Log inSign up
← Back to Learn by Concept

AP Statistics · Learn by Concept

AP Statistics Help: Graphical Representations Of Summary Statistics

Review real example questions for Graphical Representations Of Summary Statistics in AP Statistics.

Question 1 / 10

0 of 10 answered

A museum records the amounts (in dollars) donated by 60 visitors during a weekend fundraiser. The five-number summary is: min =1=1, Q1=5Q_1=5, median =12=12, Q3=25Q_3=25, max =120=120. A modified boxplot (1.5·IQR rule) is drawn. Which feature is consistent with the summary statistics?

All questions

Question 1

A museum records the amounts (in dollars) donated by 60 visitors during a weekend fundraiser. The five-number summary is: min =1=1, Q1=5Q_1=5, median =12=12, Q3=25Q_3=25, max =120=120. A modified boxplot (1.5·IQR rule) is drawn. Which feature is consistent with the summary statistics?

  1. The range is 25−5=2025-5=20 dollars.
  2. The IQR is 120−1=119120-1=119 dollars.
  3. The box extends from Q1=5Q_1=5 to Q3=25Q_3=25, with a median line at 12. (correct answer)
  4. The right whisker must extend to 120 dollars because it is the maximum donation.
  5. Because the maximum is large, the median must be greater than Q3Q_3.

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.

Question 2

All data values are multiplied by 3. What happens to the boxplot's box length?

  1. It remains unchanged
  2. It is multiplied by 3 (correct answer)
  3. It increases by 3
  4. Cannot be determined

Explanation: The box length is the interquartile range (IQR), the difference between the third and first quartiles. Multiplying every data value by 3 also multiplies both quartiles by 3, so their difference is multiplied by 3. The tempting mistake is thinking the box length stays unchanged, but scaling the data scales all measures of spread, including the IQR.

Question 3

A boxplot has Q1=40, median=50, Q3=80. Which description is most plausible?

  1. Left-skewed; mean < 50
  2. Symmetric; mean near 50
  3. Right-skewed; mean > 50 (correct answer)
  4. Bimodal; mean near 50

Explanation: The box's right half (median 50 to Q3 80) is much longer than its left half (Q1 40 to median 50), so the distribution is right-skewed. In a right-skewed distribution, the mean is pulled toward the longer tail, so it is greater than the median. Symmetric near 50 is tempting but wrong because the two box halves are clearly unequal.

Question 4

Five-number summary: 4,8,12,18,36. By the 1.5(IQR) rule,

  1. Only 18 is an outlier
  2. 4 and 36 are outliers
  3. There are no outliers
  4. Only 36 is an outlier (correct answer)

Explanation: The IQR is 18 - 8 = 10. Times 1.5 gives 15, so the fences are 8 - 15 = -7 and 18 + 15 = 33. The only value beyond a fence is 36, above 33, so only 36 is an outlier. The tempting mistake is thinking 4 is an outlier because it's far from the median, but it is well above the lower fence -7.

Question 5

Which statement about a standard boxplot is true?

  1. It shows the mean as a line
  2. It displays every data value
  3. It requires normal data
  4. It can hide multiple peaks (correct answer)

Explanation: A standard boxplot summarizes data with the median, quartiles, and whiskers, so it does not display the actual distribution shape and can hide multiple peaks. The mean is shown as a line only in some modified plots, not a standard boxplot; the center line is the median.

Question 6

Ignoring outliers, a zero-length left whisker means

  1. Minimum equals Q1 (correct answer)
  2. Q1 equals the median
  3. Median equals Q3
  4. Maximum equals Q3

Explanation: In a boxplot, the left whisker runs from the minimum to Q1. If its length is zero, those two endpoints are the same point, so the minimum equals Q1. The tempting mistake is to think it means Q1 equals the median, but that would make the left edge of the box touch the median line, not shorten the whisker.

Question 7

A school nurse records the number of absences for each of 80 students in a semester. The five-number summary (absences) is: min =0=0, Q1=1Q_1=1, median =3=3, Q3=6Q_3=6, max =15=15. A boxplot is drawn using these values (no outliers marked). Which feature is consistent with the summary statistics?

  1. The median is at 6 absences.
  2. The left whisker extends from 1 down to 0 absences. (correct answer)
  3. The interquartile range is 15−0=1515-0=15 absences.
  4. The box extends from 0 to 15 absences.
  5. The maximum value (15) must be shown as an outlier point, not as a whisker endpoint.

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.

Question 8

An environmental scientist records daily high temperatures (in ∘^\circF) for 31 days in July in one city. The five-number summary is: min =71=71, Q1=78Q_1=78, median =83=83, Q3=88Q_3=88, max =97=97. A boxplot is drawn from these values. Which feature is consistent with the summary statistics?

  1. The right whisker extends from 88 to 97 ∘^\circF. (correct answer)
  2. The median is at 88 ∘^\circF.
  3. The IQR is 97−71=2697-71=26 ∘^\circF.
  4. The box extends from 71 to 97 ∘^\circF.
  5. The left whisker ends at 78 ∘^\circF because Q1Q_1 is the minimum.

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.

Question 9

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=0.5, Q1=3Q_1=3, median =6.5=6.5, Q3=11Q_3=11, max =26=26. A boxplot is made from these statistics (no outliers marked). Which feature is consistent with the summary statistics?

  1. The median line is drawn at 11 hours.
  2. The interquartile range is 26−0.5=25.526-0.5=25.5 hours.
  3. The box extends from 3 to 11 hours. (correct answer)
  4. The left whisker ends at 3 hours.
  5. The maximum (26) must be shown as an outlier point, so the right whisker ends at 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.

Question 10

A farmer measures the weights (in pounds) of 48 pumpkins harvested from a field. The five-number summary is: minimum 66, Q1=10Q_1=10, median 1313, Q3=18Q_3=18, maximum 3333. A boxplot is drawn from these statistics (no outliers shown). Which feature is consistent with the summary statistics?

  1. The box extends from 6 to 33 pounds.
  2. The IQR is 33−6=2733-6=27 pounds.
  3. The median is located at 13 pounds, between Q1Q_1 and Q3Q_3. (correct answer)
  4. The right whisker ends at 18 pounds.
  5. The range is 18−10=818-10=8 pounds.

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.