Graphical Representations of Summary Statistics
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AP Statistics › Graphical Representations of Summary Statistics
A museum records the amounts (in dollars) donated by 60 visitors during a weekend fundraiser. The five-number summary is: min $=1$, $Q_1=5$, median $=12$, $Q_3=25$, max $=120$. A modified boxplot (1.5·IQR rule) is drawn. Which feature is consistent with the summary statistics?
The IQR is $120-1=119$ dollars.
Because the maximum is large, the median must be greater than $Q_3$.
The right whisker must extend to 120 dollars because it is the maximum donation.
The range is $25-5=20$ dollars.
The box extends from $Q_1=5$ to $Q_3=25$, with a median line at 12.
Explanation
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 = 5, Q₃ = 25, and median = 12, the box extends from 5 to 25 dollars with the median at 12 dollars, making choice C correct. Choice A incorrectly identifies the IQR as the range; IQR = Q₃ - Q₁ = 25 - 5 = 20 dollars. Choice B incorrectly calculates IQR using max - min. Choice D is incorrect because with IQR = 20, the upper fence = 25 + 1.5(20) = 55, so 120 would be plotted as an outlier point, not connected by the whisker. Choice E makes a logical error; the median cannot be greater than Q₃ by definition, as Q₃ represents the 75th percentile while the median is the 50th percentile.
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?
The left whisker extends from 1 down to 0 absences.
The interquartile range is $15-0=15$ absences.
The maximum value (15) must be shown as an outlier point, not as a whisker endpoint.
The median is at 6 absences.
The box extends from 0 to 15 absences.
Explanation
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.
An environmental scientist records daily high temperatures (in $^\circ$F) for 31 days in July in one city. The five-number summary is: min $=71$, $Q_1=78$, median $=83$, $Q_3=88$, max $=97$. A boxplot is drawn from these values. Which feature is consistent with the summary statistics?
The right whisker extends from 88 to 97 $^\circ$F.
The left whisker ends at 78 $^\circ$F because $Q_1$ is the minimum.
The median is at 88 $^\circ$F.
The IQR is $97-71=26$ $^\circ$F.
The box extends from 71 to 97 $^\circ$F.
Explanation
This question assesses understanding of whisker placement in temperature data. The right whisker extends from Q3 to the maximum value, which means from 88 to 97°F, making option A correct. Option B incorrectly places the median at Q3's position (88) when it should be at 83°F. Option C calculates max - min (97 - 71 = 26) and calls it the IQR, but the IQR should be Q3 - Q1 (88 - 78 = 10). Option D incorrectly extends the box from min to max (71 to 97) instead of Q1 to Q3 (78 to 88). Option E misunderstands the five-number summary; Q1 (78) is not the minimum value (71). Remember: whiskers show the data range beyond the quartiles, extending from the box edges to the extreme values.
A streaming service samples 100 users and records the number of hours each user watched in the past week. The five-number summary (hours) is: min $=0.5$, $Q_1=3$, median $=6.5$, $Q_3=11$, max $=26$. A boxplot is made from these statistics (no outliers marked). Which feature is consistent with the summary statistics?
The left whisker ends at 3 hours.
The maximum (26) must be shown as an outlier point, so the right whisker ends at 11 hours.
The interquartile range is $26-0.5=25.5$ hours.
The median line is drawn at 11 hours.
The box extends from 3 to 11 hours.
Explanation
This question tests identification of the box portion in a boxplot with streaming data. The box extends from Q1 to Q3, which is from 3 to 11 hours, making option C correct. Option A incorrectly places the median at Q3's value (11) instead of 6.5 hours. Option B calculates max - min (26 - 0.5 = 25.5) rather than the IQR, which is Q3 - Q1 (11 - 3 = 8). Option D states a fact about the whisker endpoint but doesn't identify the distinguishing feature of the box. Option E introduces outlier rules that don't apply since the problem explicitly states no outliers are marked. Key concept: the box in a boxplot represents the interquartile range, containing the middle 50% of observations between Q1 and Q3.
A farmer measures the weights (in pounds) of 48 pumpkins harvested from a field. The five-number summary is: minimum $6$, $Q_1=10$, median $13$, $Q_3=18$, maximum $33$. A boxplot is drawn from these statistics (no outliers shown). Which feature is consistent with the summary statistics?
The IQR is $33-6=27$ pounds.
The range is $18-10=8$ pounds.
The box extends from 6 to 33 pounds.
The right whisker ends at 18 pounds.
The median is located at 13 pounds, between $Q_1$ and $Q_3$.
Explanation
This question tests recognition of correct boxplot features from summary statistics. Given the five-number summary (min=6, Q1=10, median=13, Q3=18, max=33), we need to identify the correct statement. The box extends from Q1 to Q3 (10 to 18 pounds), not 6 to 33. The IQR is Q3-Q1 = 18-10 = 8 pounds, not 27. The median at 13 pounds is correctly positioned between Q1 (10) and Q3 (18), making choice C correct. The right whisker extends from Q3 to the maximum, ending at 33 pounds, not 18. The range is max-min = 33-6 = 27 pounds, not 8. When interpreting boxplots, the median line must always fall within the box, as it represents a value between the 25th and 75th percentiles of the distribution.
A teacher records the scores (out of 100) on a quiz for 80 students. The five-number summary is: minimum $40$, $Q_1=62$, median $70$, $Q_3=84$, maximum $98$. A boxplot is made using these values. Which feature is consistent with the summary statistics?
The box extends from 62 to 84 points.
The left whisker ends at 62 points.
The range is $84-62=22$ points.
Any score above 98 must be shown as an outlier on the boxplot.
The median line is drawn at 84 points.
Explanation
This question tests recognition of boxplot components from summary statistics. Given the five-number summary (min=40, Q1=62, median=70, Q3=84, max=98), we need to identify which statement correctly describes a boxplot feature. The box in a boxplot always extends from Q1 to Q3, which here is from 62 to 84 points, making choice B correct. The range is maximum minus minimum (98-40 = 58 points), not 22. The median line should be at 70 points, not 84. The left whisker extends from the minimum (40) to Q1 (62), not ending at 62. Without specific outlier rules given, we cannot determine if 98 would be an outlier. Remember that the box represents the interquartile range and contains the middle 50% of the data.
A librarian tracked the number of books checked out by each visitor on a particular day (population: all 60 visitors that day). The five-number summary is: min $=0$, $Q_1=1$, median $=2$, $Q_3=4$, max $=12$. A boxplot is drawn. Which feature is consistent with the summary statistics?
The maximum value 12 must be plotted as an outlier point beyond the whisker.
The range is $4-1=3$ books.
The box extends from 1 to 4 with a median line at 2.
The whiskers extend from 1 to 4, and the box extends from 0 to 12.
The median is at 4 because $Q_3=4$.
Explanation
This question assesses understanding of boxplot construction from summary statistics. The box extends from Q₁ (1) to Q₃ (4), with the median line at 2 positioned inside this box. The whiskers extend from the box edges to the minimum (0) and maximum (12) values. Choice B correctly identifies that the box extends from 1 to 4 with a median line at 2. Common misconceptions in the distractors include: confusing Q₃ with the median, calculating range instead of identifying boxplot features (range is 12-0=12, not 3), reversing box and whisker positions, and assuming extreme values must be outliers. Remember that in a standard boxplot from a five-number summary, all five values are incorporated into the plot structure.
A teacher summarizes scores (out of 50) on a quiz for 90 students. The five-number summary is: min $=11$, $Q_1=29$, median $=34$, $Q_3=41$, max $=49$. A boxplot is drawn from these values. Which feature is consistent with the summary statistics?
The box extends from 29 to 41, and the median line is at 34.
The interquartile range is $49-11=38$.
The median is drawn at 41.
The right whisker ends at 34 because the median is the largest value shown by a whisker.
The left whisker ends at 29 because $Q_1$ is the minimum.
Explanation
This question tests comprehensive boxplot interpretation with quiz scores. The box extends from Q1 to Q3 (29 to 41), and the median line is drawn at 34, making option B correct. Option A incorrectly places the median at Q3's position (41) instead of 34. Option C calculates the range (max - min = 49 - 11 = 38) rather than the IQR, which is Q3 - Q1 (41 - 29 = 12). Option D incorrectly claims Q1 equals the minimum; Q1 is 29 but the minimum is 11. Option E makes no sense as whiskers extend to extreme values, not to the median. Remember: reading a boxplot requires identifying five key values - the minimum, Q1, median, Q3, and maximum - each serving a specific role in the display.
A coffee shop measures the wait time (in minutes) for 70 customers during the morning rush. The five-number summary is: min $=2$, $Q_1=4$, median $=6$, $Q_3=9$, max $=20$. A boxplot is constructed using these values (no outliers marked). Which feature is consistent with the summary statistics?
The maximum (20) must be plotted as an outlier point rather than as the whisker endpoint.
The left whisker ends at 4 minutes.
The range is $9-4=5$ minutes.
The box extends from 4 to 9 minutes.
The median is drawn at 9 minutes.
Explanation
This question tests recognition of the box component in a boxplot. The box extends from Q1 to Q3, which is from 4 to 9 minutes, making option C correct. Option A incorrectly places the median at Q3's value (9) instead of 6. Option B calculates Q3 - Q1 (9 - 4 = 5) but mislabels it as the range rather than the IQR; the range would be max - min = 20 - 2 = 18. Option D correctly identifies where the left whisker ends but this isn't the distinguishing feature asked for. Option E introduces outlier considerations that aren't applicable since the problem states no outliers are marked. Key insight: 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 cross-country coach measured the distances (in miles) run by each athlete during a Saturday practice (population: all 32 athletes on the team). The five-number summary is: min $=2.0$, $Q_1=3.5$, median $=4.2$, $Q_3=5.0$, max $=7.5$. A boxplot is created. Which feature is consistent with the summary statistics?
Because the maximum is 7.5, 7.5 must be an outlier and not part of the whisker.
The median line is located at 4.2 within a box from 3.5 to 5.0.
The interquartile range is $7.5-2.0=5.5$ miles.
The left whisker extends from 5.0 down to 3.5.
The box extends from 2.0 to 7.5, and the whiskers extend from 3.5 to 5.0.
Explanation
This question tests recognition of correct boxplot features from given summary statistics. In a boxplot, the box spans from Q₁ (3.5) to Q₃ (5.0), with the median line drawn at 4.2 within this box. The whiskers extend from the box to the minimum (2.0) and maximum (7.5) values. Choice B correctly states that the median line is at 4.2 within a box from 3.5 to 5.0. The distractors reveal common misunderstandings: the box doesn't extend to the extremes, the IQR is 5.0-3.5=1.5 miles (not 5.5), values in a five-number summary are not outliers, and whiskers extend outward from the box (not between quartiles). A key principle: the box always contains the middle 50% of the data.