Graphical Representations of Summary Statistics - AP Statistics

An online retailer tracked delivery times (in days) for 25 randomly selected orders. The five-number summary is: min $=1$ , $Q_1=2$ , median $=3$ , $Q_3=5$ , max $=12$ . A boxplot is drawn with whiskers at the min and max. Which feature is consistent with the summary statistics?
Answer: The median line is drawn at 3 days inside the box. This question tests proficiency in graphical representations of summary statistics, specifically identifying correct boxplot elements from a five-number summary. The median line is drawn at 3 days within the box, which spans from Q1 (2 days) to Q3 (5 days), confirming choice C as consistent with the statistics. Whiskers extend from min (1 day) to Q1 and from Q3 to max (12 days). Common distractors include choice A, miscalculating the range as Q3 - Q1 = 3 instead of max - min = 11, and choice B, confusing box length with the range. A mini-lesson on boxplots: they visually summarize data distribution using five numbers, with the box highlighting the IQR for spread, the median for center, and whiskers for extremes, enabling quick assessments of variability and outliers.
A track coach records the 100-meter sprint times (in seconds) for 28 athletes during practice. The five-number summary is: min $=10.9$ , $Q_1=11.6$ , median $=12.1$ , $Q_3=12.8$ , max $=14.0$ . A standard boxplot is drawn (no modified outlier rule). Which feature is consistent with the summary statistics?
Answer: The left whisker ends at 10.9 seconds and the right whisker ends at 14.0 seconds. This question tests understanding of standard boxplot construction (without the modified outlier rule). In a standard boxplot, the box extends from Q₁ to Q₃, and the whiskers extend from the minimum to the maximum values. Given min = 10.9, Q₁ = 11.6, Q₃ = 12.8, and max = 14.0, the whiskers should end at 10.9 and 14.0 seconds, making choice B correct. Choice A incorrectly places the box at the extremes and whiskers at the quartiles. Choice C correctly calculates IQR = 12.8 - 11.6 = 1.2 seconds, not 3.1. Choice D confuses the upper quartile (Q₃) with the median; the median is given as 12.1 seconds. Choice E calculates the difference between quartiles rather than the range; the actual range = 14.0 - 10.9 = 3.1 seconds. Remember: in standard boxplots, whiskers always extend to the minimum and maximum values.
A museum recorded the ages (in years) of all 75 visitors on a particular afternoon (population: those 75 visitors). The five-number summary is: min $=4$ , $Q_1=12$ , median $=27$ , $Q_3=43$ , max $=79$ . A boxplot is constructed. Which feature is consistent with the summary statistics?
Answer: The right whisker extends from 43 to 79, and the median is at 27. This question tests understanding of boxplot components and their relationship to the five-number summary. The box extends from Q₁ (12) to Q₃ (43), with the median line at 27 positioned inside this box. The whiskers extend from the box to the minimum (4) and maximum (79) values, so the right whisker extends from 43 to 79. Choice C correctly states that the right whisker extends from 43 to 79 and the median is at 27. Common errors in distractors include: confusing box extent with range, mistaking Q₃ for the median, calculating IQR as max-min (IQR is 43-12=31, not 75), and computing range incorrectly. A fundamental concept: the box represents the interquartile range, containing the middle 50% of the data.
A delivery company tracks the number of packages delivered per route for 52 routes on a given day. The five-number summary (packages) is: min $=18$ , $Q_1=24$ , median $=27$ , $Q_3=33$ , max $=40$ . A boxplot is drawn from these values. Which feature is consistent with the summary statistics?
Answer: The right whisker extends from 33 to 40 packages. This question focuses on identifying the right whisker in a boxplot. The right whisker extends from Q3 to the maximum value, which means from 33 to 40 packages, making option D correct. Option A incorrectly calculates IQR using max - min (40 - 18 = 22) instead of Q3 - Q1 (33 - 24 = 9). Option B places the median line at Q3's position (33) rather than at the actual median (27). Option C incorrectly defines the box as extending from Q1 to median (24 to 33) instead of Q1 to Q3 (24 to 33). Option E makes a logical error claiming Q1 is the minimum when the minimum is actually 18. Remember: the box captures the middle 50% of data (from Q1 to Q3), while whiskers extend to show the full range.
A school nurse records the number of absences for each of 80 students in a semester. The five-number summary (absences) is: min $=0$ , $Q_1=1$ , median $=3$ , $Q_3=6$ , max $=15$ . A boxplot is drawn using these values (no outliers marked). Which feature is consistent with the summary statistics?
Answer: The left whisker extends from 1 down to 0 absences. This question tests understanding of whisker placement in boxplots. The left whisker should extend from Q1 down to the minimum value, which means from 1 down to 0 absences, making option B correct. Option A incorrectly places the median at Q3's value (6) instead of 3. Option C calculates a range (max - min = 15 - 0 = 15) rather than the IQR (Q3 - Q1 = 6 - 1 = 5). Option D incorrectly extends the box from min to max (0 to 15) instead of Q1 to Q3 (1 to 6). Option E misapplies outlier rules when the problem explicitly states no outliers are marked. Key concept: whiskers connect the box edges (Q1 and Q3) to the minimum and maximum values, not to each other.
A car dealership summarized the selling prices (in thousands of dollars) of 30 used cars sold last month. The five-number summary was: min $=8.5$ , $Q_1=10.0$ , median $=12.5$ , $Q_3=15.0$ , max $=22.0$ . A boxplot is drawn. Which feature is consistent with the summary statistics?
Answer: The box extends from 10.0 to 15.0 with a median line at 12.5. In AP Statistics, this question tests understanding of boxplots as visual summaries of the five-number summary. The box spans from Q1=10.0 to Q3=15.0, with the median line at 12.5 inside, whiskers to min=8.5 and max=22.0. Choice B correctly describes the box from 10.0 to 15.0 with median at 12.5, matching the data. A distractor like choice E assumes the min=8.5 is an outlier, but the boxplot uses whiskers to extremes without specifying outlier detection. For a mini-lesson: in boxplots, the box covers the IQR (15.0-10.0=5.0), representing middle 50%, while whiskers show full spread, aiding in identifying asymmetry in car prices. This graphical tool facilitates quick comparisons across dealerships or months.
A gym records the number of minutes 60 adult members spend exercising during a single visit. The five-number summary (in minutes) is: min $=12$ , $Q_1=28$ , median $=41$ , $Q_3=55$ , max $=96$ . A boxplot is drawn using these values (no outliers marked). Which feature is consistent with the summary statistics?
Answer: The box extends from 28 to 55 minutes. This question tests understanding of boxplot construction from a five-number summary. In a boxplot, the box extends from Q1 to Q3, which represents the middle 50% of the data. Given Q1 = 28 and Q3 = 55, the box should indeed extend from 28 to 55 minutes, making option A correct. Option B incorrectly places the median at Q3's position (55), when it should be at 41. Option C miscalculates the IQR as max - min (96-12=84) instead of Q3 - Q1 (55-28=27). Option D confuses Q1 with the whisker endpoint; the left whisker should end at the minimum value of 12. Option E misunderstands outlier rules; without specific outlier calculations shown, standard boxplots extend whiskers to the minimum and maximum values.
A hospital summarizes the number of hours patients spent in the emergency department for 55 visits. The five-number summary (hours) is: minimum $0.5$ , $Q_1=1.2$ , median $2.0$ , $Q_3=3.6$ , maximum $7.5$ . A boxplot is made from these values. Which feature is consistent with the summary statistics?
Answer: The box extends from $1.2$ to $3.6$ hours. This question assesses understanding of boxplot construction from summary statistics. With the five-number summary (min=0.5, Q1=1.2, median=2.0, Q3=3.6, max=7.5), we can identify boxplot features. The box extends from Q1 to Q3, which is from 1.2 to 3.6 hours, making choice A correct. The median is at 2.0 hours, not 3.6. The range is max-min = 7.5-0.5 = 7.0 hours, not 2.4. The right whisker extends from Q3 to the maximum, ending at 7.5 hours, not 3.6. Without specific outlier criteria (like 1.5×IQR rule), we cannot determine if any values would be outliers. Remember that the box in a boxplot always spans from the first quartile to the third quartile, representing where the middle half of the data falls.
A school counselor summarizes the number of colleges to which 75 seniors applied. The five-number summary is: minimum $1$ , $Q_1=3$ , median $5$ , $Q_3=7$ , maximum $15$ . A boxplot is created from these values. Which feature is consistent with the summary statistics?
Answer: The IQR is $7-3=4$ applications. This question assesses calculation of boxplot features from summary statistics. With the five-number summary (min=1, Q1=3, median=5, Q3=7, max=15), we can verify each statement. The box extends from Q1 to Q3 (3 to 7 applications), not 1 to 15. The median is at 5 applications, not 7. The left whisker extends from the minimum (1) to Q1 (3), not ending at 3. The IQR equals Q3-Q1 = 7-3 = 4 applications, which matches choice D. The range is max-min = 15-1 = 14 applications, not 4. Remember that the interquartile range (IQR) measures the spread of the middle 50% of data and is visually represented by the length of the box in a boxplot.
A teacher summarized scores on a 100-point quiz for 36 students. The five-number summary is: min $=41$ , $Q_1=55$ , median $=63$ , $Q_3=72$ , max $=90$ . A boxplot is made using whiskers at the min and max. Which feature is consistent with the summary statistics?
Answer: The left whisker extends from 41 to 55 points. This question tests the skill of verifying graphical representations of summary statistics in boxplots for quiz scores. The left whisker extends from the min (41 points) to Q1 (55 points), making choice C accurate and consistent. The box spans from 55 to 72, with median at 63, and right whisker to 90. Frequent distractors include choice A, mixing IQR (17 points) with range (49), and choice D, expanding the box to the full range. To read boxplots proficiently, recall that they encapsulate data distribution: the box for middle quartiles, median for centrality, and whiskers for extents, facilitating comparisons and identification of skewness through uneven whisker lengths.
A nutritionist records the sodium content (in milligrams) of a random sample of 40 single-serving canned soups sold at a large grocery chain. The five-number summary is: min $=180$ , $Q_1=420$ , median $=610$ , $Q_3=860$ , max $=1{,}120$ . A boxplot is drawn using the modified (1.5·IQR) rule. Which feature is consistent with these summary statistics?
Answer: The box extends from 420 mg to 860 mg, and the median line is at 610 mg. This question tests understanding of boxplot construction using the modified (1.5·IQR) rule. In a boxplot, the box always extends from Q₁ to Q₃, with the median line inside the box. Here, Q₁ = 420 mg, Q₃ = 860 mg, and median = 610 mg, so the box extends from 420 to 860 mg with the median line at 610 mg, making choice A correct. Choice B incorrectly calculates IQR as max - min (which is the range); the actual IQR = Q₃ - Q₁ = 860 - 420 = 440 mg. Choice C incorrectly assumes the median must be at the midpoint of Q₁ and Q₃, but the median's position depends on the data distribution. Choice D misunderstands the modified rule; whiskers extend to the furthest data point within 1.5·IQR of the quartiles, not necessarily to the maximum. Choice E incorrectly calculates range using the quartiles instead of min and max; the actual range = 1,120 - 180 = 940 mg.
A gym recorded the number of minutes 40 members spent on a treadmill during one visit. The five-number summary (in minutes) is: min $=12$ , $Q_1=20$ , median $=28$ , $Q_3=35$ , max $=58$ . A boxplot is drawn using the standard rule (whiskers extend to the min and max, with no outliers marked). Which feature is consistent with the summary statistics?
Answer: The left whisker extends from 12 to 20 minutes. This question assesses the skill of interpreting graphical representations of summary statistics, particularly boxplots constructed from a five-number summary. In a standard boxplot without outliers, the box spans from the first quartile (Q1 = 20 minutes) to the third quartile (Q3 = 35 minutes), with the median line at 28 minutes, while the left whisker extends from the minimum (12 minutes) to Q1 (20 minutes) and the right whisker from Q3 to the maximum (58 minutes). Choice B correctly describes the left whisker extending from 12 to 20 minutes, which matches the summary statistics. A common distractor, like choice A, mistakenly describes the box as spanning the entire range from min to max, while choice D confuses the interquartile range (IQR = Q3 - Q1 = 15 minutes) with the full range. To read a boxplot effectively, identify the five key points: min, Q1, median, Q3, and max, which divide the data into quarters and highlight the distribution's spread and central tendency. Understanding these features helps in recognizing skewness, such as the longer right whisker here indicating potential right-skewness.
A bookstore records the prices (in dollars) of 65 books sold in one day. The five-number summary is: minimum $4$ , $Q_1=9$ , median $14$ , $Q_3=22$ , maximum $35$ . A boxplot is drawn based on these statistics. Which feature is consistent with the summary statistics?
Answer: The median is shown by a line at $14$ dollars within the box. This question tests identification of correct boxplot features from summary statistics. Given the five-number summary (min=4, Q1=9, median=14, Q3=22, max=35), we need to verify each statement. The right whisker extends from Q3 (22) to the maximum (35), giving it a length of 35-22 = 13 dollars, not 8. The median line at 14 dollars is correctly positioned between Q1 (9) and Q3 (22), making choice B correct. The IQR is Q3-Q1 = 22-9 = 13 dollars, not 31. The box extends from Q1 to Q3 (9 to 22 dollars), not 4 to 35. The left whisker extends from the minimum (4) to Q1 (9), not ending at 9. In boxplots, the median line inside the box shows the 50th percentile and helps visualize data skewness.
A delivery company recorded the number of packages delivered per route for every route run on a certain Monday (population: all 36 routes that day). The five-number summary is: min $=55$ , $Q_1=70$ , median $=82$ , $Q_3=95$ , max $=140$ . A boxplot is drawn. Which feature is consistent with the summary statistics?
Answer: The median is at 82, and the whiskers extend from 55 to 70 and from 95 to 140. This question evaluates ability to identify correct boxplot features from summary statistics. In a boxplot, the box extends from Q₁ (70) to Q₃ (95), with the median line at 82 inside the box. The whiskers extend from the box edges to the minimum (55) and maximum (140) values, so whiskers run from 70 to 55 and from 95 to 140. Choice A correctly identifies that the median is at 82 and the whiskers extend from 55 to 70 and from 95 to 140. The distractors contain errors: the IQR is 95-70=25 (not 85), the range is 140-55=85 (not 25), the box doesn't span min to max, and extreme values in five-number summaries aren't automatically outliers. Remember: whiskers connect quartiles to extremes.
A real estate agent summarizes the sale prices (in thousands of dollars) for 55 homes sold in one neighborhood last year. The five-number summary is: min $=190$ , $Q_1=240$ , median $=310$ , $Q_3=400$ , max $=650$ . A modified boxplot (1.5·IQR rule) is created. Which feature is consistent with the summary statistics?
Answer: The median line is at 310 and lies inside a box from 240 to 400 (thousand dollars). This question tests understanding of modified boxplot features with the 1.5·IQR rule. The box in any boxplot extends from Q₁ to Q₃ with the median line inside. Here, Q₁ = 240, Q₃ = 400, and median = 310 (all in thousands of dollars), so the box extends from 240 to 400 with the median line at 310, making choice B correct. Choice A incorrectly calculates IQR using max - min; the actual IQR = 400 - 240 = 160. Choice C misunderstands the modified rule; with IQR = 160, the upper fence is Q₃ + 1.5·IQR = 400 + 240 = 640, so 650 would be plotted as an outlier, not connected by the whisker. Choice D incorrectly calculates range using quartiles; the actual range = 650 - 190 = 460. Choice E incorrectly assumes the median must be at the arithmetic mean of Q₁ and Q₃; median position depends on data distribution.
A wildlife researcher measured the body mass (in kg) of every adult female in a small herd of 35 deer (population: all adult females in that herd). The five-number summary is: min $=42$ , $Q_1=48$ , median $=53$ , $Q_3=57$ , max $=66$ . A boxplot is constructed. Which feature is consistent with the summary statistics?
Answer: The median line is drawn at 53 inside a box from 48 to 57. This question assesses ability to identify correct boxplot features from summary statistics. A boxplot's box extends from Q₁ (48) to Q₃ (57), with the median line drawn at 53 inside this box. The whiskers extend from the box to the minimum (42) and maximum (66) values. Choice B correctly states that the median line is at 53 inside a box from 48 to 57. The distractors contain common misconceptions: the box doesn't span from min to max, the range is 66-42=24 kg (not 9), values in the five-number summary are never outliers, and the IQR is 57-48=9 kg (not 24). Remember that in a standard boxplot, the box always represents the middle 50% of the data between the quartiles.
A wildlife biologist measures the mass (in kilograms) of 52 adult turtles in a pond. The five-number summary is: minimum $3.1$ , $Q_1=4.0$ , median $4.6$ , $Q_3=5.8$ , maximum $9.2$ . A boxplot is drawn from these values (no outliers shown). Which feature is consistent with the summary statistics?
Answer: The median is at 4.6 kg and lies between $Q_1$ and $Q_3$ . This question assesses understanding of boxplot construction and the relationship between summary statistics. With the five-number summary (min=3.1, Q1=4.0, median=4.6, Q3=5.8, max=9.2), we can verify each statement. The IQR is Q3-Q1 = 5.8-4.0 = 1.8 kg, not 6.1. The median at 4.6 kg indeed lies between Q1 (4.0) and Q3 (5.8), making choice B correct. The right whisker should extend from Q3 to the maximum, ending at 9.2 kg, not 5.8. The box extends from Q1 to Q3 (4.0 to 5.8 kg), not 3.1 to 9.2. The range is max-min = 9.2-3.1 = 6.1 kg, not 1.8. In a boxplot, the median line must always fall within the box, as it represents a value between the 25th and 75th percentiles.
A manager recorded the time (in minutes) it took each employee in a department to complete a required safety quiz (population: all 50 employees in the department). The five-number summary is: min $=6$ , $Q_1=9$ , median $=12$ , $Q_3=15$ , max $=28$ . A boxplot is drawn. Which feature is consistent with the summary statistics?
Answer: The median is at 12, and the box extends from 9 to 15. This question evaluates understanding of boxplot components and their relationship to the five-number summary. The box in a boxplot extends from Q₁ (9) to Q₃ (15), with the median line at 12 positioned inside this box. The whiskers extend from the box edges to the minimum (6) and maximum (28) values. Choice A correctly identifies that the median is at 12 and the box extends from 9 to 15. Common errors in the distractors include: confusing range with IQR (IQR is 15-9=6, not 22), misplacing whiskers, assuming the median must be at the midpoint of the range, and incorrectly identifying outliers. When reading boxplots, always remember that the box boundaries are the quartiles, not the extremes.
An engineer records the battery life (in hours) of 36 identical devices under a standardized test. The five-number summary is: min $=6.2$ , $Q_1=7.5$ , median $=8.1$ , $Q_3=8.9$ , max $=12.4$ . A modified boxplot (1.5·IQR rule) is drawn. Which feature is consistent with the summary statistics?
Answer: The box runs from 7.5 to 8.9 hours, and the median line is at 8.1 hours. This question tests understanding of modified boxplot construction with the 1.5·IQR rule. The box always extends from Q₁ to Q₃ with the median line inside. Given Q₁ = 7.5, Q₃ = 8.9, and median = 8.1, the box runs from 7.5 to 8.9 hours with the median at 8.1 hours, making choice C correct. Choice A incorrectly identifies the IQR as the range; IQR = Q₃ - Q₁ = 8.9 - 7.5 = 1.4 hours. Choice B incorrectly calculates IQR using max - min. Choice D incorrectly assumes the median must be at the arithmetic mean of Q₁ and Q₃; (7.5 + 8.9)/2 = 8.2, but the actual median is 8.1. Choice E misunderstands the modified rule; with IQR = 1.4, the upper fence = 8.9 + 1.5(1.4) = 11.0, so 12.4 would be plotted as an outlier, not connected by the whisker.
A city planner records the commute times (in minutes) for 45 randomly selected workers who drive to downtown. The five-number summary is: min $=12$ , $Q_1=18$ , median $=26$ , $Q_3=41$ , max $=58$ . A boxplot is constructed. Which feature is consistent with the summary statistics?
Answer: The median line is at 26 minutes, inside a box from 18 to 41 minutes. This question tests understanding of boxplot construction. The box in any boxplot extends from Q₁ to Q₃ with the median line positioned inside. Given Q₁ = 18, Q₃ = 41, and median = 26, the box extends from 18 to 41 minutes with the median at 26 minutes, making choice C correct. Choice A incorrectly calculates IQR using max - min = 58 - 12 = 46; the actual IQR = Q₃ - Q₁ = 41 - 18 = 23 minutes. Choice B correctly calculates the IQR (23 minutes) but calls it the range; the actual range = 58 - 12 = 46 minutes. Choice D incorrectly assumes the median must be at the midpoint of Q₁ and Q₃; (18 + 41)/2 = 29.5, but the actual median is 26. Choice E incorrectly states that whiskers end at quartiles; whiskers extend beyond the box to show the data spread.

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