. Looking at the wrong answers: Choice (A) 86 points would give an IQR of $86 - 70 = 16$, which represents about a 33% increase rather than 50%. Choice (C) 92 points creates an IQR of $92 - 70 = 22$, which is roughly an 83% increase—far too large. Choice (D) 94 points yields an IQR of $94 - 70 = 24$, representing a 100% increase (double the original), which completely misinterprets "50% larger." The correct answer is (B) 88 points. Remember that "50% larger" means you multiply the original value by 1.5, not by 0.5. This percentage increase language appears frequently on standardized tests, so practice converting phrases like "25% larger" (multiply by 1.25) or "40% smaller" (multiply by 0.6) to avoid calculation errors.

Question 10

1. Team Y must have outliers that extend the whiskers significantly beyond the box boundaries
2. Team X has more variation in the middle 50% of players, while Team Y has more extreme values (correct answer)
3. Team Y has a smaller overall range but larger interquartile range than reported in the problem
4. The median being the same means both teams have identical height distributions despite the IQR difference

Explanation: When analyzing box plots, you need to distinguish between different types of variation. The interquartile range (IQR) measures spread in the middle 50% of data, while overall consistency includes the entire distribution, including outliers. The key insight here is that "more consistent overall" doesn't necessarily mean having a smaller IQR. Team Y can be more consistent overall while having a smaller IQR than Team X if Team Y has fewer extreme outliers, even though its middle 50% might be more tightly packed. Choice B correctly explains this: Team X has more variation among its middle 50% of players (larger IQR), while Team Y's greater overall consistency comes from having fewer extreme values, even though its middle players might vary less than Team X's middle players. Choice A is backwards - if Team Y were more consistent overall, it would have shorter whiskers (fewer extreme outliers), not longer ones. Choice C contradicts the given information about Team X having the larger IQR. Choice D incorrectly assumes that identical medians mean identical distributions, which isn't true - distributions can have the same median but very different shapes and spreads. The apparent contradiction resolves when you realize that IQR and overall consistency measure different aspects of variation. A team can have a smaller IQR but still be less consistent overall due to outliers. Study tip: Remember that box plots show multiple measures of spread - IQR shows middle variation, while whiskers and outliers show extreme variation. Overall consistency depends on both components, not just the IQR.

Question 11

1. Most honors students study significantly more than regular students, but some honors students study surprisingly little (correct answer)
2. The honors class data contains measurement errors since outliers below the median are mathematically impossible
3. Honors students have more consistent study habits overall compared to students in regular classes
4. The outliers indicate that some honors students are enrolled in the wrong level of coursework for their abilities

Explanation: Box plots are powerful tools for comparing distributions because they show five key statistics: minimum, first quartile, median, third quartile, and maximum, plus any outliers. When interpreting box plots with outliers, focus on what the overall pattern reveals about the group's behavior. In this scenario, the honors class has a higher median study time, meaning the typical honors student studies more than the typical regular student. However, the outliers below the lower whisker indicate some honors students study much less than expected. This creates a clear picture: while most honors students are high achievers who study extensively, a subset studies surprisingly little yet remains in the challenging coursework. Choice A correctly captures this interpretation - the higher median shows most honors students study significantly more, while the low outliers represent those who study surprisingly little. Choice B demonstrates a fundamental misunderstanding of outliers. Outliers below the median are completely possible and mathematically valid - they simply represent data points that fall unusually far from the typical range. Choice C contradicts what the data shows. Outliers actually indicate less consistency, not more. If honors students had more consistent habits, you'd see a tighter distribution with fewer or no outliers. Choice D makes an assumption beyond what the data can tell us. While these students might be misplaced, the box plot only shows study time patterns, not academic ability or appropriate course placement. Strategy tip: When analyzing box plots with outliers, always consider what the median tells you about the typical case, then interpret outliers as interesting exceptions to that pattern.

Question 12

1. Class 3's median of 70 means about half the students scored $\le 70$ and about half scored $\ge 70$. (correct answer)
2. Class 3's median of 70 means the most common score was 70, so nearly all students earned 70.
3. Class 3's median of 70 means the average (mean) score was 70, regardless of the spread in scores.
4. Class 3's median of 70 means exactly 70% of students scored above 70 because 70 is the median.

Explanation: This question tests upper-level mathematics achievement skills, specifically interpreting data from bar graphs and box-and-whisker plots. Bar graphs display categorical data with rectangular bars, and box-and-whisker plots summarize data distribution through quartiles, showing median, quartiles, and outliers. In the Class 3 box plot, the median of 70 points is the middle value when all student scores are arranged in order. Choice A is correct because the median divides the data set in half - approximately 50% of students scored at or below 70 points, and 50% scored at or above 70 points. Choice D is incorrect because it misinterprets the median as a percentage - the median of 70 means it's the middle score value, not that 70% of students scored above it. This error occurs when students confuse the median value with percentile interpretations. To help students, emphasize that the median is the 50th percentile, meaning half the data falls below and half above this value, regardless of what the actual median value is.

Question 13

1. About half of Product B's weekly sales values are at or below 30 units and about half are at or above 30 units. (correct answer)
2. Product B's average (mean) weekly sales is exactly 30 units, regardless of the other values in the data set.
3. The most frequent weekly sales value (mode) for Product B is 30 units, so nearly all weeks were at 30 units.
4. Exactly 75% of Product B's weekly sales values are below 30 units because 30 is the median.

Explanation: This question tests upper-level mathematics achievement skills, specifically interpreting data from bar graphs and box-and-whisker plots. Bar graphs display categorical data with rectangular bars, and box-and-whisker plots summarize data distribution through quartiles, showing median, quartiles, and outliers. In the given box plot for Product B, the median value of 30 units represents the middle value when all weekly sales data are arranged in order. Choice A is correct because the median divides the data set in half - approximately 50% of values fall at or below 30 units, and 50% fall at or above 30 units. Choice B is incorrect because it confuses median with mean - the median is the middle value, not the average, and these can differ significantly in skewed distributions. This error occurs when students don't distinguish between measures of central tendency. To help students, emphasize that median is the middle value in ordered data, mean is the arithmetic average, and mode is the most frequent value - each provides different information about the data set.

Question 14

1. Determining the exact position of the median line within the box structure
2. Calculating the interquartile range since the data contains repeated values at quartile positions
3. Identifying outliers because pet ownership data typically follows a normal distribution pattern
4. Representing the discrete nature of the data on a continuous box plot scale (correct answer)

Explanation: Pet ownership data is discrete (you can't own 2.5 pets), but box plots are designed for continuous data and display quartiles as if they can fall anywhere on a continuous scale. When dealing with discrete data like pet counts, quartile positions may fall between actual data values, creating challenges in accurate representation. The other options are standard calculations that work fine with discrete data.

Question 15

1. The quiz was too difficult, causing most students to score poorly with only a few high achievers
2. Students likely fall into two distinct groups with different levels of preparation or understanding (correct answer)
3. The grading scale was inappropriate, artificially creating gaps in the middle score ranges
4. Random variation in test performance naturally creates multiple peaks in any educational assessment

Explanation: A bimodal distribution in educational assessment typically indicates the presence of two distinct subgroups within the class population. This could represent students with different levels of prior knowledge, preparation, study habits, or understanding of the material. For example, some students might have mastered the concepts (creating one peak at higher scores) while others struggled with fundamental prerequisites (creating another peak at lower scores), with fewer students falling in between these two groups.

Question 16

1. Approximately 48 minutes should be the minimum qualifying commute time
2. Approximately 52 minutes should be the minimum qualifying commute time (correct answer)
3. Approximately 57 minutes should be the minimum qualifying commute time
4. Approximately 63 minutes should be the minimum qualifying commute time

Explanation: To find the 85th percentile (top 15%), we need to count from the histogram. Total employees: 8+15+22+28+18+12+7+5 = 115. The 85th percentile corresponds to position 115 × 0.85 ≈ 98. Cumulative count: 0-10min: 8, 10-20min: 23, 20-30min: 45, 30-40min: 73, 40-50min: 91, 50-60min: 103. Since we need position 98 and we reach 103 at the 50-60 minute mark, the 85th percentile falls in the 50-60 minute range, approximately at 52 minutes.

Question 17

1. Approximately 8 minutes as the maximum wait time
2. Approximately 12 minutes as the maximum wait time
3. Approximately 15 minutes as the maximum wait time (correct answer)
4. Approximately 18 minutes as the maximum wait time

Explanation: To find the 80th percentile, we need to find where 80% of customers fall below a certain wait time. From the histogram: 0-3 min: 12 customers, 3-6 min: 18 customers, 6-9 min: 25 customers, 9-12 min: 20 customers, 12-15 min: 15 customers, 15-18 min: 8 customers, 18-21 min: 2 customers. Total customers = 100. 80% of 100 = 80 customers. Cumulative count: through 12 min = 12+18+25+20 = 75 customers. Through 15 min = 75+15 = 90 customers. Since we need 80 customers, the 80th percentile falls somewhere in the 12-15 minute range, approximately at 15 minutes.

Question 18

1. 9th graders at both schools check out approximately the same number of books per student on average
2. School B's 9th graders must check out significantly more books to compensate for lower grade performance (correct answer)
3. School A's 9th graders are more consistent readers compared to School B's 9th grade students
4. The data suggests that 9th grade represents a critical transition year for developing reading habits

Explanation: Since both schools have the same total number of books checked out across all grades, but School B shows decreases in 6th-8th grades while School A shows steady increases, School B must compensate for this deficit somewhere. The graph shows School B has an increase in 9th grade, so to balance out the lower numbers in earlier grades and match School A's total, School B's 9th graders must check out significantly more books than would be expected from a simple trend.

Question 19

1. Approximately 32% of the days in March
2. Approximately 45% of the days in March
3. Approximately 58% of the days in March (correct answer)
4. Approximately 71% of the days in March

Explanation: From the histogram, we need to find the number of days in the temperature ranges 45-50°F, 50-55°F, 55-60°F, and 60-65°F. Reading from the graph: 45-50°F has 4 days, 50-55°F has 6 days, 55-60°F has 7 days, and 60-65°F has 1 day. Total days in this range = 4 + 6 + 7 + 1 = 18 days. March has 31 days total. Percentage = (18/31) × 100% ≈ 58%.

Question 20

1. 45 students total
2. 54 students total (correct answer)
3. 60 students total
4. 72 students total

Explanation: From the double bar graph, in March School A had 18 students and School B had 36 students reading books. Since 36 = 2 × 18, School B had exactly twice as many as School A. The total for March = 18 + 36 = 54 students.